Model theory

Models of Arithmetic

Equation form expr-006404e30663f504

y≺xy \nsless x

Read as: y is less than x in the model

Means: A claim about the interpreted arithmetic operations or order stating: y is less than x in the model. This meaning is authored from the complete TR-027 source packet for expr-006404e30663f504.

3 occurrences in this chapter

Equation form expr-017dde88b4e1167d

<M\Assign{<}{M}

Read as: the interpreted less-than relation of structure M

Means: Arithmetic notation denoting: the interpreted less-than relation of structure M. This meaning is authored from the complete TR-027 source packet for expr-017dde88b4e1167d.

2 occurrences in this chapter

Equation form expr-039bfb4b8d6e9ab8

g(n+1)=⟦n+1¯⟧M by definition of g=⟦n¯′⟧M since n+1¯≡n¯'=′M(⟦n¯⟧M) by definition of ⟦t′⟧M=′M(g(n)) by definition of g=′M(h(n)) by induction hypothesis=h(′N(n)) since h is an isomorphism=h(n+1)g(n+1) & = \Value{\num{n+1}}{M} \text{ by definition of~$g$}\\ & = \Value{\num{n}'}{M} \text{ since $\num{n+1}\ident \num{n}'$}\\ & = \Assign{\prime}{M}(\Value{\num{n}}{M}) \text{ by definition of $\Value{t'}{M}$}\\ & = \Assign{\prime}{M}(g(n)) \text{ by definition of~$g$}\\ & = \Assign{\prime}{M}(h(n)) \text{ by induction hypothesis}\\ & = h(\Assign{\prime}{N}(n)) \text{ since $h$ is an isomorphism}\\ & = h(n+1)

Read as: g open parenthesis n plus one close parenthesis equals the value of the numeral for n plus one in structure M by definition of g next row equals the value of the successor of the numeral for n in structure M since the numeral for n plus one is syntactically identical to the successor of the numeral for n next row equals the interpretation of the successor operation in structure M applied to the value of the numeral for n in structure M by definition of the value of a successor term in structure M next row equals the interpretation of the successor operation in structure M applied to g of n by definition of g next row equals the interpretation of the successor operation in structure M applied to h of n by the induction hypothesis next row equals h applied to the interpretation of the successor operation in structure N applied to n since h is an isomorphism next row equals h open parenthesis n plus one close parenthesis

Means: A source-ordered calculation or table whose rows state: g open parenthesis n plus one close parenthesis equals the value of the numeral for n plus one in structure M by definition of g next row equals the value of the successor of the numeral for n in structure M since the numeral for n plus one is syntactically identical to the successor of the numeral for n next row equals the interpretation of the successor operation in structure M applied to the value of the numeral for n in structure M by definition of the value of a successor term in structure M next row equals the interpretation of the successor operation in structure M applied to g of n by definition of g next row equals the interpretation of the successor operation in structure M applied to h of n by the induction hypothesis next row equals h applied to the interpretation of the successor operation in structure N applied to n since h is an isomorphism next row equals h open parenthesis n plus one close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-039bfb4b8d6e9ab8.

1 occurrence in this chapter

Equation form expr-03c11090a6819839

⟦n+1¯⟧M=⟦n¯′⟧M\Value{\num{n+1}}{M} = \Value{\num{n}'}{M}

Read as: the value of the numeral for n plus one in structure M equals the value of the numeral for n prime in structure M

Means: Arithmetic notation denoting: the value of the numeral for n plus one in structure M equals the value of the numeral for n prime in structure M. This meaning is authored from the complete TR-027 source packet for expr-03c11090a6819839.

1 occurrence in this chapter

Equation form expr-0406bef11fd3ef6e

z≺zz \nsless z

Read as: z is less than z in the model

Means: A claim about the interpreted arithmetic operations or order stating: z is less than z in the model. This meaning is authored from the complete TR-027 source packet for expr-0406bef11fd3ef6e.

1 occurrence in this chapter

Equation form expr-043a718774c572bd

ss

Read as: s

Means: Arithmetic notation denoting: s. This meaning is authored from the complete TR-027 source packet for expr-043a718774c572bd.

4 occurrences in this chapter

Equation form expr-07895d7664c7c350

g(n)=xg(n) = x

Read as: g open parenthesis n close parenthesis equals x

Means: An arithmetic relation or equation stating: g open parenthesis n close parenthesis equals x. This meaning is authored from the complete TR-027 source packet for expr-07895d7664c7c350.

1 occurrence in this chapter

Equation form expr-08f271887ce94707

MM

Read as: M

Means: Arithmetic notation denoting: M. This meaning is authored from the complete TR-027 source packet for expr-08f271887ce94707.

3 occurrences in this chapter

Equation form expr-0978a05425bd9699

Nk\Struct{N_k}

Read as: structure N sub k

Means: Arithmetic notation denoting: structure N sub k. This meaning is authored from the complete TR-027 source packet for expr-0978a05425bd9699.

2 occurrences in this chapter

Equation form expr-0a5a0cdb7119bde3

n≠mn \neq m

Read as: n is not equal to m

Means: An arithmetic relation or equation stating: n is not equal to m. This meaning is authored from the complete TR-027 source packet for expr-0a5a0cdb7119bde3.

3 occurrences in this chapter

Equation form expr-0ad7f3c221bc0326

M1⊭Q3\Sat/{M_1}{!Q_3}

Read as: structure M sub one does not satisfy Q sub three

Means: A satisfaction claim stating: structure M sub one does not satisfy Q sub three. This meaning is authored from the complete TR-027 source packet for expr-0ad7f3c221bc0326.

1 occurrence in this chapter

Equation form expr-0b365347fe525225

cNk=k+1\Assign{c}{N_k} = k+1

Read as: the interpretation of c in structure N sub k equals k plus one

Means: Arithmetic notation denoting: the interpretation of c in structure N sub k equals k plus one. This meaning is authored from the complete TR-027 source packet for expr-0b365347fe525225.

1 occurrence in this chapter

Equation form expr-0b553c0abe4e83fb

y=n∗y = n^\nssucc

Read as: y equals n followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: y equals n followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-0b553c0abe4e83fb.

1 occurrence in this chapter

Equation form expr-0c9d5ea213d46118

b∗=bb^\nssucc = b

Read as: b followed by model successor equals b

Means: A claim about the interpreted arithmetic operations or order stating: b followed by model successor equals b. This meaning is authored from the complete TR-027 source packet for expr-0c9d5ea213d46118.

1 occurrence in this chapter

Equation form expr-0ccffa9a51d83d76

x⊕m∗=v⊕m∗=yx \nsplus m^\nssucc = v \nsplus m^\nssucc = y

Read as: x plus in the model m followed by model successor equals v plus in the model m followed by model successor equals y

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model m followed by model successor equals v plus in the model m followed by model successor equals y. This meaning is authored from the complete TR-027 source packet for expr-0ccffa9a51d83d76.

1 occurrence in this chapter

Equation form expr-0dd678d817d436c0

y∈|M|y \in \Domain{M}

Read as: y is in the domain of structure M

Means: Arithmetic notation denoting: y is in the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-0dd678d817d436c0.

1 occurrence in this chapter

Equation form expr-0e5677e25a3f86bf

|M2|\Domain{M_2}

Read as: the domain of structure M sub two

Means: Arithmetic notation denoting: the domain of structure M sub two. This meaning is authored from the complete TR-027 source packet for expr-0e5677e25a3f86bf.

1 occurrence in this chapter

Equation form expr-0fedd08b6cbc4828

u≺yu \nsless y

Read as: u is less than y in the model

Means: A claim about the interpreted arithmetic operations or order stating: u is less than y in the model. This meaning is authored from the complete TR-027 source packet for expr-0fedd08b6cbc4828.

3 occurrences in this chapter

Equation form expr-1051f75badfb2416

K⊨Q\Sat{K}{\Th{Q}}

Read as: structure K satisfies theory Q

Means: A satisfaction claim stating: structure K satisfies theory Q. This meaning is authored from the complete TR-027 source packet for expr-1051f75badfb2416.

1 occurrence in this chapter

Equation form expr-11d1d21b415e5513

z∈|M|z \in \Domain{M}

Read as: z is in the domain of structure M

Means: Arithmetic notation denoting: z is in the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-11d1d21b415e5513.

1 occurrence in this chapter

Equation form expr-1348c58056d80b83

⟨x,y⟩∈<K\tuple{x,y} \in \Assign{<}{K}

Read as: the tuple x comma y is in the interpreted less-than relation of structure K

Means: Arithmetic notation denoting: the tuple x comma y is in the interpreted less-than relation of structure K. This meaning is authored from the complete TR-027 source packet for expr-1348c58056d80b83.

1 occurrence in this chapter

Equation form expr-13713c198833258e

g(n+m)=⟦n+m¯⟧Mg(n+m) = \Value{\num{n+m}}{M}

Read as: g open parenthesis n plus m close parenthesis equals the value of the numeral for n plus m in structure M

Means: Arithmetic notation denoting: g open parenthesis n plus m close parenthesis equals the value of the numeral for n plus m in structure M. This meaning is authored from the complete TR-027 source packet for expr-13713c198833258e.

1 occurrence in this chapter

Equation form expr-139c7c04318de35e

n=0n = 0

Read as: n equals zero

Means: An arithmetic relation or equation stating: n equals zero. This meaning is authored from the complete TR-027 source packet for expr-139c7c04318de35e.

1 occurrence in this chapter

Equation form expr-13e0d4a85c4a6f89

|N|=ℕ\Domain{N} = \Nat

Read as: the domain of structure N equals the natural numbers

Means: Arithmetic notation denoting: the domain of structure N equals the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-13e0d4a85c4a6f89.

1 occurrence in this chapter

Equation form expr-143122bc4b4cc29e

y>0y > 0

Read as: y is greater than zero

Means: An arithmetic relation or equation stating: y is greater than zero. This meaning is authored from the complete TR-027 source packet for expr-143122bc4b4cc29e.

1 occurrence in this chapter

Equation form expr-14e42bdd0cd93c23

′K′(0)=0\Assign{\prime}{K'}(0) = 0

Read as: the interpretation of successor symbol in structure K prime open parenthesis zero close parenthesis equals zero

Means: Arithmetic notation denoting: the interpretation of successor symbol in structure K prime open parenthesis zero close parenthesis equals zero. This meaning is authored from the complete TR-027 source packet for expr-14e42bdd0cd93c23.

1 occurrence in this chapter

Equation form expr-159ac566157bb5d0

PA⊢∀x∃y((y+y)=x∨(y+y)′=x)\Th{PA} \Proves \lforall[x][\lexists[y][(\eq[(y+y)][x] \lor \eq[(y+y)'][x])]]

Read as: theory PA proves for every x, there exists y, open parenthesis open parenthesis y plus y close parenthesis equals x or open parenthesis y plus y close parenthesis prime equals x close parenthesis

Means: A derivability claim stating: theory PA proves for every x, there exists y, open parenthesis open parenthesis y plus y close parenthesis equals x or open parenthesis y plus y close parenthesis prime equals x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-159ac566157bb5d0.

1 occurrence in this chapter

Equation form expr-15db4a4d10c456bd

n⊕0=n+0=nn \nsplus 0 = n+0 = n

Read as: n plus in the model zero equals n plus zero equals n

Means: A claim about the interpreted arithmetic operations or order stating: n plus in the model zero equals n plus zero equals n. This meaning is authored from the complete TR-027 source packet for expr-15db4a4d10c456bd.

1 occurrence in this chapter

Equation form expr-15fd307d7f46f8c3

2¯\num{2}

Read as: the numeral for two

Means: Arithmetic notation denoting: the numeral for two. This meaning is authored from the complete TR-027 source packet for expr-15fd307d7f46f8c3.

2 occurrences in this chapter

Equation form expr-17643b1ab6160008

N⊭¬ConPA\Sat/{N}{\lnot \OCon[\Th{PA}]}

Read as: structure N does not satisfy not the formal consistency statement for theory PA

Means: An arithmetized-syntax statement denoting: structure N does not satisfy not the formal consistency statement for theory PA. This meaning is authored from the complete TR-027 source packet for expr-17643b1ab6160008.

1 occurrence in this chapter

Equation form expr-1a3aa9c185711857

g(⟦n¯⟧N)=⟦n¯⟧Mg(\Value{\num{n}}{N}) = \Value{\num{n}}{M}

Read as: g open parenthesis the value of the numeral for n in structure N close parenthesis equals the value of the numeral for n in structure M

Means: Arithmetic notation denoting: g open parenthesis the value of the numeral for n in structure N close parenthesis equals the value of the numeral for n in structure M. This meaning is authored from the complete TR-027 source packet for expr-1a3aa9c185711857.

1 occurrence in this chapter

Equation form expr-1a4057b9c504ea3e

⟦n¯⟧M\Value{\num{n}}{M}

Read as: the value of the numeral for n in structure M

Means: Arithmetic notation denoting: the value of the numeral for n in structure M. This meaning is authored from the complete TR-027 source packet for expr-1a4057b9c504ea3e.

5 occurrences in this chapter

Equation form expr-1a4913b9bbd60fea

n≺xn \nsless x

Read as: n is less than x in the model

Means: A claim about the interpreted arithmetic operations or order stating: n is less than x in the model. This meaning is authored from the complete TR-027 source packet for expr-1a4913b9bbd60fea.

1 occurrence in this chapter

Equation form expr-1a97e063166f2cca

M2\Struct{M_2}

Read as: structure M sub two

Means: Arithmetic notation denoting: structure M sub two. This meaning is authored from the complete TR-027 source packet for expr-1a97e063166f2cca.

4 occurrences in this chapter

Equation form expr-1abbec202796afbe

x⊕y=z⊕zx \nsplus y = z \nsplus z

Read as: x plus in the model y equals z plus in the model z

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y equals z plus in the model z. This meaning is authored from the complete TR-027 source packet for expr-1abbec202796afbe.

1 occurrence in this chapter

Equation form expr-1ac8df29b44efcfb

g(0)=0Mg(0) = \Assign{\Obj{0}}{M}

Read as: g open parenthesis zero close parenthesis equals the interpretation of object language symbol zero in structure M

Means: Arithmetic notation denoting: g open parenthesis zero close parenthesis equals the interpretation of object language symbol zero in structure M. This meaning is authored from the complete TR-027 source packet for expr-1ac8df29b44efcfb.

1 occurrence in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: Arithmetic notation denoting: n. This meaning is authored from the complete TR-027 source packet for expr-1b16b1df538ba12d.

19 occurrences in this chapter

Equation form expr-1ba7cc51e07e96b0

g(0N)=0Mg(\Assign{\Obj{0}}{N}) = \Assign{\Obj{0}}{M}

Read as: g open parenthesis the interpretation of object language symbol zero in structure N close parenthesis equals the interpretation of object language symbol zero in structure M

Means: Arithmetic notation denoting: g open parenthesis the interpretation of object language symbol zero in structure N close parenthesis equals the interpretation of object language symbol zero in structure M. This meaning is authored from the complete TR-027 source packet for expr-1ba7cc51e07e96b0.

1 occurrence in this chapter

Equation form expr-1cb09e48b857f437

x=cMx = \Assign{c}{M}

Read as: x equals the interpretation of c in structure M

Means: Arithmetic notation denoting: x equals the interpretation of c in structure M. This meaning is authored from the complete TR-027 source packet for expr-1cb09e48b857f437.

1 occurrence in this chapter

Equation form expr-1daa1ce23477c28c

=′M(⟦n¯⟧M)= \Assign{\prime}{M}(\Value{\num{n}}{M})

Read as: equals the interpretation of successor symbol in structure M open parenthesis the value of the numeral for n in structure M close parenthesis

Means: Arithmetic notation denoting: equals the interpretation of successor symbol in structure M open parenthesis the value of the numeral for n in structure M close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-1daa1ce23477c28c.

1 occurrence in this chapter

Equation form expr-1dc624e8ee0d484d

Q⊬∀x∀y(x+y)=(y+x)\Th{Q} \Proves/ \lforall[x][\lforall[y][\eq[(x + y)][(y+x)]]]

Read as: theory Q does not prove for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis

Means: A derivability claim stating: theory Q does not prove for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-1dc624e8ee0d484d.

1 occurrence in this chapter

Equation form expr-1f97d653b7ed2d2b

n+1n+1

Read as: n plus one

Means: Arithmetic notation denoting: n plus one. This meaning is authored from the complete TR-027 source packet for expr-1f97d653b7ed2d2b.

2 occurrences in this chapter

Equation form expr-208b35a67a1b06a7

x>0x > 0

Read as: x is greater than zero

Means: An arithmetic relation or equation stating: x is greater than zero. This meaning is authored from the complete TR-027 source packet for expr-208b35a67a1b06a7.

1 occurrence in this chapter

Equation form expr-20d4e8d8e48bc766

x∈|K|x \in \Domain{K}

Read as: x is in the domain of structure K

Means: Arithmetic notation denoting: x is in the domain of structure K. This meaning is authored from the complete TR-027 source packet for expr-20d4e8d8e48bc766.

1 occurrence in this chapter

Equation form expr-22073f71a9807028

predecessor of x^\nssucc x

Read as: predecessor of x

Means: A claim about the interpreted arithmetic operations or order stating: predecessor of x. This meaning is authored from the complete TR-027 source packet for expr-22073f71a9807028.

1 occurrence in this chapter

Equation form expr-222a04cfe2947cc3

x∗…∗=yx^{\nssucc\dots\nssucc} = y

Read as: x followed by model successor and so on model successor equals y

Means: A claim about the interpreted arithmetic operations or order stating: x followed by model successor and so on model successor equals y. This meaning is authored from the complete TR-027 source packet for expr-222a04cfe2947cc3.

1 occurrence in this chapter

Equation form expr-225055cd888818f2

K⊨∀x0≠x′\Sat{K}{\lforall[x][\eq/[\Obj 0][x']]}

Read as: structure K satisfies for every x, object language symbol zero is not equal to x prime

Means: A satisfaction claim stating: structure K satisfies for every x, object language symbol zero is not equal to x prime. This meaning is authored from the complete TR-027 source packet for expr-225055cd888818f2.

1 occurrence in this chapter

Equation form expr-225893dd7e0e2c46

[x][x]

Read as: the block of x

Means: Arithmetic notation denoting: the block of x. This meaning is authored from the complete TR-027 source packet for expr-225893dd7e0e2c46.

3 occurrences in this chapter

Equation form expr-22eafe5016ba8c2e

M1\Struct{M}_1

Read as: structure M sub one

Means: Arithmetic notation denoting: structure M sub one. This meaning is authored from the complete TR-027 source packet for expr-22eafe5016ba8c2e.

1 occurrence in this chapter

Equation form expr-237cc5a10b68c894

u≺vu \nsless v

Read as: u is less than v in the model

Means: A claim about the interpreted arithmetic operations or order stating: u is less than v in the model. This meaning is authored from the complete TR-027 source packet for expr-237cc5a10b68c894.

6 occurrences in this chapter

Equation form expr-2395f5fb0a93cb4f

⊗\nstimes

Read as: model multiplication

Means: A claim about the interpreted arithmetic operations or order stating: model multiplication. This meaning is authored from the complete TR-027 source packet for expr-2395f5fb0a93cb4f.

3 occurrences in this chapter

Equation form expr-26ffbab17785b03c

u⊕n∗≺vu \nsplus n^\nssucc \nsless v

Read as: u plus in the model n followed by model successor is less than v in the model

Means: A claim about the interpreted arithmetic operations or order stating: u plus in the model n followed by model successor is less than v in the model. This meaning is authored from the complete TR-027 source packet for expr-26ffbab17785b03c.

1 occurrence in this chapter

Equation form expr-283a62bfa767b25a

x∗x^\nssucc

Read as: x followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: x followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-283a62bfa767b25a.

2 occurrences in this chapter

Equation form expr-283ef62c2999e1c8

x⊗yx \nstimes y

Read as: x times in the model y

Means: A claim about the interpreted arithmetic operations or order stating: x times in the model y. This meaning is authored from the complete TR-027 source packet for expr-283ef62c2999e1c8.

1 occurrence in this chapter

Equation form expr-29372440ea32446f

K⊨∀x∀y(x+y′)=(x+y)′\Sat{K}{\lforall[x][\lforall[y][\eq[(x + y')][(x + y)']]]}

Read as: structure K satisfies for every x, for every y, open parenthesis x plus y prime close parenthesis equals open parenthesis x plus y close parenthesis prime

Means: A satisfaction claim stating: structure K satisfies for every x, for every y, open parenthesis x plus y prime close parenthesis equals open parenthesis x plus y close parenthesis prime. This meaning is authored from the complete TR-027 source packet for expr-29372440ea32446f.

1 occurrence in this chapter

Equation form expr-2c83a8d95120b06f

(n⊕m∗)=(n+(m+1))=(n+m)+1=(n⊕m)∗since ⊕ and ∗ agree with + and ′ on standard numbers. Now suppose x∈|K|. Then(x⊕a∗)=(x⊕a)=a=a∗=(x⊕a)∗The remaining case is if y∈|K| but x=a. Here we also have to distinguish cases according to whether y=n is standard or y=a:(a⊕n∗)=(a⊕(n+1))=a=a∗=(a⊕n)∗(a⊕a∗)=(a⊕a)=a=a∗=(a⊕a)∗(n \nsplus m^\nssucc) & = (n+(m+1)) = (n+m)+1 = (n \nsplus m)^\nssucc \intertext{since $\nsplus$ and $^\nssucc$ agree with $+$ and $\prime$ on standard numbers. Now suppose $x \in \Domain{K}$. Then} (x \nsplus a^\nssucc) & = (x \nsplus a) = a = a^\nssucc = (x \nsplus a)^\nssucc \intertext{The remaining case is if $y \in \Domain{K}$ but $x = a$. Here we also have to distinguish cases according to whether $y = n$ is standard or $y = a$:} (a \nsplus n^\nssucc) & = (a \nsplus (n+1)) = a = a^\nssucc = (a \nsplus n)^\nssucc\\ (a \nsplus a^\nssucc) & = (a \nsplus a) = a = a^\nssucc = (a \nsplus a)^\nssucc

Read as: open parenthesis n plus in the model m followed by model successor close parenthesis equals open parenthesis n plus open parenthesis m plus one close parenthesis close parenthesis equals open parenthesis n plus m close parenthesis plus one equals open parenthesis n plus in the model m close parenthesis followed by model successor since model addition and model successor agree with addition and successor on standard numbers. Now suppose x is in the domain of structure K. Then open parenthesis x plus in the model a followed by model successor close parenthesis equals open parenthesis x plus in the model a close parenthesis equals a equals a followed by model successor equals open parenthesis x plus in the model a close parenthesis followed by model successor The remaining case has x equal to a. Distinguish whether y equals the standard n or y equals a. Then open parenthesis a plus in the model n followed by model successor close parenthesis equals open parenthesis a plus in the model open parenthesis n plus one close parenthesis close parenthesis equals a equals a followed by model successor equals open parenthesis a plus in the model n close parenthesis followed by model successor next row open parenthesis a plus in the model a followed by model successor close parenthesis equals open parenthesis a plus in the model a close parenthesis equals a equals a followed by model successor equals open parenthesis a plus in the model a close parenthesis followed by model successor

Means: A source-ordered calculation or table whose rows state: open parenthesis n plus in the model m followed by model successor close parenthesis equals open parenthesis n plus open parenthesis m plus one close parenthesis close parenthesis equals open parenthesis n plus m close parenthesis plus one equals open parenthesis n plus in the model m close parenthesis followed by model successor since model addition and model successor agree with addition and successor on standard numbers. Now suppose x is in the domain of structure K. Then open parenthesis x plus in the model a followed by model successor close parenthesis equals open parenthesis x plus in the model a close parenthesis equals a equals a followed by model successor equals open parenthesis x plus in the model a close parenthesis followed by model successor The remaining case has x equal to a. Distinguish whether y equals the standard n or y equals a. Then open parenthesis a plus in the model n followed by model successor close parenthesis equals open parenthesis a plus in the model open parenthesis n plus one close parenthesis close parenthesis equals a equals a followed by model successor equals open parenthesis a plus in the model n close parenthesis followed by model successor next row open parenthesis a plus in the model a followed by model successor close parenthesis equals open parenthesis a plus in the model a close parenthesis equals a equals a followed by model successor equals open parenthesis a plus in the model a close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-2c83a8d95120b06f.

1 occurrence in this chapter

Equation form expr-2d316be2374534aa

g(n)≠g(m)g(n) \neq g(m)

Read as: g open parenthesis n close parenthesis is not equal to g open parenthesis m close parenthesis

Means: An arithmetic relation or equation stating: g open parenthesis n close parenthesis is not equal to g open parenthesis m close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-2d316be2374534aa.

1 occurrence in this chapter

Equation form expr-2d65e3b5556d0619

∀x¬x<x\lforall[x][\lnot x < x]

Read as: for every x, not x is less than x

Means: A first-order arithmetic formula stating: for every x, not x is less than x. This meaning is authored from the complete TR-027 source packet for expr-2d65e3b5556d0619.

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: Arithmetic notation denoting: x. This meaning is authored from the complete TR-027 source packet for expr-2d711642b726b044.

22 occurrences in this chapter

Equation form expr-2e4c535d9027cbf2

x⊕n∗=x⊕yx \nsplus n^\nssucc = x \nsplus y

Read as: x plus in the model n followed by model successor equals x plus in the model y

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model n followed by model successor equals x plus in the model y. This meaning is authored from the complete TR-027 source packet for expr-2e4c535d9027cbf2.

1 occurrence in this chapter

Equation form expr-2e7d2c03a9507ae2

cc

Read as: c

Means: Arithmetic notation denoting: c. This meaning is authored from the complete TR-027 source packet for expr-2e7d2c03a9507ae2.

5 occurrences in this chapter

Equation form expr-2fe51accbfe15482

¬ConPA\lnot \OCon[\Th{PA}]

Read as: not the formal consistency statement for theory PA

Means: An arithmetized-syntax statement denoting: not the formal consistency statement for theory PA. This meaning is authored from the complete TR-027 source packet for expr-2fe51accbfe15482.

1 occurrence in this chapter

Equation form expr-3036b38016cdb222

[x]≺[y][x] \nsless [y]

Read as: the block of x is less than the block of y in the model

Means: A claim about the interpreted arithmetic operations or order stating: the block of x is less than the block of y in the model. This meaning is authored from the complete TR-027 source packet for expr-3036b38016cdb222.

1 occurrence in this chapter

Equation form expr-31669d24727a80d1

PA∪{¬ConPA}\Th{PA} \cup \{\lnot \OCon[\Th{PA}]\}

Read as: theory PA union not the formal consistency statement for theory PA

Means: An arithmetized-syntax statement denoting: theory PA union not the formal consistency statement for theory PA. This meaning is authored from the complete TR-027 source packet for expr-31669d24727a80d1.

2 occurrences in this chapter

Equation form expr-330958166addde73

y=v⊕m∗≺xy = v \nsplus m^\nssucc \nsless x

Read as: y equals v plus in the model m followed by model successor is less than x in the model

Means: A claim about the interpreted arithmetic operations or order stating: y equals v plus in the model m followed by model successor is less than x in the model. This meaning is authored from the complete TR-027 source packet for expr-330958166addde73.

1 occurrence in this chapter

Equation form expr-3328b69c6fa0174e

⟨n,m⟩∈<N\tuple{n,m} \in \Assign{<}{N}

Read as: the tuple n comma m is in the interpreted less-than relation of structure N

Means: Arithmetic notation denoting: the tuple n comma m is in the interpreted less-than relation of structure N. This meaning is authored from the complete TR-027 source packet for expr-3328b69c6fa0174e.

2 occurrences in this chapter

Equation form expr-33a9c96304bcc292

|M|\Domain{M}

Read as: the domain of structure M

Means: Arithmetic notation denoting: the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-33a9c96304bcc292.

6 occurrences in this chapter

Equation form expr-347326a58919f311

a∗=aa^\nssucc = a

Read as: a followed by model successor equals a

Means: A claim about the interpreted arithmetic operations or order stating: a followed by model successor equals a. This meaning is authored from the complete TR-027 source packet for expr-347326a58919f311.

1 occurrence in this chapter

Equation form expr-349ba76d907adfab

×\times

Read as: times

Means: Arithmetic notation denoting: times. This meaning is authored from the complete TR-027 source packet for expr-349ba76d907adfab.

4 occurrences in this chapter

Equation form expr-34c01bca980af229

M⊨Q1\Sat{M}{!Q_1}

Read as: structure M satisfies Q sub one

Means: A satisfaction claim stating: structure M satisfies Q sub one. This meaning is authored from the complete TR-027 source packet for expr-34c01bca980af229.

1 occurrence in this chapter

Equation form expr-353eb8fc63aeaf0d

M⊨∀x∀y(x′=y′→x=y)\Sat{M}{\lforall[x][\lforall[y][(\eq[x'][y'] \lif \eq[x][y])]]}

Read as: structure M satisfies for every x, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis

Means: A satisfaction claim stating: structure M satisfies for every x, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-353eb8fc63aeaf0d.

1 occurrence in this chapter

Equation form expr-35ad5b7ab055aabd

ConPA\OCon[\Th{PA}]

Read as: the formal consistency statement for theory PA

Means: An arithmetized-syntax statement denoting: the formal consistency statement for theory PA. This meaning is authored from the complete TR-027 source packet for expr-35ad5b7ab055aabd.

2 occurrences in this chapter

Equation form expr-380918b946a52664

==

Read as: equals

Means: An arithmetic relation or equation stating: equals. This meaning is authored from the complete TR-027 source packet for expr-380918b946a52664.

1 occurrence in this chapter

Equation form expr-38ec0b929988bc6a

u≺xu \nsless x

Read as: u is less than x in the model

Means: A claim about the interpreted arithmetic operations or order stating: u is less than x in the model. This meaning is authored from the complete TR-027 source packet for expr-38ec0b929988bc6a.

2 occurrences in this chapter

Equation form expr-3a23ba27fcf6211e

b∗=ab^\nssucc = a

Read as: b followed by model successor equals a

Means: A claim about the interpreted arithmetic operations or order stating: b followed by model successor equals a. This meaning is authored from the complete TR-027 source packet for expr-3a23ba27fcf6211e.

1 occurrence in this chapter

Equation form expr-3bd52a5f8fc05fb1

x∈|M|x \in \Domain{M}

Read as: x is in the domain of structure M

Means: Arithmetic notation denoting: x is in the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-3bd52a5f8fc05fb1.

5 occurrences in this chapter

Equation form expr-3cf05f70af180d8b

predecessor of y≺y^\nssucc y \nsless y

Read as: predecessor of y is less than y in the model

Means: A claim about the interpreted arithmetic operations or order stating: predecessor of y is less than y in the model. This meaning is authored from the complete TR-027 source packet for expr-3cf05f70af180d8b.

1 occurrence in this chapter

Equation form expr-3d2df79065b6f165

Q\Th{Q}

Read as: theory Q

Means: Notation naming: theory Q. This meaning is authored from the complete TR-027 source packet for expr-3d2df79065b6f165.

17 occurrences in this chapter

Equation form expr-3dcb6b7ac71da736

<K′\Assign{<}{K'}

Read as: the interpreted less-than relation of structure K prime

Means: Arithmetic notation denoting: the interpreted less-than relation of structure K prime. This meaning is authored from the complete TR-027 source packet for expr-3dcb6b7ac71da736.

2 occurrences in this chapter

Equation form expr-3dddf911c3c51204

0′\Obj{0}'

Read as: object language symbol zero prime

Means: Arithmetic notation denoting: object language symbol zero prime. This meaning is authored from the complete TR-027 source packet for expr-3dddf911c3c51204.

1 occurrence in this chapter

Equation form expr-3f1b6dc519af96a7

′\prime

Read as: successor symbol

Means: Arithmetic notation denoting: successor symbol. This meaning is authored from the complete TR-027 source packet for expr-3f1b6dc519af96a7.

6 occurrences in this chapter

Equation form expr-40868eaef720dafe

[z]≠[z] \neq [y]

Read as: the block of z is not equal to the block of y

Means: An arithmetic relation or equation stating: the block of z is not equal to the block of y. This meaning is authored from the complete TR-027 source packet for expr-40868eaef720dafe.

1 occurrence in this chapter

Equation form expr-40e5bef92681d093

y≺zy \nsless z

Read as: y is less than z in the model

Means: A claim about the interpreted arithmetic operations or order stating: y is less than z in the model. This meaning is authored from the complete TR-027 source packet for expr-40e5bef92681d093.

1 occurrence in this chapter

Equation form expr-41250d4b85c7140e

[x]∩[y]=∅[x] \cap [y] = \emptyset

Read as: the block of x intersect the block of y equals the empty set

Means: An arithmetic relation or equation stating: the block of x intersect the block of y equals the empty set. This meaning is authored from the complete TR-027 source packet for expr-41250d4b85c7140e.

1 occurrence in this chapter

Equation form expr-420f248cbf45b001

x⊕y=(z⊕z)∗x \nsplus y = (z \nsplus z)^\nssucc

Read as: x plus in the model y equals open parenthesis z plus in the model z close parenthesis followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y equals open parenthesis z plus in the model z close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-420f248cbf45b001.

1 occurrence in this chapter

Equation form expr-42b8f19ae63d24a4

M\Struct{M}

Read as: structure M

Means: Arithmetic notation denoting: structure M. This meaning is authored from the complete TR-027 source packet for expr-42b8f19ae63d24a4.

32 occurrences in this chapter

Equation form expr-44a4dd050d25890f

y∗y^\nssucc

Read as: y followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: y followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-44a4dd050d25890f.

3 occurrences in this chapter

Equation form expr-453c586275863be2

0N=0′N(n)=n+1+N(n,m)=n+m×N(n,m)=n⁢m\Assign{\Obj{0}}{N} & = 0\\ \Assign{\prime}{N}(n) & = n + 1\\ \Assign{+}{N}(n, m) & = n + m\\ \Assign{\times}{N}(n, m) & = nm

Read as: the interpretation of object language symbol zero in structure N equals zero next row the interpretation of successor symbol in structure N open parenthesis n close parenthesis equals n plus one next row the interpretation of plus in structure N open parenthesis n comma m close parenthesis equals n plus m next row the interpreted multiplication operation of structure N open parenthesis n comma m close parenthesis equals n times m

Means: A source-ordered calculation or table whose rows state: the interpretation of object language symbol zero in structure N equals zero next row the interpretation of successor symbol in structure N open parenthesis n close parenthesis equals n plus one next row the interpretation of plus in structure N open parenthesis n comma m close parenthesis equals n plus m next row the interpreted multiplication operation of structure N open parenthesis n comma m close parenthesis equals n times m. This meaning is authored from the complete TR-027 source packet for expr-453c586275863be2.

1 occurrence in this chapter

Equation form expr-459fb36100506069

0K′=1\Assign{\Obj{0}}{K'} = 1

Read as: the interpretation of object language symbol zero in structure K prime equals one

Means: Arithmetic notation denoting: the interpretation of object language symbol zero in structure K prime equals one. This meaning is authored from the complete TR-027 source packet for expr-459fb36100506069.

1 occurrence in this chapter

Equation form expr-4740cf756818ec1a

u∈[y]u \in [y]

Read as: u is in the block of y

Means: An arithmetic relation or equation stating: u is in the block of y. This meaning is authored from the complete TR-027 source packet for expr-4740cf756818ec1a.

1 occurrence in this chapter

Equation form expr-49701f19cef82f09

x≠⟦n¯⟧Mx \neq \Value{\num{n}}{M}

Read as: x is not equal to the value of the numeral for n in structure M

Means: Arithmetic notation denoting: x is not equal to the value of the numeral for n in structure M. This meaning is authored from the complete TR-027 source packet for expr-49701f19cef82f09.

2 occurrences in this chapter

Equation form expr-497d497fc35ba36e

Q1!Q_1

Read as: Q sub one

Means: Arithmetic notation denoting: Q sub one. This meaning is authored from the complete TR-027 source packet for expr-497d497fc35ba36e.

1 occurrence in this chapter

Equation form expr-49efc4a74c809357

g(+N(n,m))=g(n+m)g(\Assign{+}{N}(n,m)) = g(n+m)

Read as: g open parenthesis the interpretation of plus in structure N open parenthesis n comma m close parenthesis close parenthesis equals g open parenthesis n plus m close parenthesis

Means: Arithmetic notation denoting: g open parenthesis the interpretation of plus in structure N open parenthesis n comma m close parenthesis close parenthesis equals g open parenthesis n plus m close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-49efc4a74c809357.

1 occurrence in this chapter

Equation form expr-4bd91d4ea88664d5

x⊕y∗=x=x∗=(x⊕y)∗x \nsplus y^\nssucc = x = x^\nssucc = (x \nsplus y)^\nssucc

Read as: x plus in the model y followed by model successor equals x equals x followed by model successor equals open parenthesis x plus in the model y close parenthesis followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y followed by model successor equals x equals x followed by model successor equals open parenthesis x plus in the model y close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-4bd91d4ea88664d5.

1 occurrence in this chapter

Equation form expr-4cfdffc3b4c5ca1e

′K(x)\Assign{\prime}{K}(x)

Read as: the interpretation of successor symbol in structure K open parenthesis x close parenthesis

Means: Arithmetic notation denoting: the interpretation of successor symbol in structure K open parenthesis x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-4cfdffc3b4c5ca1e.

1 occurrence in this chapter

Equation form expr-4d05672a6bd868b6

|N|\Domain{N}

Read as: the domain of structure N

Means: Arithmetic notation denoting: the domain of structure N. This meaning is authored from the complete TR-027 source packet for expr-4d05672a6bd868b6.

1 occurrence in this chapter

Equation form expr-4d63a9ed6e68cfa9

M3\Struct{M_3}

Read as: structure M sub three

Means: Arithmetic notation denoting: structure M sub three. This meaning is authored from the complete TR-027 source packet for expr-4d63a9ed6e68cfa9.

1 occurrence in this chapter

Equation form expr-4e1259f09671da63

⟦n¯⟧M∈|M|\Value{\num{n}}{M} \in \Domain{M}

Read as: the value of the numeral for n in structure M is in the domain of structure M

Means: Arithmetic notation denoting: the value of the numeral for n in structure M is in the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-4e1259f09671da63.

1 occurrence in this chapter

Equation form expr-4f1dbaf1624767fe

x≺zx \nsless z

Read as: x is less than z in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than z in the model. This meaning is authored from the complete TR-027 source packet for expr-4f1dbaf1624767fe.

3 occurrences in this chapter

Equation form expr-508401e3bdfc7cdc

n=(n−1)∗n = (n-1)^\nssucc

Read as: n equals open parenthesis n minus one close parenthesis followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: n equals open parenthesis n minus one close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-508401e3bdfc7cdc.

1 occurrence in this chapter

Equation form expr-50ef1c914f20b27a

|K|=ℕ∪{a}\Domain{K} = \Nat \cup \{a\}

Read as: the domain of structure K equals the natural numbers union a

Means: Arithmetic notation denoting: the domain of structure K equals the natural numbers union a. This meaning is authored from the complete TR-027 source packet for expr-50ef1c914f20b27a.

2 occurrences in this chapter

Equation form expr-5184679dafbcaceb

z≺yz \nsless y

Read as: z is less than y in the model

Means: A claim about the interpreted arithmetic operations or order stating: z is less than y in the model. This meaning is authored from the complete TR-027 source packet for expr-5184679dafbcaceb.

2 occurrences in this chapter

Equation form expr-52366a79f02698fb

L⊨Q5\Sat{L}{!Q_5}

Read as: structure L satisfies Q sub five

Means: A satisfaction claim stating: structure L satisfies Q sub five. This meaning is authored from the complete TR-027 source packet for expr-52366a79f02698fb.

1 occurrence in this chapter

Equation form expr-5305e79612ebbb95

n≤kn \le k

Read as: n is less than or equal to k

Means: An arithmetic relation or equation stating: n is less than or equal to k. This meaning is authored from the complete TR-027 source packet for expr-5305e79612ebbb95.

1 occurrence in this chapter

Equation form expr-53c172b5c4c61663

xx∗nn+1aax⊕y0ma00mannn+maaaaax⊗y0ma0000n0n⁢maa0aa\begin{array}{c|c} x & x^\nssucc \\ \hline n & n+1 \\ a & a \end{array} \qquad \begin{array}{c|ccc} x \nsplus y & 0 & m & a \\ \hline 0 & 0 & m & a \\ n & n & n+m & a \\ a & a & a & a \\ \end{array} \qquad \begin{array}{c|ccc} x \nstimes y & 0 & m & a \\ \hline 0 & 0 & 0 & 0 \\ n & 0 & nm & a \\ a & 0 & a & a \\ \end{array}

Read as: table begins x x followed by model successor next row table rule n n plus one next row a a table ends table begins x plus in the model y zero m a next row table rule zero zero m a next row n n n plus m a next row a a a a next row table ends table begins x times in the model y zero m a next row table rule zero zero zero zero next row n zero n times m a next row a zero a a next row table ends

Means: A source-ordered calculation or table whose rows state: table begins x x followed by model successor next row table rule n n plus one next row a a table ends table begins x plus in the model y zero m a next row table rule zero zero m a next row n n n plus m a next row a a a a next row table ends table begins x times in the model y zero m a next row table rule zero zero zero zero next row n zero n times m a next row a zero a a next row table ends. This meaning is authored from the complete TR-027 source packet for expr-53c172b5c4c61663.

1 occurrence in this chapter

Equation form expr-551dd6df4b87de67

⊕\nsplus

Read as: model addition

Means: A claim about the interpreted arithmetic operations or order stating: model addition. This meaning is authored from the complete TR-027 source packet for expr-551dd6df4b87de67.

5 occurrences in this chapter

Equation form expr-5553c013c717e9a8

Q5!Q_5

Read as: Q sub five

Means: Arithmetic notation denoting: Q sub five. This meaning is authored from the complete TR-027 source packet for expr-5553c013c717e9a8.

1 occurrence in this chapter

Equation form expr-558ceefeb1805369

M3⊨Q3\Sat{M_3}{!Q_3}

Read as: structure M sub three satisfies Q sub three

Means: A satisfaction claim stating: structure M sub three satisfies Q sub three. This meaning is authored from the complete TR-027 source packet for expr-558ceefeb1805369.

1 occurrence in this chapter

Equation form expr-55f1e7f6d4b13259

a⊕a∗=aa \nsplus a^\nssucc = a

Read as: a plus in the model a followed by model successor equals a

Means: A claim about the interpreted arithmetic operations or order stating: a plus in the model a followed by model successor equals a. This meaning is authored from the complete TR-027 source packet for expr-55f1e7f6d4b13259.

1 occurrence in this chapter

Equation form expr-5690ed5ba5408aeb

PA⊢∀x∀y∀z((x+y)=(x+z)→y=z)\Th{PA} \Proves \lforall[x][\lforall[y][\lforall[z][(\eq[(x+y)][(x+z)] \lif y = z)]]]

Read as: theory PA proves for every x, for every y, for every z, open parenthesis open parenthesis x plus y close parenthesis equals open parenthesis x plus z close parenthesis implies y equals z close parenthesis

Means: A derivability claim stating: theory PA proves for every x, for every y, for every z, open parenthesis open parenthesis x plus y close parenthesis equals open parenthesis x plus z close parenthesis implies y equals z close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-5690ed5ba5408aeb.

1 occurrence in this chapter

Equation form expr-57885e4c75965b23

Γ\Gamma

Read as: Gamma

Means: Arithmetic notation denoting: Gamma. This meaning is authored from the complete TR-027 source packet for expr-57885e4c75965b23.

6 occurrences in this chapter

Equation form expr-584217d5434c513b

[x]≠[x] \neq [z]

Read as: the block of x is not equal to the block of z

Means: An arithmetic relation or equation stating: the block of x is not equal to the block of z. This meaning is authored from the complete TR-027 source packet for expr-584217d5434c513b.

1 occurrence in this chapter

Equation form expr-58685177734bcab8

g(0N)=g(0)g(\Assign{\Obj{0}}{N}) = g(0)

Read as: g open parenthesis the interpretation of object language symbol zero in structure N close parenthesis equals g open parenthesis zero close parenthesis

Means: Arithmetic notation denoting: g open parenthesis the interpretation of object language symbol zero in structure N close parenthesis equals g open parenthesis zero close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-58685177734bcab8.

1 occurrence in this chapter

Equation form expr-5923c70e4f9e1836

|M1|\Domain{M_1}

Read as: the domain of structure M sub one

Means: Arithmetic notation denoting: the domain of structure M sub one. This meaning is authored from the complete TR-027 source packet for expr-5923c70e4f9e1836.

1 occurrence in this chapter

Equation form expr-594e519ae499312b

zz

Read as: z

Means: Arithmetic notation denoting: z. This meaning is authored from the complete TR-027 source packet for expr-594e519ae499312b.

4 occurrences in this chapter

Equation form expr-599089bbf5cc9773

g=hg = h

Read as: g equals h

Means: An arithmetic relation or equation stating: g equals h. This meaning is authored from the complete TR-027 source packet for expr-599089bbf5cc9773.

1 occurrence in this chapter

Equation form expr-5afa20230f875253

n∈ℕn \in \Nat

Read as: n is in the natural numbers

Means: An arithmetic relation or equation stating: n is in the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-5afa20230f875253.

6 occurrences in this chapter

Equation form expr-5b63a60fdb499c3a

N\Struct{N}

Read as: structure N

Means: Arithmetic notation denoting: structure N. This meaning is authored from the complete TR-027 source packet for expr-5b63a60fdb499c3a.

36 occurrences in this chapter

Equation form expr-5b99169f1ea82aec

x⊕y∉[x]x \nsplus y \notin [x]

Read as: x plus in the model y is not in the block of x

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y is not in the block of x. This meaning is authored from the complete TR-027 source packet for expr-5b99169f1ea82aec.

1 occurrence in this chapter

Equation form expr-5bb0ee0ef2e6d5c0

×K(x,y)\Assign{\times}{K}(x, y)

Read as: the interpreted multiplication operation of structure K open parenthesis x comma y close parenthesis

Means: Arithmetic notation denoting: the interpreted multiplication operation of structure K open parenthesis x comma y close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-5bb0ee0ef2e6d5c0.

1 occurrence in this chapter

Equation form expr-5cd888c50377bc38

|M|={⟦n¯⟧M:n∈ℕ}\Domain{M} = \Setabs{\Value{\num{n}}{M}}{n \in \Nat}

Read as: the domain of structure M equals the set of the value of the numeral for n in structure M such that n is in the natural numbers

Means: Arithmetic notation denoting: the domain of structure M equals the set of the value of the numeral for n in structure M such that n is in the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-5cd888c50377bc38.

3 occurrences in this chapter

Equation form expr-5d0cd28f3e4437b2

x∈|L|x \in \Domain{L}

Read as: x is in the domain of structure L

Means: Arithmetic notation denoting: x is in the domain of structure L. This meaning is authored from the complete TR-027 source packet for expr-5d0cd28f3e4437b2.

1 occurrence in this chapter

Equation form expr-5f15f283815babac

y⊕y∉[y]y \nsplus y \notin [y]

Read as: y plus in the model y is not in the block of y

Means: A claim about the interpreted arithmetic operations or order stating: y plus in the model y is not in the block of y. This meaning is authored from the complete TR-027 source packet for expr-5f15f283815babac.

1 occurrence in this chapter

Equation form expr-5f489910b4a5daaa

×M\Assign{\times}{M}

Read as: the interpreted multiplication operation of structure M

Means: Arithmetic notation denoting: the interpreted multiplication operation of structure M. This meaning is authored from the complete TR-027 source packet for expr-5f489910b4a5daaa.

2 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: Arithmetic notation denoting: zero. This meaning is authored from the complete TR-027 source packet for expr-5feceb66ffc86f38.

6 occurrences in this chapter

Equation form expr-607e495a98cb5559

ℤ\Int

Read as: the integers

Means: Arithmetic notation denoting: the integers. This meaning is authored from the complete TR-027 source packet for expr-607e495a98cb5559.

1 occurrence in this chapter

Equation form expr-61cd24e4f0d1b008

|K|\Domain{K}

Read as: the domain of structure K

Means: Arithmetic notation denoting: the domain of structure K. This meaning is authored from the complete TR-027 source packet for expr-61cd24e4f0d1b008.

1 occurrence in this chapter

Equation form expr-61e81c890ea8af25

+N\Assign{+}{N}

Read as: the interpretation of plus in structure N

Means: Arithmetic notation denoting: the interpretation of plus in structure N. This meaning is authored from the complete TR-027 source packet for expr-61e81c890ea8af25.

1 occurrence in this chapter

Equation form expr-62c66a7a5dd70c31

mm

Read as: m

Means: Arithmetic notation denoting: m. This meaning is authored from the complete TR-027 source packet for expr-62c66a7a5dd70c31.

2 occurrences in this chapter

Equation form expr-6373869f2e570647

Nk⊨c≠n¯\Sat{N_k}{\eq/[c][\num{n}]}

Read as: structure N sub k satisfies c is not equal to the numeral for n

Means: A satisfaction claim stating: structure N sub k satisfies c is not equal to the numeral for n. This meaning is authored from the complete TR-027 source packet for expr-6373869f2e570647.

1 occurrence in this chapter

Equation form expr-63a88d95adb12b74

h:ℕ→|M|h\colon \Nat \to \Domain{M}

Read as: h from the natural numbers to the domain of structure M

Means: Arithmetic notation denoting: h from the natural numbers to the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-63a88d95adb12b74.

1 occurrence in this chapter

Equation form expr-6403429ad98d3b9f

y∈[x]y \in [x]

Read as: y is in the block of x

Means: An arithmetic relation or equation stating: y is in the block of x. This meaning is authored from the complete TR-027 source packet for expr-6403429ad98d3b9f.

1 occurrence in this chapter

Equation form expr-6584d9b01c7229a9

…iterated predecessor of x≺iterated predecessor of x≺iterated predecessor of x≺x≺x∗≺x∗∗≺x∗∗∗≺…\dots ^{\nssucc\nssucc\nssucc}x \nsless ^{\nssucc\nssucc}x \nsless ^{\nssucc}x \nsless x \nsless x^{\nssucc} \nsless x^{\nssucc\nssucc} \nsless x^{\nssucc\nssucc\nssucc} \nsless \dots

Read as: and so on, the third predecessor of x is less than the second predecessor of x, which is less than the predecessor of x, which is less than x, which is less than the first successor of x, which is less than the second successor of x, which is less than the third successor of x, and so on, in the model

Means: A claim about the interpreted arithmetic operations or order stating: and so on, the third predecessor of x is less than the second predecessor of x, which is less than the predecessor of x, which is less than x, which is less than the first successor of x, which is less than the second successor of x, which is less than the third successor of x, and so on, in the model. This meaning is authored from the complete TR-027 source packet for expr-6584d9b01c7229a9.

1 occurrence in this chapter

Equation form expr-65d632215315c00d

g(n)=⟦n¯⟧Mg(n) = \Value{\num{n}}{M}

Read as: g open parenthesis n close parenthesis equals the value of the numeral for n in structure M

Means: Arithmetic notation denoting: g open parenthesis n close parenthesis equals the value of the numeral for n in structure M. This meaning is authored from the complete TR-027 source packet for expr-65d632215315c00d.

1 occurrence in this chapter

Equation form expr-67500429cdd36beb

n>0n>0

Read as: n is greater than zero

Means: An arithmetic relation or equation stating: n is greater than zero. This meaning is authored from the complete TR-027 source packet for expr-67500429cdd36beb.

3 occurrences in this chapter

Equation form expr-68c90ebed2206f6f

+K(x,y)\Assign{+}{K}(x, y)

Read as: the interpretation of plus in structure K open parenthesis x comma y close parenthesis

Means: Arithmetic notation denoting: the interpretation of plus in structure K open parenthesis x comma y close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-68c90ebed2206f6f.

1 occurrence in this chapter

Equation form expr-68d1cfddb85baf20

x⊕y∗x \nsplus y^\nssucc

Read as: x plus in the model y followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-68d1cfddb85baf20.

1 occurrence in this chapter

Equation form expr-6989b47ac8292e9b

+L=⊕\Assign{+}{L} = \nsplus

Read as: the interpretation of plus in structure L equals model addition

Means: A claim about the interpreted arithmetic operations or order stating: the interpretation of plus in structure L equals model addition. This meaning is authored from the complete TR-027 source packet for expr-6989b47ac8292e9b.

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: Arithmetic notation denoting: one. This meaning is authored from the complete TR-027 source packet for expr-6b86b273ff34fce1.

2 occurrences in this chapter

Equation form expr-6c310303f3eccfee

a≺na \nsless n

Read as: a is less than n in the model

Means: A claim about the interpreted arithmetic operations or order stating: a is less than n in the model. This meaning is authored from the complete TR-027 source packet for expr-6c310303f3eccfee.

1 occurrence in this chapter

Equation form expr-6d0bd856c59b0434

M2\mathfrak{M}_2

Read as: structure M sub two

Means: Arithmetic notation denoting: structure M sub two. This meaning is authored from the complete TR-027 source packet for expr-6d0bd856c59b0434.

1 occurrence in this chapter

Equation form expr-6d2288fc9569ee71

M2⊨Q1\Sat{M_2}{!Q_1}

Read as: structure M sub two satisfies Q sub one

Means: A satisfaction claim stating: structure M sub two satisfies Q sub one. This meaning is authored from the complete TR-027 source packet for expr-6d2288fc9569ee71.

1 occurrence in this chapter

Equation form expr-6d774ede383fb1aa

z≺uz \nsless u

Read as: z is less than u in the model

Means: A claim about the interpreted arithmetic operations or order stating: z is less than u in the model. This meaning is authored from the complete TR-027 source packet for expr-6d774ede383fb1aa.

1 occurrence in this chapter

Equation form expr-6dd77695ea5d9238

≺0{} \nsless 0

Read as: less than zero in the model

Means: A claim about the interpreted arithmetic operations or order stating: less than zero in the model. This meaning is authored from the complete TR-027 source packet for expr-6dd77695ea5d9238.

1 occurrence in this chapter

Equation form expr-6de6646caf3df8f3

a=a∗a = a^\nssucc

Read as: a equals a followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: a equals a followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-6de6646caf3df8f3.

1 occurrence in this chapter

Equation form expr-6e4cbbf82204396a

A∈TA!A \in \Th{TA}

Read as: formula A is in theory TA

Means: Notation naming: formula A is in theory TA. This meaning is authored from the complete TR-027 source packet for expr-6e4cbbf82204396a.

1 occurrence in this chapter

Equation form expr-6e597bff7579f4b0

Q⊬∀x¬x<x\Th{Q} \Proves/ \lforall[x][\lnot x < x]

Read as: theory Q does not prove for every x, not x is less than x

Means: A derivability claim stating: theory Q does not prove for every x, not x is less than x. This meaning is authored from the complete TR-027 source packet for expr-6e597bff7579f4b0.

1 occurrence in this chapter

Equation form expr-6e788ae1b2e697c9

g(n+1)=⟦n+1¯⟧Mg(n+1) = \Value{\num{n+1}}{M}

Read as: g open parenthesis n plus one close parenthesis equals the value of the numeral for n plus one in structure M

Means: Arithmetic notation denoting: g open parenthesis n plus one close parenthesis equals the value of the numeral for n plus one in structure M. This meaning is authored from the complete TR-027 source packet for expr-6e788ae1b2e697c9.

1 occurrence in this chapter

Equation form expr-6f5de70bcb4734dd

PA⊢∀x(y≠0→x<(x+y))\Th{PA} \Proves \lforall[x][(y \neq \Obj{0} \lif x < (x+y))]

Read as: theory PA proves for every x, open parenthesis y is not equal to object language symbol zero implies x is less than open parenthesis x plus y close parenthesis close parenthesis

Means: A derivability claim stating: theory PA proves for every x, open parenthesis y is not equal to object language symbol zero implies x is less than open parenthesis x plus y close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-6f5de70bcb4734dd.

1 occurrence in this chapter

Equation form expr-7031abeb145ec3b6

(x⊕y)∗(x \nsplus y)^\nssucc

Read as: open parenthesis x plus in the model y close parenthesis followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: open parenthesis x plus in the model y close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-7031abeb145ec3b6.

1 occurrence in this chapter

Equation form expr-70843ec890890170

(y⊕y)∗∉[y](y \nsplus y)^\nssucc \notin [y]

Read as: open parenthesis y plus in the model y close parenthesis followed by model successor is not in the block of y

Means: A claim about the interpreted arithmetic operations or order stating: open parenthesis y plus in the model y close parenthesis followed by model successor is not in the block of y. This meaning is authored from the complete TR-027 source packet for expr-70843ec890890170.

1 occurrence in this chapter

Equation form expr-72111011ea259203

g(0)=ag(0) = a

Read as: g open parenthesis zero close parenthesis equals a

Means: An arithmetic relation or equation stating: g open parenthesis zero close parenthesis equals a. This meaning is authored from the complete TR-027 source packet for expr-72111011ea259203.

1 occurrence in this chapter

Equation form expr-729ded21f178dd5b

Q⊢n+m¯=(n¯+m¯)\Th{Q} \Proves \num{n+m} = (\num{n} + \num{m})

Read as: theory Q proves the numeral for n plus m equals open parenthesis the numeral for n plus the numeral for m close parenthesis

Means: A derivability claim stating: theory Q proves the numeral for n plus m equals open parenthesis the numeral for n plus the numeral for m close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-729ded21f178dd5b.

1 occurrence in this chapter

Equation form expr-72fcd7dafa1f7007

n+1¯\num{n+1}

Read as: the numeral for n plus one

Means: Arithmetic notation denoting: the numeral for n plus one. This meaning is authored from the complete TR-027 source packet for expr-72fcd7dafa1f7007.

1 occurrence in this chapter

Equation form expr-732d2b595bb1b287

s:M→M∖{z}s\colon M \to M \setminus \{z\}

Read as: s from M to M set minus z

Means: Arithmetic notation denoting: s from M to M set minus z. This meaning is authored from the complete TR-027 source packet for expr-732d2b595bb1b287.

1 occurrence in this chapter

Equation form expr-74bb1ccd6fcdeca0

g(n)=n−1g(n) = n-1

Read as: g open parenthesis n close parenthesis equals n minus one

Means: An arithmetic relation or equation stating: g open parenthesis n close parenthesis equals n minus one. This meaning is authored from the complete TR-027 source packet for expr-74bb1ccd6fcdeca0.

1 occurrence in this chapter

Equation form expr-758f984f4043d173

Q⊢∀x¬x<0\Th{Q} \Proves \lforall[x][\lnot x<0]

Read as: theory Q proves for every x, not x is less than zero

Means: A derivability claim stating: theory Q proves for every x, not x is less than zero. This meaning is authored from the complete TR-027 source packet for expr-758f984f4043d173.

1 occurrence in this chapter

Equation form expr-759ee9455773a9db

x≺x⊕yx \nsless x \nsplus y

Read as: x is less than the model sum of x and y

Means: A claim about the interpreted arithmetic operations or order stating: x is less than the model sum of x and y. This meaning is authored from the complete TR-027 source packet for expr-759ee9455773a9db.

2 occurrences in this chapter

Equation form expr-7639b12da173ed30

K′\Struct{K'}

Read as: structure K prime

Means: Arithmetic notation denoting: structure K prime. This meaning is authored from the complete TR-027 source packet for expr-7639b12da173ed30.

2 occurrences in this chapter

Equation form expr-783819f2e7b1882c

N⊨Q\Sat{N}{\Th{Q}}

Read as: structure N satisfies theory Q

Means: A satisfaction claim stating: structure N satisfies theory Q. This meaning is authored from the complete TR-027 source packet for expr-783819f2e7b1882c.

1 occurrence in this chapter

Equation form expr-79d4c7f9c8579543

ℕ\Nat

Read as: the natural numbers

Means: Arithmetic notation denoting: the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-79d4c7f9c8579543.

16 occurrences in this chapter

Equation form expr-7a19604a81b74d4e

y≺(y⊕y)∗y \nsless (y \nsplus y)^\nssucc

Read as: y is less than the model successor of the model sum of y and y

Means: A claim about the interpreted arithmetic operations or order stating: y is less than the model successor of the model sum of y and y. This meaning is authored from the complete TR-027 source packet for expr-7a19604a81b74d4e.

1 occurrence in this chapter

Equation form expr-7ac6071350b06b46

g:ℕ→|K|g\colon \Nat \to \Domain{K}

Read as: g from the natural numbers to the domain of structure K

Means: Arithmetic notation denoting: g from the natural numbers to the domain of structure K. This meaning is authored from the complete TR-027 source packet for expr-7ac6071350b06b46.

1 occurrence in this chapter

Equation form expr-7bb0687703a6f9db

M⊭n¯<m¯\Sat/{M}{\num{n} < \num{m}}

Read as: structure M does not satisfy the numeral for n is less than the numeral for m

Means: A satisfaction claim stating: structure M does not satisfy the numeral for n is less than the numeral for m. This meaning is authored from the complete TR-027 source packet for expr-7bb0687703a6f9db.

1 occurrence in this chapter

Equation form expr-7d8f56a89cb46e85

M1\mathfrak{M}_1

Read as: structure M sub one

Means: Arithmetic notation denoting: structure M sub one. This meaning is authored from the complete TR-027 source packet for expr-7d8f56a89cb46e85.

1 occurrence in this chapter

Equation form expr-7e351a02195c132d

M1⊨Q2\Sat{M_1}{!Q_2}

Read as: structure M sub one satisfies Q sub two

Means: A satisfaction claim stating: structure M sub one satisfies Q sub two. This meaning is authored from the complete TR-027 source packet for expr-7e351a02195c132d.

1 occurrence in this chapter

Equation form expr-7e901baa2083cacd

x⊕y∈[x]x \nsplus y \in [x]

Read as: x plus in the model y is in the block of x

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y is in the block of x. This meaning is authored from the complete TR-027 source packet for expr-7e901baa2083cacd.

1 occurrence in this chapter

Equation form expr-7f024b2d7f1db4d4

n¯\num{n}

Read as: the numeral for n

Means: Arithmetic notation denoting: the numeral for n. This meaning is authored from the complete TR-027 source packet for expr-7f024b2d7f1db4d4.

5 occurrences in this chapter

Equation form expr-7f8a70388b5fe1dc

g(′N(n))=′M(g(n))g(\Assign{\prime}{N}(n)) = \Assign{\prime}{M}(g(n))

Read as: g open parenthesis the interpretation of successor symbol in structure N open parenthesis n close parenthesis close parenthesis equals the interpretation of successor symbol in structure M open parenthesis g open parenthesis n close parenthesis close parenthesis

Means: Arithmetic notation denoting: g open parenthesis the interpretation of successor symbol in structure N open parenthesis n close parenthesis close parenthesis equals the interpretation of successor symbol in structure M open parenthesis g open parenthesis n close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-7f8a70388b5fe1dc.

1 occurrence in this chapter

Equation form expr-80fce15b99d444a5

Q⊢n¯≠m¯\Th{Q} \Proves \eq/[\num{n}][\num{m}]

Read as: theory Q proves the numeral for n is not equal to the numeral for m

Means: A derivability claim stating: theory Q proves the numeral for n is not equal to the numeral for m. This meaning is authored from the complete TR-027 source packet for expr-80fce15b99d444a5.

1 occurrence in this chapter

Equation form expr-8238c028f61fc0f7

A!A

Read as: formula A

Means: Arithmetic notation denoting: formula A. This meaning is authored from the complete TR-027 source packet for expr-8238c028f61fc0f7.

2 occurrences in this chapter

Equation form expr-8254c329a92850f6

kk

Read as: k

Means: Arithmetic notation denoting: k. This meaning is authored from the complete TR-027 source packet for expr-8254c329a92850f6.

2 occurrences in this chapter

Equation form expr-82e7769d356674f8

a⊕a∗=b=b∗=(a⊕a)∗b⊕b∗=a=a∗=(b⊕b)∗b⊕a∗=b=b∗=(b⊕a)∗a⊕b∗=a=a∗=(a⊕b)∗If x is standard, but y is non-standard, we haven⊕a∗=n⊕a=b=b∗=(n⊕a)∗n⊕b∗=n⊕b=a=a∗=(n⊕b)∗& a \nsplus a^\nssucc = b = b^\nssucc = (a \nsplus a)^\nssucc\\ & b \nsplus b^\nssucc = a = a^\nssucc = (b \nsplus b)^\nssucc\\ & b \nsplus a^\nssucc = b = b^\nssucc = (b \nsplus a)^\nssucc\\ & a \nsplus b^\nssucc = a = a^\nssucc = (a \nsplus b)^\nssucc\\ \intertext{If $x$ is standard, but $y$ is non-standard, we have} & n \nsplus a^\nssucc = n \nsplus a = b = b^\nssucc = (n \nsplus a)^\nssucc\\ & n \nsplus b^\nssucc = n \nsplus b = a = a^\nssucc = (n \nsplus b)^\nssucc

Read as: a plus in the model a followed by model successor equals b equals b followed by model successor equals open parenthesis a plus in the model a close parenthesis followed by model successor next row b plus in the model b followed by model successor equals a equals a followed by model successor equals open parenthesis b plus in the model b close parenthesis followed by model successor next row b plus in the model a followed by model successor equals b equals b followed by model successor equals open parenthesis b plus in the model a close parenthesis followed by model successor next row a plus in the model b followed by model successor equals a equals a followed by model successor equals open parenthesis a plus in the model b close parenthesis followed by model successor next row If x is standard comma but y is nonstandard comma we have n plus in the model a followed by model successor equals n plus in the model a equals b equals b followed by model successor equals open parenthesis n plus in the model a close parenthesis followed by model successor next row n plus in the model b followed by model successor equals n plus in the model b equals a equals a followed by model successor equals open parenthesis n plus in the model b close parenthesis followed by model successor

Means: A source-ordered calculation or table whose rows state: a plus in the model a followed by model successor equals b equals b followed by model successor equals open parenthesis a plus in the model a close parenthesis followed by model successor next row b plus in the model b followed by model successor equals a equals a followed by model successor equals open parenthesis b plus in the model b close parenthesis followed by model successor next row b plus in the model a followed by model successor equals b equals b followed by model successor equals open parenthesis b plus in the model a close parenthesis followed by model successor next row a plus in the model b followed by model successor equals a equals a followed by model successor equals open parenthesis a plus in the model b close parenthesis followed by model successor next row If x is standard comma but y is nonstandard comma we have n plus in the model a followed by model successor equals n plus in the model a equals b equals b followed by model successor equals open parenthesis n plus in the model a close parenthesis followed by model successor next row n plus in the model b followed by model successor equals n plus in the model b equals a equals a followed by model successor equals open parenthesis n plus in the model b close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-82e7769d356674f8.

1 occurrence in this chapter

Equation form expr-83dfe28eff40ebc8

x⊕0=xx \nsplus 0 = x

Read as: x plus in the model zero equals x

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model zero equals x. This meaning is authored from the complete TR-027 source packet for expr-83dfe28eff40ebc8.

1 occurrence in this chapter

Equation form expr-84b7ce4ca30c3a69

[x]=[y][x] = [y]

Read as: the block of x equals the block of y

Means: An arithmetic relation or equation stating: the block of x equals the block of y. This meaning is authored from the complete TR-027 source packet for expr-84b7ce4ca30c3a69.

1 occurrence in this chapter

Equation form expr-84d7c29eb5648796

M1\Struct{M_1}

Read as: structure M sub one

Means: Arithmetic notation denoting: structure M sub one. This meaning is authored from the complete TR-027 source packet for expr-84d7c29eb5648796.

3 occurrences in this chapter

Equation form expr-84e3d6b9a99b9671

Q⊢¬n¯<m¯\Th{Q} \Proves \lnot \num{n} < \num{m}

Read as: theory Q proves not the numeral for n is less than the numeral for m

Means: A derivability claim stating: theory Q proves not the numeral for n is less than the numeral for m. This meaning is authored from the complete TR-027 source packet for expr-84e3d6b9a99b9671.

1 occurrence in this chapter

Equation form expr-8553d2666d3f9c70

g(n)=g(⟦n¯⟧N)=⟦n¯⟧Mg(n) = g(\Value{\num{n}}{N}) = \Value{\num{n}}{M}

Read as: g open parenthesis n close parenthesis equals g open parenthesis the value of the numeral for n in structure N close parenthesis equals the value of the numeral for n in structure M

Means: Arithmetic notation denoting: g open parenthesis n close parenthesis equals g open parenthesis the value of the numeral for n in structure N close parenthesis equals the value of the numeral for n in structure M. This meaning is authored from the complete TR-027 source packet for expr-8553d2666d3f9c70.

1 occurrence in this chapter

Equation form expr-8721b86268d9e793

x≠yx \neq y

Read as: x is not equal to y

Means: An arithmetic relation or equation stating: x is not equal to y. This meaning is authored from the complete TR-027 source packet for expr-8721b86268d9e793.

1 occurrence in this chapter

Equation form expr-8a0db34747ba3a70

x≠vx \neq v

Read as: x is not equal to v

Means: An arithmetic relation or equation stating: x is not equal to v. This meaning is authored from the complete TR-027 source packet for expr-8a0db34747ba3a70.

1 occurrence in this chapter

Equation form expr-8a232bbbd111f249

|L|=ℕ∪{a,b}\Domain{L} = \Nat \cup \{a, b\}

Read as: the domain of structure L equals the natural numbers union a comma b

Means: Arithmetic notation denoting: the domain of structure L equals the natural numbers union a comma b. This meaning is authored from the complete TR-027 source packet for expr-8a232bbbd111f249.

1 occurrence in this chapter

Equation form expr-8b9ac2febf771814

x≺xx \nsless x

Read as: x is less than x in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than x in the model. This meaning is authored from the complete TR-027 source packet for expr-8b9ac2febf771814.

1 occurrence in this chapter

Equation form expr-8c2fb3955b2ad990

u∈[x]u \in [x]

Read as: u is in the block of x

Means: An arithmetic relation or equation stating: u is in the block of x. This meaning is authored from the complete TR-027 source packet for expr-8c2fb3955b2ad990.

5 occurrences in this chapter

Equation form expr-8c347d705767246f

′L=∗\Assign{\prime}{L} = \nssucc

Read as: the interpretation of successor symbol in structure L equals model successor

Means: A claim about the interpreted arithmetic operations or order stating: the interpretation of successor symbol in structure L equals model successor. This meaning is authored from the complete TR-027 source packet for expr-8c347d705767246f.

1 occurrence in this chapter

Equation form expr-8d2cacefc75ba038

∅\emptyset

Read as: the empty set

Means: Arithmetic notation denoting: the empty set. This meaning is authored from the complete TR-027 source packet for expr-8d2cacefc75ba038.

1 occurrence in this chapter

Equation form expr-8e6faa0d85c79ab4

z∈Mz \in M

Read as: z is in M

Means: An arithmetic relation or equation stating: z is in M. This meaning is authored from the complete TR-027 source packet for expr-8e6faa0d85c79ab4.

1 occurrence in this chapter

Equation form expr-8eb7afb84080c1e7

g(0)=⟦0¯⟧Mg(0) = \Value{\num{0}}{M}

Read as: g open parenthesis zero close parenthesis equals the value of the numeral for zero in structure M

Means: Arithmetic notation denoting: g open parenthesis zero close parenthesis equals the value of the numeral for zero in structure M. This meaning is authored from the complete TR-027 source packet for expr-8eb7afb84080c1e7.

1 occurrence in this chapter

Equation form expr-8feac84deb26f044

+K′(x,y)=x+y−1if x>0 and y>00otherwise×K′(x,y)=1if x=1 or y=1x⁢y−x−y+2if x>1 and y>10otherwise\Assign{+}{K'}(x, y) & = \begin{cases} x+y-1 & \text{if $x$, $y >0$}\\ 0 & \text{otherwise} \end{cases}\\ \Assign{\times}{K'}(x, y) & = \begin{cases} 1 & \text{if $x = 1$ or $y = 1$}\\ xy - x - y + 2 & \text{if $x$, $y > 1$}\\ 0 & \text{otherwise}\\ \end{cases}

Read as: the interpretation of plus in structure K prime open parenthesis x comma y close parenthesis equals cases begin x plus y minus one if x and y are both greater than zero next row zero otherwise table ends next row the interpreted multiplication operation of structure K prime open parenthesis x comma y close parenthesis equals cases begin one if x equals one or y equals one next row x times y minus x minus y plus two if x and y are both greater than one next row zero otherwise next row table ends

Means: A source-ordered calculation or table whose rows state: the interpretation of plus in structure K prime open parenthesis x comma y close parenthesis equals cases begin x plus y minus one if x and y are both greater than zero next row zero otherwise table ends next row the interpreted multiplication operation of structure K prime open parenthesis x comma y close parenthesis equals cases begin one if x equals one or y equals one next row x times y minus x minus y plus two if x and y are both greater than one next row zero otherwise next row table ends. This meaning is authored from the complete TR-027 source packet for expr-8feac84deb26f044.

1 occurrence in this chapter

Equation form expr-914ad04bc7458404

Q⊢∀x(x<n¯′→(x=0¯∨x=1¯∨…∨x=n¯))\Th{Q} \Proves \lforall[x][(x < \num{n}' \lif (\eq[x][\num{0}] \lor \eq[x][\num{1}] \lor \dots \lor \eq[x][\num{n}]))]

Read as: theory Q proves for every x, open parenthesis x is less than the numeral for n prime implies open parenthesis x equals the numeral for zero or x equals the numeral for one or and so on or x equals the numeral for n close parenthesis close parenthesis

Means: A derivability claim stating: theory Q proves for every x, open parenthesis x is less than the numeral for n prime implies open parenthesis x equals the numeral for zero or x equals the numeral for one or and so on or x equals the numeral for n close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-914ad04bc7458404.

1 occurrence in this chapter

Equation form expr-91596408b1c392b6

g:ℕ→|M|g\colon \Nat \to \Domain{M}

Read as: g from the natural numbers to the domain of structure M

Means: Arithmetic notation denoting: g from the natural numbers to the domain of structure M. This meaning is authored from the complete TR-027 source packet for expr-91596408b1c392b6.

3 occurrences in this chapter

Equation form expr-91910acfc0bcd121

[u]≠[u] \neq [v]

Read as: the block of u is not equal to the block of v

Means: An arithmetic relation or equation stating: the block of u is not equal to the block of v. This meaning is authored from the complete TR-027 source packet for expr-91910acfc0bcd121.

1 occurrence in this chapter

Equation form expr-92fcc8fed3e157e4

x≺ux \nsless u

Read as: x is less than u in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than u in the model. This meaning is authored from the complete TR-027 source packet for expr-92fcc8fed3e157e4.

2 occurrences in this chapter

Equation form expr-95a688d1cb8a0175

K⊨∀x(x=0∨∃yx=y′)\Sat{K}{\lforall[x][(\eq[x][\Obj 0] \lor \lexists[y][\eq[x][y']])]}

Read as: structure K satisfies for every x, open parenthesis x equals object language symbol zero or there exists y, x equals y prime close parenthesis

Means: A satisfaction claim stating: structure K satisfies for every x, open parenthesis x equals object language symbol zero or there exists y, x equals y prime close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-95a688d1cb8a0175.

1 occurrence in this chapter

Equation form expr-961b6dd3ede3cb8e

aaaa

Read as: a followed by a

Means: Arithmetic notation denoting: a followed by a. This meaning is authored from the complete TR-027 source packet for expr-961b6dd3ede3cb8e.

1 occurrence in this chapter

Equation form expr-963871c44bfd6245

M⊨n¯<m¯\Sat{M}{\num{n} < \num{m}}

Read as: structure M satisfies the numeral for n is less than the numeral for m

Means: A satisfaction claim stating: structure M satisfies the numeral for n is less than the numeral for m. This meaning is authored from the complete TR-027 source packet for expr-963871c44bfd6245.

1 occurrence in this chapter

Equation form expr-977f19219c49b665

0M\Assign{\Obj{0}}{M}

Read as: the interpretation of object language symbol zero in structure M

Means: Arithmetic notation denoting: the interpretation of object language symbol zero in structure M. This meaning is authored from the complete TR-027 source packet for expr-977f19219c49b665.

2 occurrences in this chapter

Equation form expr-9829f2e34e8f0b11

1¯\num{1}

Read as: the numeral for one

Means: Arithmetic notation denoting: the numeral for one. This meaning is authored from the complete TR-027 source packet for expr-9829f2e34e8f0b11.

2 occurrences in this chapter

Equation form expr-9834876dcfb05cb1

aaaaaa

Read as: a followed by a followed by a

Means: Arithmetic notation denoting: a followed by a followed by a. This meaning is authored from the complete TR-027 source packet for expr-9834876dcfb05cb1.

1 occurrence in this chapter

Equation form expr-9a0a248c7cf1bc9a

h(0)=h(0N)=0Mh(0) = h(\Assign{\Obj{0}}{N}) =\Assign{\Obj{0}}{M}

Read as: h open parenthesis zero close parenthesis equals h open parenthesis the interpretation of object language symbol zero in structure N close parenthesis equals the interpretation of object language symbol zero in structure M

Means: Arithmetic notation denoting: h open parenthesis zero close parenthesis equals h open parenthesis the interpretation of object language symbol zero in structure N close parenthesis equals the interpretation of object language symbol zero in structure M. This meaning is authored from the complete TR-027 source packet for expr-9a0a248c7cf1bc9a.

1 occurrence in this chapter

Equation form expr-9b7ba6bb774a5ef4

0′′\Obj{0}''

Read as: object language symbol zero prime prime

Means: Arithmetic notation denoting: object language symbol zero prime prime. This meaning is authored from the complete TR-027 source packet for expr-9b7ba6bb774a5ef4.

1 occurrence in this chapter

Equation form expr-9be604010dbe3b65

c≠k¯∈Γ0\eq/[c][\num{k}] \in \Gamma_0

Read as: the formula c is not equal to the numeral for k belongs to Gamma sub zero

Means: An arithmetic relation or equation stating: the formula c is not equal to the numeral for k belongs to Gamma sub zero. This meaning is authored from the complete TR-027 source packet for expr-9be604010dbe3b65.

1 occurrence in this chapter

Equation form expr-9d1357a55d31c1dc

⟦n¯⟧M≠⟦m¯⟧M\Value{\num{n}}{M} \neq \Value{\num{m}}{M}

Read as: the value of the numeral for n in structure M is not equal to the value of the numeral for m in structure M

Means: Arithmetic notation denoting: the value of the numeral for n in structure M is not equal to the value of the numeral for m in structure M. This meaning is authored from the complete TR-027 source packet for expr-9d1357a55d31c1dc.

1 occurrence in this chapter

Equation form expr-9d69f823e877748c

0Mi\Assign{\Obj{0}}{M_i}

Read as: the interpretation of object language symbol zero in structure M sub i

Means: Arithmetic notation denoting: the interpretation of object language symbol zero in structure M sub i. This meaning is authored from the complete TR-027 source packet for expr-9d69f823e877748c.

1 occurrence in this chapter

Equation form expr-9d7103ac3526e8f6

m∈ℕm \in \Nat

Read as: m is in the natural numbers

Means: An arithmetic relation or equation stating: m is in the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-9d7103ac3526e8f6.

1 occurrence in this chapter

Equation form expr-9d752b1bdf8ee617

|L′|=ℕ\Domain{L'} = \Nat

Read as: the domain of structure L prime equals the natural numbers

Means: Arithmetic notation denoting: the domain of structure L prime equals the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-9d752b1bdf8ee617.

1 occurrence in this chapter

Equation form expr-9e8238f66180fc0d

0K=0′K(x)=x+1if x∈ℕaif x=a+K(x,y)=x+yif x∈ℕ and y∈ℕaotherwise×K(x,y)=x⁢yif x∈ℕ and y∈ℕ0if x=0 or y=0aotherwise<K={⟨x,y⟩:x∈ℕ and y∈ℕ and x<y}∪{⟨x,a⟩:x∈|K|}\Assign{\Obj{0}}{K} & = 0\\ \Assign{\prime}{K}(x) & = \begin{cases} x+1 & \text{if $x\in \Nat$}\\ a & \text{if $x = a$} \end{cases}\\ \Assign{+}{K}(x, y) & = \begin{cases} x+y & \text{if $x$, $y \in\Nat$}\\ a & \text{otherwise} \end{cases}\\ \Assign{\times}{K}(x, y) & = \begin{cases} xy & \text{if $x$, $y \in\Nat$}\\ 0 & \text{if $x=0$ or $y=0$}\\ a & \text{otherwise}\\ \end{cases}\\ \Assign{<}{K} & = \Setabs{\tuple{x,y}}{x, y \in \Nat \text{ and } x<y} \cup \Setabs{\tuple{x,a}}{x \in \Domain{K}}

Read as: the interpretation of object language symbol zero in structure K equals zero next row the interpretation of successor symbol in structure K open parenthesis x close parenthesis equals cases begin x plus one if x is in the natural numbers next row a if x equals a table ends next row the interpretation of plus in structure K open parenthesis x comma y close parenthesis equals cases begin x plus y if x and y are in the natural numbers next row a otherwise table ends next row the interpreted multiplication operation of structure K open parenthesis x comma y close parenthesis equals cases begin x times y if x and y are in the natural numbers next row zero if x equals zero or y equals zero next row a otherwise next row table ends next row the interpreted less-than relation of structure K equals the set of the tuple x comma y such that x and y are in the natural numbers and x is less than y union the set of the tuple x comma a such that x is in the domain of structure K

Means: A source-ordered calculation or table whose rows state: the interpretation of object language symbol zero in structure K equals zero next row the interpretation of successor symbol in structure K open parenthesis x close parenthesis equals cases begin x plus one if x is in the natural numbers next row a if x equals a table ends next row the interpretation of plus in structure K open parenthesis x comma y close parenthesis equals cases begin x plus y if x and y are in the natural numbers next row a otherwise table ends next row the interpreted multiplication operation of structure K open parenthesis x comma y close parenthesis equals cases begin x times y if x and y are in the natural numbers next row zero if x equals zero or y equals zero next row a otherwise next row table ends next row the interpreted less-than relation of structure K equals the set of the tuple x comma y such that x and y are in the natural numbers and x is less than y union the set of the tuple x comma a such that x is in the domain of structure K. This meaning is authored from the complete TR-027 source packet for expr-9e8238f66180fc0d.

1 occurrence in this chapter

Equation form expr-9f3504db545ac1c7

Q3!Q_3

Read as: Q sub three

Means: Arithmetic notation denoting: Q sub three. This meaning is authored from the complete TR-027 source packet for expr-9f3504db545ac1c7.

1 occurrence in this chapter

Equation form expr-a05ec4216c67a667

predecessor of y^\nssucc y

Read as: predecessor of y

Means: A claim about the interpreted arithmetic operations or order stating: predecessor of y. This meaning is authored from the complete TR-027 source packet for expr-a05ec4216c67a667.

1 occurrence in this chapter

Equation form expr-a0d301e223047dbc

0N=0\Assign{\Obj{0}}{N} = 0

Read as: the interpretation of object language symbol zero in structure N equals zero

Means: Arithmetic notation denoting: the interpretation of object language symbol zero in structure N equals zero. This meaning is authored from the complete TR-027 source packet for expr-a0d301e223047dbc.

1 occurrence in this chapter

Equation form expr-a1d6db5e88eabe33

x≺nx \nsless n

Read as: x is less than n in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than n in the model. This meaning is authored from the complete TR-027 source packet for expr-a1d6db5e88eabe33.

1 occurrence in this chapter

Equation form expr-a1f6a667c564fd72

0M=∅′M(s)=s⌢a+M(n,m)=an+m×M(n,m)=an⁢m\Assign{\Obj{0}}{M} & = \emptyset\\ \Assign{\prime}{M}(s) & = s \concat a\\ \Assign{+}{M}(n, m) & = a^{n + m}\\ \Assign{\times}{M}(n, m) & = a^{nm}

Read as: the interpretation of object language symbol zero in structure M equals the empty set next row the interpretation of successor symbol in structure M open parenthesis s close parenthesis equals s concatenated with a next row the interpretation of plus in structure M open parenthesis n comma m close parenthesis equals a superscript n plus m next row the interpreted multiplication operation of structure M open parenthesis n comma m close parenthesis equals a superscript n times m

Means: A source-ordered calculation or table whose rows state: the interpretation of object language symbol zero in structure M equals the empty set next row the interpretation of successor symbol in structure M open parenthesis s close parenthesis equals s concatenated with a next row the interpretation of plus in structure M open parenthesis n comma m close parenthesis equals a superscript n plus m next row the interpreted multiplication operation of structure M open parenthesis n comma m close parenthesis equals a superscript n times m. This meaning is authored from the complete TR-027 source packet for expr-a1f6a667c564fd72.

1 occurrence in this chapter

Equation form expr-a1fce4363854ff88

yy

Read as: y

Means: Arithmetic notation denoting: y. This meaning is authored from the complete TR-027 source packet for expr-a1fce4363854ff88.

15 occurrences in this chapter

Equation form expr-a25d9c9866c45520

Γ0⊆Γ\Gamma_0 \subseteq \Gamma

Read as: Gamma sub zero is a subset of Gamma

Means: An arithmetic relation or equation stating: Gamma sub zero is a subset of Gamma. This meaning is authored from the complete TR-027 source packet for expr-a25d9c9866c45520.

1 occurrence in this chapter

Equation form expr-a318c24216defe20

++

Read as: plus

Means: Arithmetic notation denoting: plus. This meaning is authored from the complete TR-027 source packet for expr-a318c24216defe20.

5 occurrences in this chapter

Equation form expr-a4cd1fc4fcb08699

∀xx≠x′\lforall[x][\eq/[x][x']]

Read as: for every x, x is not equal to x prime

Means: A first-order arithmetic formula stating: for every x, x is not equal to x prime. This meaning is authored from the complete TR-027 source packet for expr-a4cd1fc4fcb08699.

1 occurrence in this chapter

Equation form expr-a512bb4cdc87ca19

∀x∀y∀z((x<y∧y<z)→x<z)\lforall[x][\lforall[y][\lforall[z][((x < y \land y < z) \lif x < z)]]]

Read as: for every x, for every y, for every z, open parenthesis open parenthesis x is less than y and y is less than z close parenthesis implies x is less than z close parenthesis

Means: A first-order arithmetic formula stating: for every x, for every y, for every z, open parenthesis open parenthesis x is less than y and y is less than z close parenthesis implies x is less than z close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-a512bb4cdc87ca19.

1 occurrence in this chapter

Equation form expr-a6bee7fe74bd9fbc

a⊕z=aa \nsplus z = a

Read as: a plus in the model z equals a

Means: A claim about the interpreted arithmetic operations or order stating: a plus in the model z equals a. This meaning is authored from the complete TR-027 source packet for expr-a6bee7fe74bd9fbc.

1 occurrence in this chapter

Equation form expr-a70f68efa165fda2

K\Struct{K}

Read as: structure K

Means: Arithmetic notation denoting: structure K. This meaning is authored from the complete TR-027 source packet for expr-a70f68efa165fda2.

13 occurrences in this chapter

Equation form expr-a87ea6068a057b68

∀x0≠x′\lforall[x][\eq/[\Obj 0][x']]

Read as: for every x, object language symbol zero is not equal to x prime

Means: A first-order arithmetic formula stating: for every x, object language symbol zero is not equal to x prime. This meaning is authored from the complete TR-027 source packet for expr-a87ea6068a057b68.

1 occurrence in this chapter

Equation form expr-a9d9783f1e8a9800

∀x(x<n¯′→(x=0¯∨x=1¯∨…∨x=n¯))\lforall[x][(x < \num{n}' \lif (\eq[x][\num{0}] \lor \eq[x][\num{1}] \lor \dots \lor \eq[x][\num{n}]))]

Read as: for every x, open parenthesis x is less than the numeral for n prime implies open parenthesis x equals the numeral for zero or x equals the numeral for one or and so on or x equals the numeral for n close parenthesis close parenthesis

Means: A first-order arithmetic formula stating: for every x, open parenthesis x is less than the numeral for n prime implies open parenthesis x equals the numeral for zero or x equals the numeral for one or and so on or x equals the numeral for n close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-a9d9783f1e8a9800.

1 occurrence in this chapter

Equation form expr-aaa481df509e0b59

K⊭∀x¬x<x\Sat/{K}{\lforall[x][\lnot x<x]}

Read as: structure K does not satisfy for every x, not x is less than x

Means: A satisfaction claim stating: structure K does not satisfy for every x, not x is less than x. This meaning is authored from the complete TR-027 source packet for expr-aaa481df509e0b59.

1 occurrence in this chapter

Equation form expr-aaa9402664f1a41f

hh

Read as: h

Means: Arithmetic notation denoting: h. This meaning is authored from the complete TR-027 source packet for expr-aaa9402664f1a41f.

2 occurrences in this chapter

Equation form expr-aaf312d2d7202d5e

y≺vy \nsless v

Read as: y is less than v in the model

Means: A claim about the interpreted arithmetic operations or order stating: y is less than v in the model. This meaning is authored from the complete TR-027 source packet for expr-aaf312d2d7202d5e.

2 occurrences in this chapter

Equation form expr-aafda2cc5bdbea48

|M|=ℕ\Domain{M} = \Nat

Read as: the domain of structure M equals the natural numbers

Means: Arithmetic notation denoting: the domain of structure M equals the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-aafda2cc5bdbea48.

1 occurrence in this chapter

Equation form expr-abada9758f3a5636

Nk⊨A\Sat{N_k}{!A}

Read as: structure N sub k satisfies formula A

Means: A satisfaction claim stating: structure N sub k satisfies formula A. This meaning is authored from the complete TR-027 source packet for expr-abada9758f3a5636.

1 occurrence in this chapter

Equation form expr-abb8457b17402949

M3⊨Q2\Sat{M_3}{!Q_2}

Read as: structure M sub three satisfies Q sub two

Means: A satisfaction claim stating: structure M sub three satisfies Q sub two. This meaning is authored from the complete TR-027 source packet for expr-abb8457b17402949.

1 occurrence in this chapter

Equation form expr-abbabd08b7256c52

c≠n¯\eq/[c][\num{n}]

Read as: c is not equal to the numeral for n

Means: An arithmetic relation or equation stating: c is not equal to the numeral for n. This meaning is authored from the complete TR-027 source packet for expr-abbabd08b7256c52.

1 occurrence in this chapter

Equation form expr-adedc2ab1c653ff0

n¯′\num{n}'

Read as: the numeral for n prime

Means: Arithmetic notation denoting: the numeral for n prime. This meaning is authored from the complete TR-027 source packet for expr-adedc2ab1c653ff0.

1 occurrence in this chapter

Equation form expr-af2142063ac551b1

z∈[y]z \in [y]

Read as: z is in the block of y

Means: An arithmetic relation or equation stating: z is in the block of y. This meaning is authored from the complete TR-027 source packet for expr-af2142063ac551b1.

1 occurrence in this chapter

Equation form expr-b0c5509775e74bbe

⟨g(n),g(m)⟩∈<M\tuple{g(n), g(m)} \in \Assign{<}{M}

Read as: the tuple g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis is in the interpreted less-than relation of structure M

Means: Arithmetic notation denoting: the tuple g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis is in the interpreted less-than relation of structure M. This meaning is authored from the complete TR-027 source packet for expr-b0c5509775e74bbe.

2 occurrences in this chapter

Equation form expr-b2854ea5ad90299c

⟦m¯⟧M=g(m)\Value{\num{m}}{M} = g(m)

Read as: the value of the numeral for m in structure M equals g open parenthesis m close parenthesis

Means: Arithmetic notation denoting: the value of the numeral for m in structure M equals g open parenthesis m close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-b2854ea5ad90299c.

1 occurrence in this chapter

Equation form expr-b2c5dd670eb86662

x⊕n∗≺vx \nsplus n^\nssucc \nsless v

Read as: x plus in the model n followed by model successor is less than v in the model

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model n followed by model successor is less than v in the model. This meaning is authored from the complete TR-027 source packet for expr-b2c5dd670eb86662.

1 occurrence in this chapter

Equation form expr-b2c77c7c4e824567

x≺ax \nsless a

Read as: x is less than a in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than a in the model. This meaning is authored from the complete TR-027 source packet for expr-b2c77c7c4e824567.

1 occurrence in this chapter

Equation form expr-b49424a3e015731b

=+M(⟦n¯⟧M,⟦m¯⟧M)= \Assign{+}{M}(\Value{\num{n}}{M},\Value{\num{m}}{M})

Read as: equals the interpretation of plus in structure M open parenthesis the value of the numeral for n in structure M comma the value of the numeral for m in structure M close parenthesis

Means: Arithmetic notation denoting: equals the interpretation of plus in structure M open parenthesis the value of the numeral for n in structure M comma the value of the numeral for m in structure M close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-b49424a3e015731b.

1 occurrence in this chapter

Equation form expr-b568c602ff399c22

∗\nssucc

Read as: model successor

Means: A claim about the interpreted arithmetic operations or order stating: model successor. This meaning is authored from the complete TR-027 source packet for expr-b568c602ff399c22.

9 occurrences in this chapter

Equation form expr-b620f7fd0a05aeb7

0¯\num{0}

Read as: the numeral for zero

Means: Arithmetic notation denoting: the numeral for zero. This meaning is authored from the complete TR-027 source packet for expr-b620f7fd0a05aeb7.

3 occurrences in this chapter

Equation form expr-b6c5ccae5ba847ef

N⊨ConPA\Sat{N}{\OCon[\Th{PA}]}

Read as: structure N satisfies the formal consistency statement for theory PA

Means: An arithmetized-syntax statement denoting: structure N satisfies the formal consistency statement for theory PA. This meaning is authored from the complete TR-027 source packet for expr-b6c5ccae5ba847ef.

1 occurrence in this chapter

Equation form expr-b6e5b1770fd606ae

M⊨TA\Sat{M}{\Th{TA}}

Read as: structure M satisfies theory TA

Means: A satisfaction claim stating: structure M satisfies theory TA. This meaning is authored from the complete TR-027 source packet for expr-b6e5b1770fd606ae.

1 occurrence in this chapter

Equation form expr-b7358701e22a7a6c

y=0y = 0

Read as: y equals zero

Means: An arithmetic relation or equation stating: y equals zero. This meaning is authored from the complete TR-027 source packet for expr-b7358701e22a7a6c.

1 occurrence in this chapter

Equation form expr-b7a6f691649904e4

+M\Assign{+}{M}

Read as: the interpretation of plus in structure M

Means: Arithmetic notation denoting: the interpretation of plus in structure M. This meaning is authored from the complete TR-027 source packet for expr-b7a6f691649904e4.

2 occurrences in this chapter

Equation form expr-b823156ebb35cc81

x≺vx \nsless v

Read as: x is less than v in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than v in the model. This meaning is authored from the complete TR-027 source packet for expr-b823156ebb35cc81.

1 occurrence in this chapter

Equation form expr-b9a7f1a2870ae925

⟨⟦n¯⟧M,⟦m¯⟧M⟩∈<M\tuple{\Value{\num{n}}{M}, \Value{\num{m}}{M}} \in \Assign{<}{M}

Read as: the tuple the value of the numeral for n in structure M comma the value of the numeral for m in structure M is in the interpreted less-than relation of structure M

Means: Arithmetic notation denoting: the tuple the value of the numeral for n in structure M comma the value of the numeral for m in structure M is in the interpreted less-than relation of structure M. This meaning is authored from the complete TR-027 source packet for expr-b9a7f1a2870ae925.

1 occurrence in this chapter

Equation form expr-ba5936fc01df4efc

v∈[y]v \in [y]

Read as: v is in the block of y

Means: An arithmetic relation or equation stating: v is in the block of y. This meaning is authored from the complete TR-027 source packet for expr-ba5936fc01df4efc.

3 occurrences in this chapter

Equation form expr-ba7f48f632bd9b6b

⟦n¯⟧M=g(n)\Value{\num{n}}{M} = g(n)

Read as: the value of the numeral for n in structure M equals g open parenthesis n close parenthesis

Means: Arithmetic notation denoting: the value of the numeral for n in structure M equals g open parenthesis n close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-ba7f48f632bd9b6b.

2 occurrences in this chapter

Equation form expr-bbeadda1f333e3cd

a⊕0≠0⊕aa \nsplus 0 \neq 0 \nsplus a

Read as: a plus in the model zero is not equal to zero plus in the model a

Means: A claim about the interpreted arithmetic operations or order stating: a plus in the model zero is not equal to zero plus in the model a. This meaning is authored from the complete TR-027 source packet for expr-bbeadda1f333e3cd.

1 occurrence in this chapter

Equation form expr-bca5ed895ceeecf9

PA⊢A\Th{PA} \Proves !A

Read as: theory PA proves formula A

Means: A derivability claim stating: theory PA proves formula A. This meaning is authored from the complete TR-027 source packet for expr-bca5ed895ceeecf9.

1 occurrence in this chapter

Equation form expr-bcf24cbd57d12979

y∉[x]y \notin [x]

Read as: y is not in the block of x

Means: An arithmetic relation or equation stating: y is not in the block of x. This meaning is authored from the complete TR-027 source packet for expr-bcf24cbd57d12979.

1 occurrence in this chapter

Equation form expr-bd8f49440da17235

a∗=ba^\nssucc = b

Read as: a followed by model successor equals b

Means: A claim about the interpreted arithmetic operations or order stating: a followed by model successor equals b. This meaning is authored from the complete TR-027 source packet for expr-bd8f49440da17235.

1 occurrence in this chapter

Equation form expr-be58d8ae65b26403

0\Obj{0}

Read as: object language symbol zero

Means: Arithmetic notation denoting: object language symbol zero. This meaning is authored from the complete TR-027 source packet for expr-be58d8ae65b26403.

4 occurrences in this chapter

Equation form expr-bec16f173b11c6e6

|M|={a}*\Domain{M} = \{a\}^*

Read as: the domain of structure M equals the Kleene star of the singleton alphabet containing a

Means: Arithmetic notation denoting: the domain of structure M equals the Kleene star of the singleton alphabet containing a. This meaning is authored from the complete TR-027 source packet for expr-bec16f173b11c6e6.

1 occurrence in this chapter

Equation form expr-bf751aee037ca7f2

M3⊭Q1\Sat/{M_3}{!Q_1}

Read as: structure M sub three does not satisfy Q sub one

Means: A satisfaction claim stating: structure M sub three does not satisfy Q sub one. This meaning is authored from the complete TR-027 source packet for expr-bf751aee037ca7f2.

1 occurrence in this chapter

Equation form expr-c06a5a25579ef1ae

L\Struct{L}

Read as: structure L

Means: Arithmetic notation denoting: structure L. This meaning is authored from the complete TR-027 source packet for expr-c06a5a25579ef1ae.

5 occurrences in this chapter

Equation form expr-c083db45c1828d6e

K⊨∀x(x+0)=x\Sat{K}{\lforall[x][\eq[(x + \Obj 0)][x]]}

Read as: structure K satisfies for every x, open parenthesis x plus object language symbol zero close parenthesis equals x

Means: A satisfaction claim stating: structure K satisfies for every x, open parenthesis x plus object language symbol zero close parenthesis equals x. This meaning is authored from the complete TR-027 source packet for expr-c083db45c1828d6e.

1 occurrence in this chapter

Equation form expr-c08c20bd9d545010

Mc⊨TA\Sat{M^c}{\Th{TA}}

Read as: structure M superscript c satisfies theory TA

Means: A satisfaction claim stating: structure M superscript c satisfies theory TA. This meaning is authored from the complete TR-027 source packet for expr-c08c20bd9d545010.

1 occurrence in this chapter

Equation form expr-c17b77e97a7ee2ac

∀xx′≠0\lforall[x][\eq/[x'][\Obj{0}]]

Read as: for every x, x prime is not equal to object language symbol zero

Means: A first-order arithmetic formula stating: for every x, x prime is not equal to object language symbol zero. This meaning is authored from the complete TR-027 source packet for expr-c17b77e97a7ee2ac.

1 occurrence in this chapter

Equation form expr-c1881d6c6b847110

≺y\nsless y

Read as: less than y in the model

Means: A claim about the interpreted arithmetic operations or order stating: less than y in the model. This meaning is authored from the complete TR-027 source packet for expr-c1881d6c6b847110.

2 occurrences in this chapter

Equation form expr-c377e14aefa77170

M2⊨Q3\Sat{M_2}{!Q_3}

Read as: structure M sub two satisfies Q sub three

Means: A satisfaction claim stating: structure M sub two satisfies Q sub three. This meaning is authored from the complete TR-027 source packet for expr-c377e14aefa77170.

1 occurrence in this chapter

Equation form expr-c41149824f043317

L′\Struct{L'}

Read as: structure L prime

Means: Arithmetic notation denoting: structure L prime. This meaning is authored from the complete TR-027 source packet for expr-c41149824f043317.

1 occurrence in this chapter

Equation form expr-c5140ca684997fc0

K⊨∃xx<x\Sat{K}{\lexists[x][x < x]}

Read as: structure K satisfies there exists x, x is less than x

Means: A satisfaction claim stating: structure K satisfies there exists x, x is less than x. This meaning is authored from the complete TR-027 source packet for expr-c5140ca684997fc0.

1 occurrence in this chapter

Equation form expr-c63c8e0f289a448b

n>0n > 0

Read as: n is greater than zero

Means: An arithmetic relation or equation stating: n is greater than zero. This meaning is authored from the complete TR-027 source packet for expr-c63c8e0f289a448b.

1 occurrence in this chapter

Equation form expr-c6569e52b7ade4dd

<N\Assign{<}{N}

Read as: the interpreted less-than relation of structure N

Means: Arithmetic notation denoting: the interpreted less-than relation of structure N. This meaning is authored from the complete TR-027 source packet for expr-c6569e52b7ade4dd.

1 occurrence in this chapter

Equation form expr-c738cff2cb5da3e7

n≮mn \not< m

Read as: n is not less than m

Means: An arithmetic relation or equation stating: n is not less than m. This meaning is authored from the complete TR-027 source packet for expr-c738cff2cb5da3e7.

1 occurrence in this chapter

Equation form expr-c75669186bb89f88

Mc\Struct{M^c}

Read as: structure M superscript c

Means: Arithmetic notation denoting: structure M superscript c. This meaning is authored from the complete TR-027 source packet for expr-c75669186bb89f88.

2 occurrences in this chapter

Equation form expr-c89068b466452277

y≺y∗y \nsless y^\nssucc

Read as: y is less than the model successor of y

Means: A claim about the interpreted arithmetic operations or order stating: y is less than the model successor of y. This meaning is authored from the complete TR-027 source packet for expr-c89068b466452277.

2 occurrences in this chapter

Equation form expr-c9ad2bbe9b9f9327

x⊕n=yx \nsplus n = y

Read as: x plus in the model n equals y

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model n equals y. This meaning is authored from the complete TR-027 source packet for expr-c9ad2bbe9b9f9327.

1 occurrence in this chapter

Equation form expr-c9c35d45f3c4a346

⟨x,y⟩∈<K′\tuple{x, y} \in \Assign{<}{K'}

Read as: the tuple x comma y is in the interpreted less-than relation of structure K prime

Means: Arithmetic notation denoting: the tuple x comma y is in the interpreted less-than relation of structure K prime. This meaning is authored from the complete TR-027 source packet for expr-c9c35d45f3c4a346.

1 occurrence in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: a

Means: Arithmetic notation denoting: a. This meaning is authored from the complete TR-027 source packet for expr-ca978112ca1bbdca.

4 occurrences in this chapter

Equation form expr-cc04574ab4892531

PA⊢¬PrfPA(n¯,⌜⊥⌝)\Th{PA} \Proves \lnot \OPrf[\Th{PA}](\num{n}, \gn{\lfalse})

Read as: theory PA proves not the formal proof predicate for theory PA open parenthesis the numeral for n comma the Goedel number of falsum close parenthesis

Means: An arithmetized-syntax statement denoting: theory PA proves not the formal proof predicate for theory PA open parenthesis the numeral for n comma the Goedel number of falsum close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-cc04574ab4892531.

1 occurrence in this chapter

Equation form expr-cc1dbf63b81157ee

PA⊢∀x∀y(x<y→(x′<y∨x′=y))\Th{PA} \Proves \lforall[x][\lforall[y][(x < y \lif (x' < y \lor x' = y))]]

Read as: theory PA proves for every x, for every y, open parenthesis x is less than y implies open parenthesis x prime is less than y or x prime equals y close parenthesis close parenthesis

Means: A derivability claim stating: theory PA proves for every x, for every y, open parenthesis x is less than y implies open parenthesis x prime is less than y or x prime equals y close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-cc1dbf63b81157ee.

1 occurrence in this chapter

Equation form expr-ccd0d23b56b6a247

0K=0′K(x)=x+1if x∈ℕaif x=a+K(x,y)=x+yif x∈ℕ and y∈ℕaotherwise×K(x,y)=x⁢yif x∈ℕ and y∈ℕ0if x=0 or y=0aotherwise<K={⟨x,y⟩:x∈ℕ and y∈ℕ and x<y}∪{⟨x,a⟩:x∈|K|}\Assign{\Obj{0}}{K} & = 0\\ \Assign{\prime}{K}(x) & = \begin{cases} x+1 & \text{if $x\in \Nat$}\\ a & \text{if $x = a$} \end{cases}\\ \Assign{+}{K}(x, y) & = \begin{cases} x+y & \text{if $x$, $y \in\Nat$}\\ a & \text{otherwise} \end{cases}\\ \Assign{\times}{K}(x, y) & = \begin{cases} xy & \text{if $x$, $y \in\Nat$}\\ 0 & \text{if $x = 0$ or $y = 0$}\\ a & \text{otherwise}\\ \end{cases}\\ \Assign{<}{K} & = \Setabs{\tuple{x,y}}{x, y \in \Nat \text{ and } x<y} \cup \Setabs{\tuple{x,a}}{x \in \Domain{K}}

Read as: the interpretation of object language symbol zero in structure K equals zero next row the interpretation of successor symbol in structure K open parenthesis x close parenthesis equals cases begin x plus one if x is in the natural numbers next row a if x equals a table ends next row the interpretation of plus in structure K open parenthesis x comma y close parenthesis equals cases begin x plus y if x and y are in the natural numbers next row a otherwise table ends next row the interpreted multiplication operation of structure K open parenthesis x comma y close parenthesis equals cases begin x times y if x and y are in the natural numbers next row zero if x equals zero or y equals zero next row a otherwise next row table ends next row the interpreted less-than relation of structure K equals the set of the tuple x comma y such that x and y are in the natural numbers and x is less than y union the set of the tuple x comma a such that x is in the domain of structure K

Means: A source-ordered calculation or table whose rows state: the interpretation of object language symbol zero in structure K equals zero next row the interpretation of successor symbol in structure K open parenthesis x close parenthesis equals cases begin x plus one if x is in the natural numbers next row a if x equals a table ends next row the interpretation of plus in structure K open parenthesis x comma y close parenthesis equals cases begin x plus y if x and y are in the natural numbers next row a otherwise table ends next row the interpreted multiplication operation of structure K open parenthesis x comma y close parenthesis equals cases begin x times y if x and y are in the natural numbers next row zero if x equals zero or y equals zero next row a otherwise next row table ends next row the interpreted less-than relation of structure K equals the set of the tuple x comma y such that x and y are in the natural numbers and x is less than y union the set of the tuple x comma a such that x is in the domain of structure K. This meaning is authored from the complete TR-027 source packet for expr-ccd0d23b56b6a247.

1 occurrence in this chapter

Equation form expr-cd0aa9856147b6c5

gg

Read as: g

Means: Arithmetic notation denoting: g. This meaning is authored from the complete TR-027 source packet for expr-cd0aa9856147b6c5.

13 occurrences in this chapter

Equation form expr-ce9368c8dae3b133

x≠nx \neq n

Read as: x is not equal to n

Means: An arithmetic relation or equation stating: x is not equal to n. This meaning is authored from the complete TR-027 source packet for expr-ce9368c8dae3b133.

1 occurrence in this chapter

Equation form expr-cf441df67cb93045

M1⊨Q1\Sat{M_1}{!Q_1}

Read as: structure M sub one satisfies Q sub one

Means: A satisfaction claim stating: structure M sub one satisfies Q sub one. This meaning is authored from the complete TR-027 source packet for expr-cf441df67cb93045.

1 occurrence in this chapter

Equation form expr-cf7cfe5cf35ef9ef

x=(y⊕y)∗x = (y \nsplus y)^\nssucc

Read as: x equals open parenthesis y plus in the model y close parenthesis followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: x equals open parenthesis y plus in the model y close parenthesis followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-cf7cfe5cf35ef9ef.

1 occurrence in this chapter

Equation form expr-cfd32f3e0dec89aa

xx∗nn+1aabbx⊕ymabnn+mbaaababbba\begin{array}{c|c} x & x^\nssucc \\ \hline n & n+1 \\ a & a\\ b & b \end{array} \qquad \begin{array}{c|ccc} x \nsplus y & m & a & b\\ \hline n & n+m & b & a\\ a & a & b & a\\ b & b & b & a \end{array}

Read as: table begins x x followed by model successor next row table rule n n plus one next row a a next row b b table ends table begins x plus in the model y m a b next row table rule n n plus m b a next row a a b a next row b b b a table ends

Means: A source-ordered calculation or table whose rows state: table begins x x followed by model successor next row table rule n n plus one next row a a next row b b table ends table begins x plus in the model y m a b next row table rule n n plus m b a next row a a b a next row b b b a table ends. This meaning is authored from the complete TR-027 source packet for expr-cfd32f3e0dec89aa.

1 occurrence in this chapter

Equation form expr-d03543adcda36df1

Γ0\Gamma_0

Read as: Gamma sub zero

Means: Arithmetic notation denoting: Gamma sub zero. This meaning is authored from the complete TR-027 source packet for expr-d03543adcda36df1.

1 occurrence in this chapter

Equation form expr-d0590eaf020b9397

[x]≠[x] \neq [y]

Read as: the block of x is not equal to the block of y

Means: An arithmetic relation or equation stating: the block of x is not equal to the block of y. This meaning is authored from the complete TR-027 source packet for expr-d0590eaf020b9397.

3 occurrences in this chapter

Equation form expr-d0bca111f8628137

[0][0]

Read as: the block of zero

Means: Arithmetic notation denoting: the block of zero. This meaning is authored from the complete TR-027 source packet for expr-d0bca111f8628137.

1 occurrence in this chapter

Equation form expr-d112c3f30fc91b94

′M\Assign{\prime}{M}

Read as: the interpretation of successor symbol in structure M

Means: Arithmetic notation denoting: the interpretation of successor symbol in structure M. This meaning is authored from the complete TR-027 source packet for expr-d112c3f30fc91b94.

4 occurrences in this chapter

Equation form expr-d1730165aac18061

g(′N(n))=g(n+1)g(\Assign{\prime}{N}(n)) = g(n+1)

Read as: g open parenthesis the interpretation of successor symbol in structure N open parenthesis n close parenthesis close parenthesis equals g open parenthesis n plus one close parenthesis

Means: Arithmetic notation denoting: g open parenthesis the interpretation of successor symbol in structure N open parenthesis n close parenthesis close parenthesis equals g open parenthesis n plus one close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-d1730165aac18061.

1 occurrence in this chapter

Equation form expr-d2385b32131f4418

g(×N(n,m))=×M(g(n),g(m))g(\Assign{\times}{N}(n, m)) = \Assign{\times}{M}(g(n), g(m))

Read as: g open parenthesis the interpreted multiplication operation of structure N open parenthesis n comma m close parenthesis close parenthesis equals the interpreted multiplication operation of structure M open parenthesis g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis close parenthesis

Means: Arithmetic notation denoting: g open parenthesis the interpreted multiplication operation of structure N open parenthesis n comma m close parenthesis close parenthesis equals the interpreted multiplication operation of structure M open parenthesis g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-d2385b32131f4418.

1 occurrence in this chapter

Equation form expr-d26ef72670d8de2a

M⊨n¯≠m¯\Sat{M}{\eq/[\num{n}][\num{m}]}

Read as: structure M satisfies the numeral for n is not equal to the numeral for m

Means: A satisfaction claim stating: structure M satisfies the numeral for n is not equal to the numeral for m. This meaning is authored from the complete TR-027 source packet for expr-d26ef72670d8de2a.

1 occurrence in this chapter

Equation form expr-d27e20ea5ad139aa

∀x(x=0∨∃yy′=x)\lforall[x][(\eq[x][\Obj 0] \lor \lexists[y][\eq[y'][x]])]

Read as: for every x, open parenthesis x equals object language symbol zero or there exists y, y prime equals x close parenthesis

Means: A first-order arithmetic formula stating: for every x, open parenthesis x equals object language symbol zero or there exists y, y prime equals x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-d27e20ea5ad139aa.

1 occurrence in this chapter

Equation form expr-d2e40107b4474ff8

y≺y⊕yy \nsless y \nsplus y

Read as: y is less than the model sum of y and y

Means: A claim about the interpreted arithmetic operations or order stating: y is less than the model sum of y and y. This meaning is authored from the complete TR-027 source packet for expr-d2e40107b4474ff8.

1 occurrence in this chapter

Equation form expr-d33d5093ec487a6e

Q4!Q_4

Read as: Q sub four

Means: Arithmetic notation denoting: Q sub four. This meaning is authored from the complete TR-027 source packet for expr-d33d5093ec487a6e.

1 occurrence in this chapter

Equation form expr-d438a18a7da07bf0

x<yx < y

Read as: x is less than y

Means: An arithmetic relation or equation stating: x is less than y. This meaning is authored from the complete TR-027 source packet for expr-d438a18a7da07bf0.

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: Arithmetic notation denoting: two. This meaning is authored from the complete TR-027 source packet for expr-d4735e3a265e16ee.

1 occurrence in this chapter

Equation form expr-d71d4e9daa035bf4

Q⊢n¯<m¯\Th{Q} \Proves \num{n} < \num{m}

Read as: theory Q proves the numeral for n is less than the numeral for m

Means: A derivability claim stating: theory Q proves the numeral for n is less than the numeral for m. This meaning is authored from the complete TR-027 source packet for expr-d71d4e9daa035bf4.

1 occurrence in this chapter

Equation form expr-d79de11401d04ba7

′Mi\Assign{\prime}{M_i}

Read as: the interpretation of successor symbol in structure M sub i

Means: Arithmetic notation denoting: the interpretation of successor symbol in structure M sub i. This meaning is authored from the complete TR-027 source packet for expr-d79de11401d04ba7.

1 occurrence in this chapter

Equation form expr-d7f383f004b491a2

L⊭∀x∀y(x+y)=(y+x)\Sat/{L}{\lforall[x][\lforall[y][\eq[(x+y)][(y+x)]]]}

Read as: structure L does not satisfy for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis

Means: A satisfaction claim stating: structure L does not satisfy for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-d7f383f004b491a2.

1 occurrence in this chapter

Equation form expr-d87bd6818219a861

≺\nsless

Read as: the model order relation

Means: A claim about the interpreted arithmetic operations or order stating: the model order relation. This meaning is authored from the complete TR-027 source packet for expr-d87bd6818219a861.

12 occurrences in this chapter

Equation form expr-d89ad7af62404a57

PA\Th{PA}

Read as: theory PA

Means: Notation naming: theory PA. This meaning is authored from the complete TR-027 source packet for expr-d89ad7af62404a57.

20 occurrences in this chapter

Equation form expr-d91c636a7ce3b46c

v≺xv \nsless x

Read as: v is less than x in the model

Means: A claim about the interpreted arithmetic operations or order stating: v is less than x in the model. This meaning is authored from the complete TR-027 source packet for expr-d91c636a7ce3b46c.

1 occurrence in this chapter

Equation form expr-d91cdf33b3f39f3e

n≺mn \nsless m

Read as: n is less than m in the model

Means: A claim about the interpreted arithmetic operations or order stating: n is less than m in the model. This meaning is authored from the complete TR-027 source packet for expr-d91cdf33b3f39f3e.

1 occurrence in this chapter

Equation form expr-da8100dd647be368

M2⊭Q2\Sat/{M_2}{!Q_2}

Read as: structure M sub two does not satisfy Q sub two

Means: A satisfaction claim stating: structure M sub two does not satisfy Q sub two. This meaning is authored from the complete TR-027 source packet for expr-da8100dd647be368.

1 occurrence in this chapter

Equation form expr-dabd3aff769f07eb

<<

Read as: the less-than symbol

Means: An arithmetic relation or equation stating: the less-than symbol. This meaning is authored from the complete TR-027 source packet for expr-dabd3aff769f07eb.

5 occurrences in this chapter

Equation form expr-dbe1b003f68155cb

x∈M∖{z}x \in M \setminus \{z\}

Read as: x is in M set minus z

Means: An arithmetic relation or equation stating: x is in M set minus z. This meaning is authored from the complete TR-027 source packet for expr-dbe1b003f68155cb.

1 occurrence in this chapter

Equation form expr-dc048264506c2d20

∃xPrfPA(x,⌜⊥⌝)\lexists[x][\OPrf[\Th{PA}](x, \gn{\lfalse})]

Read as: there exists x, the formal proof predicate for theory PA open parenthesis x comma the Goedel number of falsum close parenthesis

Means: An arithmetized-syntax statement denoting: there exists x, the formal proof predicate for theory PA open parenthesis x comma the Goedel number of falsum close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-dc048264506c2d20.

1 occurrence in this chapter

Equation form expr-dd7d690b005e955c

M⊨A\Sat{M}{!A}

Read as: structure M satisfies formula A

Means: A satisfaction claim stating: structure M satisfies formula A. This meaning is authored from the complete TR-027 source packet for expr-dd7d690b005e955c.

2 occurrences in this chapter

Equation form expr-de9daef818c950d3

x≺yx \nsless y

Read as: x is less than y in the model

Means: A claim about the interpreted arithmetic operations or order stating: x is less than y in the model. This meaning is authored from the complete TR-027 source packet for expr-de9daef818c950d3.

9 occurrences in this chapter

Equation form expr-df1bf796764e65b7

TA⊆Γ\Th{TA} \subseteq \Gamma

Read as: theory TA is a subset of Gamma

Means: Notation naming: theory TA is a subset of Gamma. This meaning is authored from the complete TR-027 source packet for expr-df1bf796764e65b7.

1 occurrence in this chapter

Equation form expr-e09761ce764b32fc

⟦c⟧Nk=k+1\Value{c}{N_k} = k+1

Read as: the value of c in structure N sub k equals k plus one

Means: Arithmetic notation denoting: the value of c in structure N sub k equals k plus one. This meaning is authored from the complete TR-027 source packet for expr-e09761ce764b32fc.

1 occurrence in this chapter

Equation form expr-e1692d3a86e0857f

u=x⊕n∗u = x\nsplus n^\nssucc

Read as: u equals x plus in the model n followed by model successor

Means: A claim about the interpreted arithmetic operations or order stating: u equals x plus in the model n followed by model successor. This meaning is authored from the complete TR-027 source packet for expr-e1692d3a86e0857f.

1 occurrence in this chapter

Equation form expr-e290e5a48eb0ada3

∀x∀y(x+y)=(y+x)\lforall[x][\lforall[y][\eq[(x + y)][(y+x)]]]

Read as: for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis

Means: A first-order arithmetic formula stating: for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-e290e5a48eb0ada3.

1 occurrence in this chapter

Equation form expr-e2b0ae2042a4f135

∀x∀y(x′=y′→x=y)row label Q1∀x0≠x′row label Q2∀x(x=0∨∃yx=y′)row label Q3& \lforall[x][\lforall[y][(\eq[x'][y'] \lif \eq[x][y])]] \tag{$!Q_1$}\\ & \lforall[x][\eq/[\Obj 0][x']] \tag{$!Q_2$}\\ & \lforall[x][(\eq[x][\Obj 0] \lor \lexists[y][\eq[x][y']])] \tag{$!Q_3$}

Read as: for every x, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis row labeled Q sub one next row for every x, object language symbol zero is not equal to x prime row labeled Q sub two next row for every x, open parenthesis x equals object language symbol zero or there exists y, x equals y prime close parenthesis row labeled Q sub three

Means: A source-ordered calculation or table whose rows state: for every x, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis row labeled Q sub one next row for every x, object language symbol zero is not equal to x prime row labeled Q sub two next row for every x, open parenthesis x equals object language symbol zero or there exists y, x equals y prime close parenthesis row labeled Q sub three. This meaning is authored from the complete TR-027 source packet for expr-e2b0ae2042a4f135.

1 occurrence in this chapter

Equation form expr-e31bc64d3cd9db17

x≺x∗x \nsless x^\nssucc

Read as: x is less than the model successor of x

Means: A claim about the interpreted arithmetic operations or order stating: x is less than the model successor of x. This meaning is authored from the complete TR-027 source packet for expr-e31bc64d3cd9db17.

1 occurrence in this chapter

Equation form expr-e3b22a412737ce02

g(0)=h(0)g(0) = h(0)

Read as: g open parenthesis zero close parenthesis equals h open parenthesis zero close parenthesis

Means: An arithmetic relation or equation stating: g open parenthesis zero close parenthesis equals h open parenthesis zero close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-e3b22a412737ce02.

1 occurrence in this chapter

Equation form expr-e3d3dea30e3a6e26

y∗…∗=xy^{\nssucc\dots\nssucc} = x

Read as: y followed by model successor and so on model successor equals x

Means: A claim about the interpreted arithmetic operations or order stating: y followed by model successor and so on model successor equals x. This meaning is authored from the complete TR-027 source packet for expr-e3d3dea30e3a6e26.

2 occurrences in this chapter

Equation form expr-e3ec9bf763c3dd61

∀x∀y((x<y∨y<x)∨x=y))\lforall[x][\lforall[y][((x < y \lor y < x) \lor \eq[x][y]))]]

Read as: for every x, for every y, open parenthesis open parenthesis x is less than y or y is less than x close parenthesis or x equals y close parenthesis close parenthesis

Means: A first-order arithmetic formula stating: for every x, for every y, open parenthesis open parenthesis x is less than y or y is less than x close parenthesis or x equals y close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-e3ec9bf763c3dd61.

1 occurrence in this chapter

Equation form expr-e441d80407d5fd85

Γ=TA∪{c≠0¯,c≠1¯,c≠2¯,…}\Gamma = \Th{TA} \cup \{\eq/[c][\num{0}], \eq/[c][\num{1}], \eq/[c][\num{2}], \dots\}

Read as: Gamma equals theory TA union c is not equal to the numeral for zero comma c is not equal to the numeral for one comma c is not equal to the numeral for two comma and so on

Means: Notation naming: Gamma equals theory TA union c is not equal to the numeral for zero comma c is not equal to the numeral for one comma c is not equal to the numeral for two comma and so on. This meaning is authored from the complete TR-027 source packet for expr-e441d80407d5fd85.

1 occurrence in this chapter

Equation form expr-e45a7db4b0dbd77f

y⊕n=xy \nsplus n = x

Read as: y plus in the model n equals x

Means: A claim about the interpreted arithmetic operations or order stating: y plus in the model n equals x. This meaning is authored from the complete TR-027 source packet for expr-e45a7db4b0dbd77f.

2 occurrences in this chapter

Equation form expr-e64eeeddb7abd26a

g(+N(n,m))=+M(g(n),g(m))g(\Assign{+}{N}(n, m)) = \Assign{+}{M}(g(n), g(m))

Read as: g open parenthesis the interpretation of plus in structure N open parenthesis n comma m close parenthesis close parenthesis equals the interpretation of plus in structure M open parenthesis g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis close parenthesis

Means: Arithmetic notation denoting: g open parenthesis the interpretation of plus in structure N open parenthesis n comma m close parenthesis close parenthesis equals the interpretation of plus in structure M open parenthesis g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-e64eeeddb7abd26a.

1 occurrence in this chapter

Equation form expr-e748c72d469dd4ac

K⊨∀x∀y(x′=y′→x=y)\Sat{K}{\lforall[x][\lforall[y][(\eq[x'][y'] \lif \eq[x][y])]]}

Read as: structure K satisfies for every x, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis

Means: A satisfaction claim stating: structure K satisfies for every x, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-e748c72d469dd4ac.

1 occurrence in this chapter

Equation form expr-e8ab09ef10f0f9d8

x=y⊕yx = y \nsplus y

Read as: x equals y plus in the model y

Means: A claim about the interpreted arithmetic operations or order stating: x equals y plus in the model y. This meaning is authored from the complete TR-027 source packet for expr-e8ab09ef10f0f9d8.

1 occurrence in this chapter

Equation form expr-e8c1c380abe0ebe9

n<mn < m

Read as: n is less than m

Means: An arithmetic relation or equation stating: n is less than m. This meaning is authored from the complete TR-027 source packet for expr-e8c1c380abe0ebe9.

2 occurrences in this chapter

Equation form expr-ea57f1b2ecb2ac97

′N\Assign{\prime}{N}

Read as: the interpretation of successor symbol in structure N

Means: Arithmetic notation denoting: the interpretation of successor symbol in structure N. This meaning is authored from the complete TR-027 source packet for expr-ea57f1b2ecb2ac97.

3 occurrences in this chapter

Equation form expr-eaf752fb46657da6

a⊕0=aa\nsplus 0 = a

Read as: a plus in the model zero equals a

Means: A claim about the interpreted arithmetic operations or order stating: a plus in the model zero equals a. This meaning is authored from the complete TR-027 source packet for expr-eaf752fb46657da6.

1 occurrence in this chapter

Equation form expr-edf17661d9f030f8

v⊕m∗=yv \nsplus m^\nssucc = y

Read as: v plus in the model m followed by model successor equals y

Means: A claim about the interpreted arithmetic operations or order stating: v plus in the model m followed by model successor equals y. This meaning is authored from the complete TR-027 source packet for expr-edf17661d9f030f8.

1 occurrence in this chapter

Equation form expr-ee0aa86822cf4ecd

a≺aa \nsless a

Read as: a is less than a in the model

Means: A claim about the interpreted arithmetic operations or order stating: a is less than a in the model. This meaning is authored from the complete TR-027 source packet for expr-ee0aa86822cf4ecd.

1 occurrence in this chapter

Equation form expr-ee4ebdda4c0c38a5

∀x(x×0)=0row label Q6∀x∀y(x×y′)=((x×y)+x)row label Q7∀x∀y(x<y↔∃z(z′+x)=y)row label Q8& \lforall[x][\eq[(x \times \Obj 0)][\Obj 0]] \tag{$!Q_6$}\\ & \lforall[x][\lforall[y][\eq[(x \times y')][((x \times y) + x)]]] \tag{$!Q_7$}\\ & \lforall[x][\lforall[y][(x < y \liff \lexists[z][\eq[(z' + x)][y]])]] \tag{$!Q_8$}

Read as: for every x, open parenthesis x times object language symbol zero close parenthesis equals object language symbol zero row labeled Q sub six next row for every x, for every y, open parenthesis x times y prime close parenthesis equals open parenthesis open parenthesis x times y close parenthesis plus x close parenthesis row labeled Q sub seven next row for every x, for every y, open parenthesis x is less than y if and only if there exists z, open parenthesis z prime plus x close parenthesis equals y close parenthesis row labeled Q sub eight

Means: A source-ordered calculation or table whose rows state: for every x, open parenthesis x times object language symbol zero close parenthesis equals object language symbol zero row labeled Q sub six next row for every x, for every y, open parenthesis x times y prime close parenthesis equals open parenthesis open parenthesis x times y close parenthesis plus x close parenthesis row labeled Q sub seven next row for every x, for every y, open parenthesis x is less than y if and only if there exists z, open parenthesis z prime plus x close parenthesis equals y close parenthesis row labeled Q sub eight. This meaning is authored from the complete TR-027 source packet for expr-ee4ebdda4c0c38a5.

1 occurrence in this chapter

Equation form expr-ee677c8abaa4c9ad

x∈[y]x \in [y]

Read as: x is in the block of y

Means: An arithmetic relation or equation stating: x is in the block of y. This meaning is authored from the complete TR-027 source packet for expr-ee677c8abaa4c9ad.

1 occurrence in this chapter

Equation form expr-eee1c230c6f27590

TA\Th{TA}

Read as: theory TA

Means: Notation naming: theory TA. This meaning is authored from the complete TR-027 source packet for expr-eee1c230c6f27590.

12 occurrences in this chapter

Equation form expr-f17b5cafdf6c80da

x⊕yx \nsplus y

Read as: x plus in the model y

Means: A claim about the interpreted arithmetic operations or order stating: x plus in the model y. This meaning is authored from the complete TR-027 source packet for expr-f17b5cafdf6c80da.

2 occurrences in this chapter

Equation form expr-f24501cd355c999c

TA={A:N⊨A}\Th{TA} = \Setabs{!A}{\Sat{N}{!A}}

Read as: theory TA equals the set of formula A such that structure N satisfies formula A

Means: Notation naming: theory TA equals the set of formula A such that structure N satisfies formula A. This meaning is authored from the complete TR-027 source packet for expr-f24501cd355c999c.

1 occurrence in this chapter

Equation form expr-f37e06896eecca0a

M⊨Q\Sat{M}{\Th{Q}}

Read as: structure M satisfies theory Q

Means: A satisfaction claim stating: structure M satisfies theory Q. This meaning is authored from the complete TR-027 source packet for expr-f37e06896eecca0a.

2 occurrences in this chapter

Equation form expr-f396f29509427e56

LA\Lang{L_A}

Read as: language L sub A

Means: Arithmetic notation denoting: language L sub A. This meaning is authored from the complete TR-027 source packet for expr-f396f29509427e56.

16 occurrences in this chapter

Equation form expr-f39dfa197e69c9c0

¬∃xPrfPA(x,⌜⊥⌝)\lnot \lexists[x][\OPrf[\Th{PA}](x, \gn{\lfalse})]

Read as: not there exists x, the formal proof predicate for theory PA open parenthesis x comma the Goedel number of falsum close parenthesis

Means: An arithmetized-syntax statement denoting: not there exists x, the formal proof predicate for theory PA open parenthesis x comma the Goedel number of falsum close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-f39dfa197e69c9c0.

1 occurrence in this chapter

Equation form expr-f48cc60070f00c24

[z][z]

Read as: the block of z

Means: Arithmetic notation denoting: the block of z. This meaning is authored from the complete TR-027 source packet for expr-f48cc60070f00c24.

1 occurrence in this chapter

Equation form expr-f564121517eb7a6a

⟨g(n),g(m)⟩∉<M\tuple{g(n), g(m)} \notin \Assign{<}{M}

Read as: the tuple g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis is not in the interpreted less-than relation of structure M

Means: Arithmetic notation denoting: the tuple g open parenthesis n close parenthesis comma g open parenthesis m close parenthesis is not in the interpreted less-than relation of structure M. This meaning is authored from the complete TR-027 source packet for expr-f564121517eb7a6a.

1 occurrence in this chapter

Equation form expr-f658d14a295d231e

Nk⊨Γ0\Sat{N_k}{\Gamma_0}

Read as: structure N sub k satisfies Gamma sub zero

Means: A satisfaction claim stating: structure N sub k satisfies Gamma sub zero. This meaning is authored from the complete TR-027 source packet for expr-f658d14a295d231e.

1 occurrence in this chapter

Equation form expr-f722ebf867640c5c

⟦n+m¯⟧M=⟦n¯+m¯⟧M\Value{\num{n+m}}{M} = \Value{\num{n}+\num{m}}{M}

Read as: the value of the numeral for n plus m in structure M equals the value of the numeral for n plus the numeral for m in structure M

Means: Arithmetic notation denoting: the value of the numeral for n plus m in structure M equals the value of the numeral for n plus the numeral for m in structure M. This meaning is authored from the complete TR-027 source packet for expr-f722ebf867640c5c.

1 occurrence in this chapter

Equation form expr-f8ef16f248355710

|N|={⟦n¯⟧N:n∈ℕ}\Domain{N} = \Setabs{\Value{\num{n}}{N}}{n \in \Nat}

Read as: the domain of structure N equals the set of the value of the numeral for n in structure N such that n is in the natural numbers

Means: Arithmetic notation denoting: the domain of structure N equals the set of the value of the numeral for n in structure N such that n is in the natural numbers. This meaning is authored from the complete TR-027 source packet for expr-f8ef16f248355710.

1 occurrence in this chapter

Equation form expr-f91c5260de97716d

PA⊢∀xx′…′=(x+n¯)\Th{PA} \Proves \lforall[x][\eq[x^{\prime\dots\prime}][(x + \num{n})]]

Read as: theory PA proves for every x, x superscript successor symbol and so on successor symbol equals open parenthesis x plus the numeral for n close parenthesis

Means: A derivability claim stating: theory PA proves for every x, x superscript successor symbol and so on successor symbol equals open parenthesis x plus the numeral for n close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-f91c5260de97716d.

1 occurrence in this chapter

Equation form expr-f9759825d81d4ea6

∀x∀y(x+y)=(y+x)\lforall[x][\lforall[y][\eq[(x+y)][(y+x)]]]

Read as: for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis

Means: A first-order arithmetic formula stating: for every x, for every y, open parenthesis x plus y close parenthesis equals open parenthesis y plus x close parenthesis. This meaning is authored from the complete TR-027 source packet for expr-f9759825d81d4ea6.

1 occurrence in this chapter

Equation form expr-faa7280221a24d1c

≺𝘇\nsless \nszero

Read as: less than model zero

Means: A claim about the interpreted arithmetic operations or order stating: less than model zero. This meaning is authored from the complete TR-027 source packet for expr-faa7280221a24d1c.

1 occurrence in this chapter

Equation form expr-fb365187c776daa2

y∗=xy^\nssucc = x

Read as: y followed by model successor equals x

Means: A claim about the interpreted arithmetic operations or order stating: y followed by model successor equals x. This meaning is authored from the complete TR-027 source packet for expr-fb365187c776daa2.

1 occurrence in this chapter

Equation form expr-fd301322eabfd16b

∀x(x×0)=0row label Q6∀x∀y(x×y′)=((x×y)+x)row label Q7∀x∀y(x<y↔∃z(z′+x)=y)row label Q8& \lforall[x][\eq[(x \times \Obj 0)][\Obj 0]] \tag{$!Q_6$}\\ & \lforall[x][\lforall[y][\eq[(x \times y')][((x \times y) + x)]]] \tag{$!Q_7$}\\ & \lforall[x][\lforall[y][(x < y \liff \lexists[z][\eq[(z'+x)][y]])]] \tag{$!Q_8$}

Read as: for every x, open parenthesis x times object language symbol zero close parenthesis equals object language symbol zero row labeled Q sub six next row for every x, for every y, open parenthesis x times y prime close parenthesis equals open parenthesis open parenthesis x times y close parenthesis plus x close parenthesis row labeled Q sub seven next row for every x, for every y, open parenthesis x is less than y if and only if there exists z, open parenthesis z prime plus x close parenthesis equals y close parenthesis row labeled Q sub eight

Means: A source-ordered calculation or table whose rows state: for every x, open parenthesis x times object language symbol zero close parenthesis equals object language symbol zero row labeled Q sub six next row for every x, for every y, open parenthesis x times y prime close parenthesis equals open parenthesis open parenthesis x times y close parenthesis plus x close parenthesis row labeled Q sub seven next row for every x, for every y, open parenthesis x is less than y if and only if there exists z, open parenthesis z prime plus x close parenthesis equals y close parenthesis row labeled Q sub eight. This meaning is authored from the complete TR-027 source packet for expr-fd301322eabfd16b.

1 occurrence in this chapter

Equation form expr-fd876049c4809e5a

z∈[x]z \in [x]

Read as: z is in the block of x

Means: An arithmetic relation or equation stating: z is in the block of x. This meaning is authored from the complete TR-027 source packet for expr-fd876049c4809e5a.

1 occurrence in this chapter

Equation form expr-fe2826cc3bc75bdd

×N\Assign{\times}{N}

Read as: the interpreted multiplication operation of structure N

Means: Arithmetic notation denoting: the interpreted multiplication operation of structure N. This meaning is authored from the complete TR-027 source packet for expr-fe2826cc3bc75bdd.

1 occurrence in this chapter

Equation form expr-fe4bfa85965f8d89

𝘇\nszero

Read as: model zero

Means: A claim about the interpreted arithmetic operations or order stating: model zero. This meaning is authored from the complete TR-027 source packet for expr-fe4bfa85965f8d89.

4 occurrences in this chapter

Equation form expr-ff2a63ac44a2d5a2

n<mn<m

Read as: n is less than m

Means: An arithmetic relation or equation stating: n is less than m. This meaning is authored from the complete TR-027 source packet for expr-ff2a63ac44a2d5a2.

1 occurrence in this chapter

Equation form expr-ff85001e894570fd

⟦0⟧M=0M\Value{\Obj{0}}{M} = \Assign{\Obj{0}}{M}

Read as: the value of object language symbol zero in structure M equals the interpretation of object language symbol zero in structure M

Means: Arithmetic notation denoting: the value of object language symbol zero in structure M equals the interpretation of object language symbol zero in structure M. This meaning is authored from the complete TR-027 source packet for expr-ff85001e894570fd.

1 occurrence in this chapter

Equation form expr-ffef7f79544747c9

y≠𝘇y \neq \nszero

Read as: y is not equal to model zero

Means: A claim about the interpreted arithmetic operations or order stating: y is not equal to model zero. This meaning is authored from the complete TR-027 source packet for expr-ffef7f79544747c9.

1 occurrence in this chapter

Interpretations in the standard model

Four source-ordered equations interpret zero, successor, addition, and multiplication in the natural numbers.

Source

Interpretations in the finite-string copy

Four equations interpret arithmetic on finite strings of the symbol a and exhibit a copy isomorphic to the standard model.

Source

Definition of a standard arithmetic structure

A structure for the language of arithmetic is standard exactly when it is isomorphic to the standard natural-number structure.

Source

Standard structures are exhausted by numeral values

The domain of every standard arithmetic structure consists exactly of the values of its standard numerals.

Source

Exercise on the converse domain claim

Unsolved exercise asking for a numeral-generated arithmetic structure that is not isomorphic to the standard model.

Source

A numeral-generated model of Q is standard

A model of Robinson arithmetic whose domain is exhausted by standard numeral values is isomorphic to the standard model.

Source

Uniqueness of the standard isomorphism

The numeral-value map is the unique isomorphism from the standard model to any standard arithmetic structure.

Source

Induction calculation for uniqueness

A seven-row equality calculation proves that the two candidate isomorphisms agree at the successor step.

Source

Definition of standard and nonstandard numbers

The definition separates elements equal to standard numeral values from the remaining nonstandard elements of a nonstandard structure.

Source

A nonstandard element forces a nonstandard structure

Any arithmetic structure containing an element outside all standard numeral values cannot be isomorphic to the standard model.

Source

Exercise separating the first three Q axioms

Unsolved exercise asking for three structures that each falsify a different one of the first three axioms of Robinson arithmetic.

Source

First three axioms of Robinson arithmetic

Three displayed source-order axioms state successor injectivity, zero outside the successor range, and the predecessor-or-zero alternative.

Source

True arithmetic has an enumerable nonstandard model

Compactness yields an enumerable nonstandard model of the complete theory of the standard natural numbers.

Source

Example of the K model of Q

The full example defines K, tabulates its operations, and checks the first five axioms of Robinson arithmetic while exhibiting a sentence false in the standard model.

Source

Operations of the K model of Q

A case definition specifies zero, successor, addition, multiplication, and order on the natural numbers with one extra element.

Source

Successor-addition calculation in K

A source-order case calculation verifies the recursive addition axiom for standard and nonstandard arguments in K.

Source

Exercise completing the verification of K

Unsolved exercise asking to verify the remaining Q axioms in K and find a successor-only sentence distinguishing K from the standard model.

Source

Remaining arithmetic axioms for K

Three displayed axioms specify multiplication by zero, recursive multiplication, and the definition of less than.

Source

Example of the L model of Q

The example defines successor and addition on a two-extra-element structure and verifies the recursive addition axiom before showing addition is noncommutative.

Source

Nonstandard addition cases in L

Six source-ordered equations check recursive addition for the two extra elements a and b and for mixed standard arguments.

Source

Exercise expanding L to a full model of Q

Unsolved exercise asking for multiplication and order interpretations that make L satisfy the remaining axioms of Robinson arithmetic.

Source

Remaining arithmetic axioms for L

The multiplication and order axioms left to interpret in L are displayed in source order.

Source

Exercise on a two-cycle successor

Unsolved exercise asking whether a model of Robinson arithmetic can exchange two nonstandard elements under successor.

Source

The interpreted order in a model of PA

The model order is irreflexive, transitive, and total on unequal elements, hence a strict linear order.

Source

Discreteness of the model order

Model zero is least, each successor is the immediate next element, and each nonzero element has a unique predecessor.

Source

Exercise axiomatizing discreteness

Unsolved exercise asking for arithmetic sentences derivable in PA that guarantee the stated zero, successor, and order properties.

Source

Standard elements precede nonstandard elements

Every standard numeral value lies below every nonstandard element in the interpreted order.

Source

Definition and characterization of a nonstandard block

A nonstandard element belongs to a two-way successor chain with no endpoints, characterized by finite model additions in either direction.

Source

Distinct blocks are uniformly ordered

If one representative of a block is below a representative of another distinct block, every element of the first block is below every element of the second.

Source

Distinct blocks are disjoint

Two unequal blocks have empty intersection.

Source

Adding nonstandard elements leaves the original block

For nonstandard x and y, x is below their model sum and that sum is not in the block of x.

Source

No least nonstandard block

Every nonstandard block has a strictly smaller nonstandard block.

Source

No largest block

The block ordering in a nonstandard model of Peano arithmetic has no greatest block.

Source

Exercise proving there is no largest block

Unsolved exercise asking for a proof that a nonstandard model of PA has no greatest block.

Source

Density of the block ordering

Between any two distinct ordered blocks there is a third block.

Source

Exercise completing the density proof

Unsolved exercise asking for the required PA sentence and a detailed verification that the average element lies in a genuinely intermediate block.

Source

Definition of a computable arithmetic structure

A structure on the natural numbers is computable when successor, addition, and multiplication are computable and its order relation is decidable.

Source

A computable nonstandard model of Q

The example transports K to a natural-number domain through a corrected bijection and explicitly gives the resulting computable operations and decidable order.

Source

Operations of K recalled for transport

A case display recalls the full arithmetic interpretation on the one-extra-element model K.

Source

Transported addition and multiplication on K prime

Two case definitions give the transported computable operations on the natural-number presentation of K.

Source

Exercise transporting the L model

Unsolved exercise asking for a natural-number presentation isomorphic to the two-extra-element model L.

Source

Tennenbaum's theorem

Every computable model of Peano arithmetic is standard and therefore isomorphic to the standard natural-number structure.

Source

Cross-reference reference-000511

reference prop:standard-domain

Source occurrence

Cross-reference reference-000512

reference prop:thq-standard

Source occurrence

Cross-reference reference-000513

reference prop:standard-domain

Source occurrence

Cross-reference reference-000514

reference ex:model-K-of-Q

Source occurrence

Cross-reference reference-000515

reference ex:model-L-of-Q

Source occurrence

Cross-reference reference-000516

reference ex:model-L-of-Q

Source occurrence

Cross-reference reference-000517

reference lem:less-zero

Source occurrence

Cross-reference reference-000518

reference lem:less-nsucc

Source occurrence

Cross-reference reference-000519

reference prop:M-discrete

Source occurrence

Cross-reference reference-000520

reference prop:blocks-dense

Source occurrence

Cross-reference reference-000521

reference ex:model-K-of-Q

Source occurrence

Cross-reference reference-000522

reference ex:model-L-of-Q

Source occurrence

Cross-reference reference-000523

Reference ex:comp-model-q

Source occurrence

Source disclosures