Incompleteness

Introduction to Incompleteness

Equation form expr-00fe9b383f124db5

yny_n

Read as: y sub n

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: y sub n

3 occurrences in this chapter

Equation form expr-0392a7f58dc6f6ba

0\Obj 0

Read as: object language zero

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: object language zero

1 occurrence in this chapter

Equation form expr-04290de733e78ab2

A(x1,…,xk)!A(x_1, \dots, x_k)

Read as: formula A open parenthesis x sub one comma and so on comma x sub k close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A open parenthesis x sub one comma and so on comma x sub k close parenthesis

1 occurrence in this chapter

Equation form expr-10e072fde0c2a074

A[t/x]\Subst{!A}{t}{x}

Read as: the result of substituting term t for variable x in formula A

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: the result of substituting term t for variable x in formula A

1 occurrence in this chapter

Equation form expr-12d9b1e9f8e5096a

N,s⊨A(x)\Sat{N}{!A(x)}[s]

Read as: structure N satisfies formula A open parenthesis x close parenthesis at assignment s

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: structure N satisfies formula A open parenthesis x close parenthesis at assignment s

1 occurrence in this chapter

Equation form expr-189fd3281c2dba40

XAX_{!A}

Read as: X sub A

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: X sub A

2 occurrences in this chapter

Equation form expr-18ac3e7343f01689

dd

Read as: d

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: d

2 occurrences in this chapter

Equation form expr-194dc7fcadcedaea

A(x1,…,xk,y)!A(x_1, \dots, x_k, y)

Read as: formula A open parenthesis x sub one comma and so on comma x sub k comma y close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A open parenthesis x sub one comma and so on comma x sub k comma y close parenthesis

1 occurrence in this chapter

Equation form expr-19e9c269d8cb4bcf

s(x)=ns(x) = n

Read as: s open parenthesis x close parenthesis equals n

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: s open parenthesis x close parenthesis equals n

1 occurrence in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: n

3 occurrences in this chapter

Equation form expr-1b8340cbc4763167

f:ℕk→ℕf\colon \Nat^k \to \Nat

Read as: f from the natural numbers superscript k to the natural numbers

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: f from the natural numbers superscript k to the natural numbers

1 occurrence in this chapter

Equation form expr-1d5c7a4a8bd57efb

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

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

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: the interpretation of object language symbol successor symbol in structure N open parenthesis n close parenthesis equals n plus one

1 occurrence in this chapter

Equation form expr-1dcbe3fa05542df9

n∈Dn \in D

Read as: number n belongs to the diagonal set D

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: number n belongs to the diagonal set D

2 occurrences in this chapter

Equation form expr-1ddc4a673cfde62f

A∧¬A!A \land \lnot !A

Read as: formula A and not formula A

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A and not formula A

1 occurrence in this chapter

Equation form expr-1f97d653b7ed2d2b

n+1n+1

Read as: n plus one

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: n plus one

1 occurrence in this chapter

Equation form expr-21cb0b6872f81e31

TA⊨¬A\Th{TA} \Entails \lnot !A

Read as: true arithmetic semantically entails not formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: true arithmetic semantically entails not formula A

1 occurrence in this chapter

Equation form expr-2276f20a68e8bc6c

d∉Dd \notin D

Read as: diagonal index d does not belong to the diagonal set D

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: diagonal index d does not belong to the diagonal set D

2 occurrences in this chapter

Equation form expr-233c3669cce10c28

N\Struct N

Read as: structure N

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: structure N

1 occurrence in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: f

2 occurrences in this chapter

Equation form expr-284c47be48fb0c66

¬\lnot

Read as: negation symbol

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: negation symbol

1 occurrence in this chapter

Equation form expr-2a1dc4189af4e832

×N(n,m)=n·m\Assign{\Obj \times}{N}(n, m) = n\cdot m

Read as: the interpretation of object language symbol times in structure N open parenthesis n comma m close parenthesis equals n times m

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: the interpretation of object language symbol times in structure N open parenthesis n comma m close parenthesis equals n times m

1 occurrence in this chapter

Equation form expr-2d511c187d84d509

Γ⊢∀y(A(n1¯,…,nk¯,y)→y=m¯)\Gamma \Proves \lforall[y](!A(\num{n_1}, \dots, \num{n_k}, y) \lif y = \num{m})

Read as: Gamma proves for every y open parenthesis formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k comma y close parenthesis implies y equals the numeral for m close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves for every y open parenthesis formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k comma y close parenthesis implies y equals the numeral for m close parenthesis

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: x

5 occurrences in this chapter

Equation form expr-305103140e202ac0

D⊆ℕD \subseteq \Nat

Read as: D is a subset of the natural numbers

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: D is a subset of the natural numbers

1 occurrence in this chapter

Equation form expr-33b1d4adba206718

R⊆ℕkR \subseteq \Nat^k

Read as: relation R is a subset of the k-fold Cartesian power of the natural numbers

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: relation R is a subset of the k-fold Cartesian power of the natural numbers

1 occurrence in this chapter

Equation form expr-342776565b5bf0e6

x×yx \times y

Read as: x times y

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: x times y

1 occurrence in this chapter

Equation form expr-349ba76d907adfab

×\times

Read as: times symbol

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: times symbol

6 occurrences in this chapter

Equation form expr-37303b54a174a80a

Q={A:{Q1,…,Q8}⊨A}\Th{Q} = \Setabs{!A}{\{!Q_1, \dots, !Q_8\} \Entails !A}

Read as: Robinson arithmetic Q equals the set of formula A such that Q sub one comma and so on comma Q sub eight semantically entails formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Robinson arithmetic Q equals the set of formula A such that Q sub one comma and so on comma Q sub eight semantically entails formula A

1 occurrence in this chapter

Equation form expr-3d2df79065b6f165

Q\Th{Q}

Read as: Robinson arithmetic Q

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: Robinson arithmetic Q

30 occurrences in this chapter

Equation form expr-3e02d29541b9dc8d

(2¯+3¯)=5¯(\num{2} + \num{3}) = \num{5}

Read as: open parenthesis the numeral for two plus the numeral for three close parenthesis equals the numeral for five

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: open parenthesis the numeral for two plus the numeral for three close parenthesis equals the numeral for five

1 occurrence in this chapter

Equation form expr-3f1b6dc519af96a7

′\prime

Read as: successor symbol

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: successor symbol

2 occurrences in this chapter

Equation form expr-3f38f9099b7b27bb

∀y1…∀yn((A(0)∧∀x(A(x)→A(x′)))→∀xA(x))\lforall[y_1][\dots\lforall[y_n][((!A(\Obj 0) \land \lforall[x][(!A(x) \lif !A(x'))]) \lif \lforall[x][!A(x)])]]

Read as: for every y sub one, and so on for every y sub n, open parenthesis open parenthesis formula A open parenthesis object language symbol zero close parenthesis and for every x, open parenthesis formula A open parenthesis x close parenthesis implies formula A open parenthesis x prime close parenthesis close parenthesis close parenthesis implies for every x, formula A open parenthesis x close parenthesis close parenthesis

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: for every y sub one, and so on for every y sub n, open parenthesis open parenthesis formula A open parenthesis object language symbol zero close parenthesis and for every x, open parenthesis formula A open parenthesis x close parenthesis implies formula A open parenthesis x prime close parenthesis close parenthesis close parenthesis implies for every x, formula A open parenthesis x close parenthesis close parenthesis

1 occurrence in this chapter

Equation form expr-3f39d5c348e5b79d

DD

Read as: capital D

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: capital D

9 occurrences in this chapter

Equation form expr-4624e4c302dec30e

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

Read as: true arithmetic equals the set of formula A such that structure N satisfies formula A

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: true arithmetic equals the set of formula A such that structure N satisfies formula A

1 occurrence in this chapter

Equation form expr-4a5f9b28c7dae2c7

ProvΓ(x)\OProv[\Gamma](x)

Read as: the arithmetized provability predicate for Gamma applied to x

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: the arithmetized provability predicate for Gamma applied to x

1 occurrence in this chapter

Equation form expr-4b68ab3847feda7d

XX

Read as: capital X

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: capital X

4 occurrences in this chapter

Equation form expr-4be9f56ddfff63ff

Γ⊢A(n1¯,…,nk¯,m¯)\Gamma \Proves !A(\num{n_1}, \dots, \num{n_k}, \num{m})

Read as: Gamma proves formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k comma the numeral for m close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k comma the numeral for m close parenthesis

1 occurrence in this chapter

Equation form expr-4e07408562bedb8b

33

Read as: three

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: three

2 occurrences in this chapter

Equation form expr-52e4b1e57e932dad

0∈XA0 \in X_{!A}

Read as: zero is in X sub A

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: zero is in X sub A

2 occurrences in this chapter

Equation form expr-53f8d2db25f75cf3

An(n¯)!A_n(\num{n})

Read as: formula A sub n open parenthesis the numeral for n close parenthesis

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: formula A sub n open parenthesis the numeral for n close parenthesis

1 occurrence in this chapter

Equation form expr-57885e4c75965b23

Γ\Gamma

Read as: Gamma

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: Gamma

44 occurrences in this chapter

Equation form expr-57b011c61ad335dd

Q⊢Af(n1¯,n2¯,m¯)\Th{Q} \Proves !A_f(\num{n_1}, \num{n_2}, \num{m})

Read as: Robinson arithmetic Q proves formula A sub f open parenthesis the numeral for n sub one comma the numeral for n sub two comma the numeral for m close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Robinson arithmetic Q proves formula A sub f open parenthesis the numeral for n sub one comma the numeral for n sub two comma the numeral for m close parenthesis

1 occurrence in this chapter

Equation form expr-583292a11ba24381

n1n_1

Read as: n sub one

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: n sub one

2 occurrences in this chapter

Equation form expr-5afa20230f875253

n∈ℕn \in \Nat

Read as: n is in the natural numbers

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: n is in the natural numbers

3 occurrences in this chapter

Equation form expr-5b63a60fdb499c3a

N\Struct{N}

Read as: structure N

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: structure N

4 occurrences in this chapter

Equation form expr-5c41998a533c3083

n∈XAn \in X_{!A}

Read as: n is in X sub A

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: n is in X sub A

2 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: zero

1 occurrence in this chapter

Equation form expr-6193864491e4d7b1

N⊨A\Sat{N}{!A}

Read as: structure N satisfies formula A

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: structure N satisfies formula A

2 occurrences in this chapter

Equation form expr-62c66a7a5dd70c31

mm

Read as: m

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: m

2 occurrences in this chapter

Equation form expr-63077942e4d90ef5

Γ={A:Γ0⊨A}\Gamma = \Setabs{!A}{\Gamma_0 \Entails !A}

Read as: Gamma equals the set of formula A such that Gamma sub zero semantically entails formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma equals the set of formula A such that Gamma sub zero semantically entails formula A

1 occurrence in this chapter

Equation form expr-643fbf6e4b47c099

Γ⊢¬A(n1¯,…,nk¯)\Gamma \Proves \lnot !A(\num{n_1}, \dots, \num{n_k})

Read as: Gamma proves not formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves not formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k close parenthesis

1 occurrence in this chapter

Equation form expr-6562529bf967e9a8

f(n1,…,nk)=mf(n_1, \dots, n_k) = m

Read as: f open parenthesis n sub one comma and so on comma n sub k close parenthesis equals m

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: f open parenthesis n sub one comma and so on comma n sub k close parenthesis equals m

1 occurrence in this chapter

Equation form expr-6637867ac78daee3

¬An(n¯)\lnot !A_n(\num{n})

Read as: not formula A sub n open parenthesis the numeral for n close parenthesis

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: not formula A sub n open parenthesis the numeral for n close parenthesis

1 occurrence in this chapter

Equation form expr-668f5d0c2102ac90

¬An(n¯)∈Γ\lnot !A_n(\num{n}) \in \Gamma

Read as: the negated formula A sub n applied to the numeral for n belongs to theory Gamma

Means: The negation of the n-th enumerated formula, with numeral n substituted for its free variable, is a theorem of Gamma. This is the membership test for diagonal set D.

1 occurrence in this chapter

Equation form expr-66985e1aab72eb72

1∈XA1 \in X_{!A}

Read as: one is in X sub A

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: one is in X sub A

2 occurrences in this chapter

Equation form expr-6750634c0fe2bd68

f(n1,n2)=mf(n_1, n_2) = m

Read as: f open parenthesis n sub one comma n sub two close parenthesis equals m

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: f open parenthesis n sub one comma n sub two close parenthesis equals m

1 occurrence in this chapter

Equation form expr-68a0237f0c73a77c

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

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

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: the interpretation of object language symbol plus in structure N open parenthesis n comma m close parenthesis equals n plus m

1 occurrence in this chapter

Equation form expr-69db2efcf937c1ef

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

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

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: the interpretation of object language symbol zero in structure N equals zero

1 occurrence in this chapter

Equation form expr-6a99751d93986ab4

=0= 0

Read as: equals zero

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: equals zero

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: one

1 occurrence in this chapter

Equation form expr-6c610d09a35efbf8

Γ⊢¬A\Gamma \Proves \lnot !A

Read as: Gamma proves not formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves not formula A

1 occurrence in this chapter

Equation form expr-6e4cbbf82204396a

A∈TA!A \in \Th{TA}

Read as: formula A is in true arithmetic

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: formula A is in true arithmetic

1 occurrence in this chapter

Equation form expr-71983740200349d8

∀xA(x)\lforall[x][!A(x)]

Read as: for every x, formula A open parenthesis x close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: for every x, formula A open parenthesis x close parenthesis

1 occurrence in this chapter

Equation form expr-71c52326c9b7e234

Γ⊬¬A\Gamma \Proves/ \lnot !A

Read as: Gamma does not prove not formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma does not prove not formula A

1 occurrence in this chapter

Equation form expr-729f2cd8398e9960

∈\in

Read as: the membership relation symbol

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: the membership relation symbol

1 occurrence in this chapter

Equation form expr-73c66f31256bd49a

D={n:Γ⊢¬An(n¯)}D = \Setabs{n}{\Gamma \Proves \lnot !A_n(\num{n})}

Read as: the diagonal set D equals the set of numbers n such that theory Gamma proves not formula A sub n applied to the numeral for n

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: the diagonal set D equals the set of numbers n such that theory Gamma proves not formula A sub n applied to the numeral for n

1 occurrence in this chapter

Equation form expr-74c0e74537bf75aa

∀x∀y(x′=y′→x=y)row label Q1∀x0≠x′row label Q2∀x(x=0∨∃yx=y′)row label Q3∀x(x+0)=xrow label Q4∀x∀y(x+y′)=(x+y)′row label Q5∀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][\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{$!Q_3$}\\ & \lforall[x][\eq[(x + \Obj 0)][x]] \tag{$!Q_4$}\\ & \lforall[x][\lforall[y][\eq[(x + y')][(x + y)']]] \tag{$!Q_5$}\\ & \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, for every y, open parenthesis x prime equals y prime implies x equals y close parenthesis row labeled Q sub one next row then for every x, object language symbol zero is not equal to x prime row labeled Q sub two next row then 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 next row then for every x, open parenthesis x plus object language symbol zero close parenthesis equals x row labeled Q sub four next row then for every x, for every y, open parenthesis x plus y prime close parenthesis equals open parenthesis x plus y close parenthesis prime row labeled Q sub five next row then 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 then 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 then 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: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. 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 then for every x, object language symbol zero is not equal to x prime row labeled Q sub two next row then 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 next row then for every x, open parenthesis x plus object language symbol zero close parenthesis equals x row labeled Q sub four next row then for every x, for every y, open parenthesis x plus y prime close parenthesis equals open parenthesis x plus y close parenthesis prime row labeled Q sub five next row then 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 then 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 then 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

1 occurrence in this chapter

Equation form expr-74ff7b7ac2470720

<N={⟨n,m⟩:n∈ℕ,m∈ℕ,n<m}\Assign{\Obj <}{N} = \Setabs{\tuple{n, m}}{n \in \Nat, m \in \Nat, n < m}

Read as: the interpretation of the less-than relation symbol in structure N equals the set of ordered pairs n comma m such that n and m are natural numbers and n is less than m

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: the interpretation of the less-than relation symbol in structure N equals the set of ordered pairs n comma m such that n and m are natural numbers and n is less than m

1 occurrence in this chapter

Equation form expr-77e6f9505f66fa91

2+3=52+3 = 5

Read as: two plus three equals five

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: two plus three equals five

1 occurrence in this chapter

Equation form expr-79d4c7f9c8579543

ℕ\Nat

Read as: the natural numbers

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: the natural numbers

5 occurrences in this chapter

Equation form expr-7a9a8f9dd6debc33

x∈Xx \in X

Read as: variable x belongs to set X

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: variable x belongs to set X

1 occurrence in this chapter

Equation form expr-7df1a618a9888478

d∈Dd \in D

Read as: diagonal index d belongs to the diagonal set D

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: diagonal index d belongs to the diagonal set D

3 occurrences in this chapter

Equation form expr-7f024b2d7f1db4d4

n¯\num{n}

Read as: the numeral for n

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: the numeral for n

1 occurrence in this chapter

Equation form expr-814745bd35a2a37b

A(0,y1,…,yn)∧∀x(A(x,y1,…,yn)→A(x′,y1,…,yn))!A(\Obj 0, y_1, \dots, y_n) \land \lforall[x][(!A(x, y_1, \dots, y_n) \lif !A(x', y_1, \dots, y_n))]

Read as: formula A open parenthesis object language symbol zero comma y sub one comma and so on comma y sub n close parenthesis and for every x, open parenthesis formula A open parenthesis x comma y sub one comma and so on comma y sub n close parenthesis implies formula A open parenthesis x prime comma y sub one comma and so on comma y sub n close parenthesis close parenthesis

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: formula A open parenthesis object language symbol zero comma y sub one comma and so on comma y sub n close parenthesis and for every x, open parenthesis formula A open parenthesis x comma y sub one comma and so on comma y sub n close parenthesis implies formula A open parenthesis x prime comma y sub one comma and so on comma y sub n close parenthesis close parenthesis

1 occurrence in this chapter

Equation form expr-81aa5a490db29cc9

A(x)!A(x)

Read as: formula A open parenthesis x close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A open parenthesis x close parenthesis

2 occurrences in this chapter

Equation form expr-8238c028f61fc0f7

A!A

Read as: formula A

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A

15 occurrences in this chapter

Equation form expr-8779068734acaece

·\cdot

Read as: dot symbol

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: dot symbol

3 occurrences in this chapter

Equation form expr-8c2169ebe37c4966

A0(x)!A_0(x)

Read as: formula A sub zero open parenthesis x close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A sub zero open parenthesis x close parenthesis

2 occurrences in this chapter

Equation form expr-8efa5ea4fe923a45

∀xA(x,y1,…,yn)\lforall[x][!A(x, y_1, \dots, y_n)]

Read as: for every x, formula A open parenthesis x comma y sub one comma and so on comma y sub n close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: for every x, formula A open parenthesis x comma y sub one comma and so on comma y sub n close parenthesis

1 occurrence in this chapter

Equation form expr-90d79610990ddcb2

n∉Dn \notin D

Read as: number n does not belong to the diagonal set D

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: number n does not belong to the diagonal set D

1 occurrence in this chapter

Equation form expr-98571ab199ee3a75

N⊨¬A\Sat{N}{\lnot !A}

Read as: structure N satisfies not formula A

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: structure N satisfies not formula A

1 occurrence in this chapter

Equation form expr-9af4148d0e745f37

2∈XA2 \in X_{!A}

Read as: two is in X sub A

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: two is in X sub A

1 occurrence in this chapter

Equation form expr-a0ab927e8683220d

Con\OCon

Read as: the formal consistency statement

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: the formal consistency statement

1 occurrence in this chapter

Equation form expr-a12b4cca9a048807

Γ⊢Ad(d¯)\Gamma \Proves !A_d(\num{d})

Read as: Gamma proves formula A sub d open parenthesis the numeral for d close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves formula A sub d open parenthesis the numeral for d close parenthesis

1 occurrence in this chapter

Equation form expr-a318c24216defe20

++

Read as: plus symbol

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: plus symbol

5 occurrences in this chapter

Equation form expr-a63b58fbb7050d47

f21\Obj f^2_1

Read as: object language symbol f superscript two sub one

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: object language symbol f superscript two sub one

1 occurrence in this chapter

Equation form expr-a8e0639bf1645146

R(n1,…,nk)R(n_1, \dots, n_k)

Read as: relation R holds of n sub one through n sub k

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: relation R holds of n sub one through n sub k

2 occurrences in this chapter

Equation form expr-ab733e02bed54e75

Γ⊨A\Gamma \Entails !A

Read as: Gamma semantically entails formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma semantically entails formula A

2 occurrences in this chapter

Equation form expr-adedb03de35ed98c

Γ\Gamma

Read as: Gamma

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: Gamma

1 occurrence in this chapter

Equation form expr-af414b0457e53cda

Γ⊢A(n1¯,…,nk¯)\Gamma \Proves !A(\num{n_1}, \dots, \num{n_k})

Read as: Gamma proves formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves formula A open parenthesis the numeral for n sub one comma and so on comma the numeral for n sub k close parenthesis

1 occurrence in this chapter

Equation form expr-b31f9dce7c89d4e2

N⊨TA\Sat{N}{\Th{TA}}

Read as: structure N satisfies true arithmetic

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: structure N satisfies true arithmetic

1 occurrence in this chapter

Equation form expr-b6b337ece4924770

n2n_2

Read as: n sub two

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: n sub two

2 occurrences in this chapter

Equation form expr-bc9367794d2761fe

A∈Γ!A \in \Gamma

Read as: formula A belongs to theory Gamma

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: formula A belongs to theory Gamma

2 occurrences in this chapter

Equation form expr-be58d8ae65b26403

0\Obj{0}

Read as: object language zero

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: object language zero

1 occurrence in this chapter

Equation form expr-c63f9557f464c93a

¬A\lnot !A

Read as: not formula A

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: not formula A

3 occurrences in this chapter

Equation form expr-c806cdc5d3b3372b

χX(B)=1\Char{X}(!B) = 1

Read as: the characteristic function of X open parenthesis formula B close parenthesis equals one

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: the characteristic function of X open parenthesis formula B close parenthesis equals one

2 occurrences in this chapter

Equation form expr-c887b3828dd4ba10

y1y_1

Read as: y sub one

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: y sub one

3 occurrences in this chapter

Equation form expr-caff5d1112939253

{Q1,…,Q8}\{!Q_1, \dots, !Q_8\}

Read as: the set containing axioms Q sub one through Q sub eight

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: the set containing axioms Q sub one through Q sub eight

1 occurrence in this chapter

Equation form expr-cb40773102c92888

Γ⊨¬A\Gamma \Entails \lnot !A

Read as: Gamma semantically entails not formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma semantically entails not formula A

1 occurrence in this chapter

Equation form expr-cbbba670a3f47b53

Γ⊢A\Gamma \Proves !A

Read as: Gamma proves formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves formula A

4 occurrences in this chapter

Equation form expr-cbf770cbe9a2d200

Γ⊢¬Ad(d¯)\Gamma \Proves \lnot !A_d(\num{d})

Read as: Gamma proves not formula A sub d open parenthesis the numeral for d close parenthesis

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves not formula A sub d open parenthesis the numeral for d close parenthesis

2 occurrences in this chapter

Equation form expr-cd37871db1ba340f

A∨¬A!A \lor \lnot !A

Read as: formula A or not formula A

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A or not formula A

1 occurrence in this chapter

Equation form expr-cdc2ed7d3b3d72c2

⊥\lfalse

Read as: falsum

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: falsum

1 occurrence in this chapter

Equation form expr-cde346439cd0151b

Af(x1,x2,y)!A_f(x_1, x_2, y)

Read as: formula A sub f open parenthesis x sub one comma x sub two comma y close parenthesis

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: formula A sub f open parenthesis x sub one comma x sub two comma y close parenthesis

1 occurrence in this chapter

Equation form expr-d03543adcda36df1

Γ0\Gamma_0

Read as: Gamma sub zero

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: Gamma sub zero

9 occurrences in this chapter

Equation form expr-d055ee4dbcdd0c8b

B!B

Read as: formula B

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula B

6 occurrences in this chapter

Equation form expr-d203ba01eef4198c

δ\delta

Read as: delta

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: delta

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: two

2 occurrences in this chapter

Equation form expr-d4a0cfb83b24d498

∀x\lforall[x]

Read as: for every x

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: for every x

1 occurrence in this chapter

Equation form expr-d6af7a2a04de9b37

f:ℕ→Xf \colon \Nat \to X

Read as: f from the natural numbers to X

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: f from the natural numbers to X

1 occurrence in this chapter

Equation form expr-d6f7e36999a71407

|N|=ℕ\Domain N = \Nat

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

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: the domain of structure N equals the natural numbers

1 occurrence in this chapter

Equation form expr-d7a17ee1489cf9bc

LA\Lang L_A

Read as: language L sub A

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: language L sub A

15 occurrences in this chapter

Equation form expr-d89ad7af62404a57

PA\Th{PA}

Read as: Peano arithmetic

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: Peano arithmetic

13 occurrences in this chapter

Equation form expr-daadff1066574ea0

GΓ!G_\Gamma

Read as: G sub Gamma

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: G sub Gamma

4 occurrences in this chapter

Equation form expr-dabd3aff769f07eb

<<

Read as: the less-than symbol

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: the less-than symbol

1 occurrence in this chapter

Equation form expr-dc9f6d82c234a7e1

f(x1,x2)f(x_1, x_2)

Read as: f open parenthesis x sub one comma x sub two close parenthesis

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: f open parenthesis x sub one comma x sub two close parenthesis

1 occurrence in this chapter

Equation form expr-dcc10690963a4de2

A1(x)!A_1(x)

Read as: formula A sub one open parenthesis x close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A sub one open parenthesis x close parenthesis

2 occurrences in this chapter

Equation form expr-e0fca11a04a2a41c

An(x)!A_n(x)

Read as: formula A sub n open parenthesis x close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A sub n open parenthesis x close parenthesis

2 occurrences in this chapter

Equation form expr-e11ab910214796e7

((A(0)∧∀x(A(x)→A(x′)))→∀xA(x))((!A(\Obj 0) \land \lforall[x][(!A(x) \lif !A(x'))]) \lif \lforall[x][!A(x)])

Read as: open parenthesis open parenthesis formula A open parenthesis object language symbol zero close parenthesis and for every x, open parenthesis formula A open parenthesis x close parenthesis implies formula A open parenthesis x prime close parenthesis close parenthesis close parenthesis implies for every x, formula A open parenthesis x close parenthesis close parenthesis

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: open parenthesis open parenthesis formula A open parenthesis object language symbol zero close parenthesis and for every x, open parenthesis formula A open parenthesis x close parenthesis implies formula A open parenthesis x prime close parenthesis close parenthesis close parenthesis implies for every x, formula A open parenthesis x close parenthesis close parenthesis

1 occurrence in this chapter

Equation form expr-e492b02824f52f65

n,m∈ℕn, m \in \Nat

Read as: n and m are natural numbers

Means: A set, relation, or membership condition in the Introduction to Incompleteness context. Read as: n and m are natural numbers

2 occurrences in this chapter

Equation form expr-e9096094fbd36747

PA=TA\Th{PA} = \Th{TA}

Read as: Peano arithmetic equals true arithmetic

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: Peano arithmetic equals true arithmetic

1 occurrence in this chapter

Equation form expr-e9e3fcec583a9837

Γ⊢ProvΓ(n¯)\Gamma \Proves \Prov[\Gamma](\num{n})

Read as: Gamma proves that the provability predicate for Gamma holds of the numeral for n

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma proves that the provability predicate for Gamma holds of the numeral for n

1 occurrence in this chapter

Equation form expr-eee1c230c6f27590

TA\Th{TA}

Read as: true arithmetic

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: true arithmetic

7 occurrences in this chapter

Equation form expr-ef2d127de37b942b

55

Read as: five

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: five

2 occurrences in this chapter

Equation form expr-f01ac326f8366d0e

¬ProvΓ(n¯)\lnot \Prov[\Gamma](\num{n})

Read as: the provability predicate for Gamma does not hold of the numeral for n

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: the provability predicate for Gamma does not hold of the numeral for n

1 occurrence in this chapter

Equation form expr-f064f2952aed96a2

χX\Char{X}

Read as: the characteristic function of X

Means: A function, characteristic function, or numerical computation condition in the Introduction to Incompleteness context. Read as: the characteristic function of X

1 occurrence in this chapter

Equation form expr-f0706d38015789d6

Pres={A∈LA+:N⊨A}\Th{Pres} = \Setabs{!A \in \Lang{L_{A^+}}}{\Sat{N}{!A}}

Read as: Presburger arithmetic equals the set of formulas A in language L sub A plus that are satisfied by structure N

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: Presburger arithmetic equals the set of formulas A in language L sub A plus that are satisfied by structure N

1 occurrence in this chapter

Equation form expr-f13ffed6ab9c214e

TA⊨A\Th{TA} \Entails !A

Read as: true arithmetic semantically entails formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: true arithmetic semantically entails formula A

2 occurrences in this chapter

Equation form expr-f24501cd355c999c

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

Read as: true arithmetic equals the set of formula A such that structure N satisfies formula A

Means: An interpretation or satisfaction assertion about the standard arithmetic structure N in the Introduction to Incompleteness context. Read as: true arithmetic equals the set of formula A such that structure N satisfies formula A

1 occurrence in this chapter

Equation form expr-f396f29509427e56

LA\Lang{L_A}

Read as: language L sub A

Means: A theory, language, or metatheoretic variable used in the incompleteness discussion in the Introduction to Incompleteness context. Read as: language L sub A

1 occurrence in this chapter

Equation form expr-f6ac033cc1d1554a

Γ⊬A\Gamma \Proves/ !A

Read as: Gamma does not prove formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma does not prove formula A

2 occurrences in this chapter

Equation form expr-f8653f213f88cac0

Γ={A:Γ⊨A}\Gamma = \Setabs{!A}{\Gamma \Entails !A}

Read as: Gamma equals the set of formula A such that Gamma semantically entails formula A

Means: A syntactic provability, semantic consequence, or formal consistency assertion in the Introduction to Incompleteness context. Read as: Gamma equals the set of formula A such that Gamma semantically entails formula A

1 occurrence in this chapter

Equation form expr-fba00f8a62ca2905

Ad(x)!A_d(x)

Read as: formula A sub d open parenthesis x close parenthesis

Means: A formula, schema parameter, variable, or logical construction in the Introduction to Incompleteness context. Read as: formula A sub d open parenthesis x close parenthesis

3 occurrences in this chapter

Equation form expr-fe3756fb5599cf1a

n·mn \cdot m

Read as: n times m

Means: An arithmetic symbol, numeral, value, or equation in the Introduction to Incompleteness context. Read as: n times m

1 occurrence in this chapter

Introduction to incompleteness definition

Defines a theory as a set of sentences closed under semantic consequence, and states the closure equation in source order.

Source

Introduction to incompleteness definition

Defines the standard model of arithmetic by giving its natural number domain and the interpretations of zero, successor, addition, multiplication, and less than in source order.

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Introduction to incompleteness definition

Defines true arithmetic as the set of sentences satisfied by the standard model of arithmetic, with display punctuation kept outside navigable mathematics in the reader projection.

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Introduction to incompleteness definition

Defines when a theory Gamma is axiomatized by an axiom set Gamma sub zero, using closure under semantic consequence.

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Introduction to incompleteness definition

Defines Robinson arithmetic Q from its eight displayed axioms and then identifies the theory as every sentence semantically entailed by those axioms.

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Introduction to incompleteness displayed formal object

States the eight axioms of Robinson arithmetic Q in order, from injectivity and nonzeroness of successor through the recursive clauses for addition, multiplication, and less than.

Source

Introduction to incompleteness definition

Defines the induction schema for a formula with the displayed base case, successor step, and universal conclusion, then defines Peano arithmetic as Robinson arithmetic plus every induction instance.

Source

Introduction to incompleteness definition

Defines a complete theory as one that semantically entails each sentence in its language or semantically entails its negation.

Source

Introduction to incompleteness definition

Defines a decidable set by a computational procedure returning one for membership and zero otherwise.

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Introduction to incompleteness definition

Defines an axiomatizable theory as one axiomatized by a decidable set of axioms.

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Introduction to incompleteness example

Explains why finitely and schematically axiomatized theories are axiomatizable, and gives a source order decision procedure for recognizing Peano arithmetic axioms and induction instances.

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Introduction to incompleteness definition

Defines a computably enumerable set as an empty set or a set having a computable enumeration.

Source

Introduction to incompleteness definition

Defines when a formula represents a numerical function in a theory by requiring proofs of the correct value and of its uniqueness for every input tuple.

Source

Introduction to incompleteness definition

Defines when a formula represents a relation in a theory by requiring proofs of its positive numeral instances and proofs of their negations when the relation fails.

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Introduction to incompleteness theorem

States the incompleteness result that a consistent axiomatizable arithmetic theory representing all computable functions and decidable relations is not complete.

Source

Introduction to incompleteness theorem

States that a consistent theory representing every decidable relation cannot itself be decidable.

Source

Introduction to incompleteness theorem

States that a theory which is both axiomatizable and complete is decidable.

Source

Introduction to incompleteness corollary

Derives that a consistent axiomatizable theory representing every decidable property is not complete.

Source

Unsolved incompleteness exercise

Asks the reader to show that true arithmetic is not axiomatizable, while permitting the assumption that it represents all decidable properties. The exercise is preserved as a task and intentionally remains unsolved.

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Source disclosures