Set Theory

Cardinals

Equation form expr-00ab483201161488

a\cardfont{a}

Read as: the cardinal a

Means: the cardinal a

4 occurrences in this chapter

Equation form expr-0149f87ca2dcf995

|A|=|B|\card{A} = \card{B}

Read as: the cardinality of capital A equals the cardinality of capital B

Means: the cardinality of capital A equals the cardinality of capital B

1 occurrence in this chapter

Equation form expr-076359e21c8f03ce

a∈b\cardfont{a} \in \cardfont{b}

Read as: the cardinal a belongs to the cardinal b

Means: the cardinal a belongs to the cardinal b

1 occurrence in this chapter

Equation form expr-082746415f78746d

|A|→|B|\card{A} \to \card{B}

Read as: the cardinality of capital A maps to the cardinality of capital B

Means: the cardinality of capital A maps to the cardinality of capital B

1 occurrence in this chapter

Equation form expr-0883e11e3376067e

b⊆⋃X\cardfont{b} \subseteq \bigcup X

Read as: the cardinal b is a subset of the union of capital X

Means: the cardinal b is a subset of the union of capital X

1 occurrence in this chapter

Equation form expr-090035fbe2da68be

n≈m\cardeq{n}{m}

Read as: n is equinumerous with m

Means: n is equinumerous with m

2 occurrences in this chapter

Equation form expr-093c96a2275b4297

⋃C\bigcup C

Read as: the union of capital C

Means: the union of capital C

1 occurrence in this chapter

Equation form expr-099959186b7f6cac

n=mn = m

Read as: n equals m

Means: n equals m

1 occurrence in this chapter

Equation form expr-0a1cfa102be6b21a

a<|℘(a)|\cardfont{a} < \card{\Pow{\cardfont{a}}}

Read as: the cardinal a is less than the cardinality of the power set of the cardinal a

Means: the cardinal a is less than the cardinality of the power set of the cardinal a

1 occurrence in this chapter

Equation form expr-0a5a0cdb7119bde3

n≠mn \neq m

Read as: n is not equal to m

Means: n is not equal to m

1 occurrence in this chapter

Equation form expr-0ae2677c65b5868e

γ=|γ|\gamma = \card{\gamma}

Read as: gamma equals the cardinality of gamma

Means: gamma equals the cardinality of gamma

2 occurrences in this chapter

Equation form expr-0e663e0f57f57a74

n∈ωn \in \omega

Read as: n belongs to omega

Means: n belongs to omega

2 occurrences in this chapter

Equation form expr-0f0e1602c643459b

δ∈γ\delta \in \gamma

Read as: delta belongs to gamma

Means: delta belongs to gamma

1 occurrence in this chapter

Equation form expr-11baa595827a4e0f

ω\omega

Read as: omega

Means: omega

6 occurrences in this chapter

Equation form expr-1270972fba1fbceb

β≈β+1\cardeq{\beta}{\beta \ordplus 1}

Read as: beta is equinumerous with beta ordinal plus one

Means: beta is equinumerous with beta ordinal plus one

1 occurrence in this chapter

Equation form expr-15a013fc11339f39

1≠21 \neq 2

Read as: one is not equal to two

Means: one is not equal to two

1 occurrence in this chapter

Equation form expr-1624cb0c7e6c8527

ord(A,<)=ord(B,⋖) iff ⟨A,<⟩≅⟨B,⋖⟩.\ordtype{A, <} = \ordtype{B, \lessdot} \text{ iff } \tuple{A, <} \isomorphic \tuple{B, \lessdot}.

Read as: the order type of capital A is less than equals the order type of capital B is strictly smaller than if and only if the ordered pair capital A, then is less than is isomorphic to the ordered pair capital B, then is strictly smaller than

Means: the order type of capital A is less than equals the order type of capital B is strictly smaller than if and only if the ordered pair capital A, then is less than is isomorphic to the ordered pair capital B, then is strictly smaller than

1 occurrence in this chapter

Equation form expr-181dc47425f7c51c

A=∅A = \emptyset

Read as: capital A equals the empty set

Means: capital A equals the empty set

1 occurrence in this chapter

Equation form expr-1b16b1df538ba12d

nn

Read as: n

Means: n

2 occurrences in this chapter

Equation form expr-1d55d2489f35fa31

#x(x≠x)\# x( x\neq x)

Read as: the number of x such that x is not equal to x

Means: the number of x such that x is not equal to x

1 occurrence in this chapter

Equation form expr-24bed2b8115326c2

A≼B\cardle{A}{B}

Read as: capital A has cardinality at most that of capital B

Means: capital A has cardinality at most that of capital B

3 occurrences in this chapter

Equation form expr-27f2abfaef6b532f

ω≤|A|\omega \leq \card{A}

Read as: omega is less than or equal to the cardinality of capital A

Means: omega is less than or equal to the cardinality of capital A

1 occurrence in this chapter

Equation form expr-29b2a2a0ca4ae64c

ω≤a\omega \leq \cardfont{a}

Read as: omega is less than or equal to the cardinal a

Means: omega is less than or equal to the cardinal a

1 occurrence in this chapter

Equation form expr-2a5696277fc7abba

ϵxF(x)=ϵxG(x)iff ∀x(F(x)↔G(x)).\fregeext{x}{F(x)} = \fregeext{x}{G(x)} \text{iff } \lforall[x][(F(x) \liff G(x))].

Read as: the Frege extension of x capital F of x equals the Frege extension of x capital G of x, iff, for every x, open scope, capital F of x if and only if capital G of x, close scope

Means: the Frege extension of x capital F of x equals the Frege extension of x capital G of x, iff, for every x, open scope, capital F of x if and only if capital G of x, close scope

1 occurrence in this chapter

Equation form expr-2fd098660be1f33e

ω≈n\cardeq{\omega}{n}

Read as: omega is equinumerous with n

Means: omega is equinumerous with n

1 occurrence in this chapter

Equation form expr-2ff65ba39788855e

A≈γ\cardeq{A}{\gamma}

Read as: capital A is equinumerous with gamma

Means: capital A is equinumerous with gamma

2 occurrences in this chapter

Equation form expr-333e0a1e27815d0c

GG

Read as: capital G

Means: capital G

5 occurrences in this chapter

Equation form expr-343121a3dd57d29d

b∈C\cardfont{b} \in C

Read as: the cardinal b belongs to capital C

Means: the cardinal b belongs to capital C

1 occurrence in this chapter

Equation form expr-44f56945a45a657f

b≼⋃X\cardle{\cardfont{b}}{\bigcup X}

Read as: the cardinal b has cardinality at most that of the union of capital X

Means: the cardinal b has cardinality at most that of the union of capital X

1 occurrence in this chapter

Equation form expr-4553b7f3532fddef

⟨A,R⟩\tuple{A, R}

Read as: the ordered pair capital A, then capital R

Means: the ordered pair capital A, then capital R

2 occurrences in this chapter

Equation form expr-4638f7b280414004

|B|\card{B}

Read as: the cardinality of capital B

Means: the cardinality of capital B

2 occurrences in this chapter

Equation form expr-47f27e36721ed78c

∃R(∀v∀y(Rvy→(Fv∧Gy))∧∀v(Fv→∃!yRvy)∧∀y(Gy→∃!vRvy))\exists R(&\forall v\forall y(Rvy \lif (Fv \land Gy)) \land {}\\ &\forall v(Fv \lif \lexists![y][Rvy]) \land {}\\ &\forall y(Gy \lif \lexists![v][Rvy]))

Read as: there exists a relation capital R such that three conditions hold. First, whenever capital R relates v to y, capital F holds of v and capital G holds of y. Second, every v satisfying capital F is related by capital R to exactly one y. Third, every y satisfying capital G is related by capital R to exactly one v. End conditions

Means: there exists a relation capital R such that three conditions hold. First, whenever capital R relates v to y, capital F holds of v and capital G holds of y. Second, every v satisfying capital F is related by capital R to exactly one y. Third, every y satisfying capital G is related by capital R to exactly one v. End conditions

1 occurrence in this chapter

Equation form expr-4824594a76d5ed04

|X|\card{X}

Read as: the cardinality of capital X

Means: the cardinality of capital X

2 occurrences in this chapter

Equation form expr-4893e9df8b5496eb

α\alpha

Read as: alpha

Means: alpha

1 occurrence in this chapter

Equation form expr-4b2c9ecd3e3cdcda

0≠10 \neq 1

Read as: zero is not equal to one

Means: zero is not equal to one

1 occurrence in this chapter

Equation form expr-4b68ab3847feda7d

XX

Read as: capital X

Means: capital X

3 occurrences in this chapter

Equation form expr-4d2eb930d95d1423

b\cardfont{b}

Read as: the cardinal b

Means: the cardinal b

2 occurrences in this chapter

Equation form expr-522df6ad7d1bb5ca

Z−\Zminus

Read as: set theory Z minus

Means: set theory Z minus

1 occurrence in this chapter

Equation form expr-52a3f81b9be81af1

F∼ΦF \sim \Phi

Read as: capital F is equivalent to capital Phi

Means: capital F is equivalent to capital Phi

1 occurrence in this chapter

Equation form expr-52c6aa20f5dea62f

|A|=|B| iff A≈B.This is very similar to what is now called Humes Principle, which says:#xF(x)=#xG(x) iff F∼G\card{A} = \card{B} & \text{ iff } A \approx B. \intertext{This is very similar to what is now called \emph{Hume's Principle}, which says:} \fregenum{x} {F(x)} = \fregenum{x}{G(x)} & \text{ iff } F \sim G

Read as: Source-ordered display. the cardinality of capital A equals the cardinality of capital B if and only if capital A is approximately equal to capital B. Then, This is very similar to what is now called Hume's Principle , which says:. Then, the Frege number of x capital F of x equals the Frege number of x capital G of x if and only if capital F is equivalent to capital G. End display

Means: Source-ordered display. the cardinality of capital A equals the cardinality of capital B if and only if capital A is approximately equal to capital B. Then, This is very similar to what is now called Hume's Principle , which says:. Then, the Frege number of x capital F of x equals the Frege number of x capital G of x if and only if capital F is equivalent to capital G. End display

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

22 occurrences in this chapter

Equation form expr-585831256f78241c

|A|≈A\cardeq{\card{A}}{A}

Read as: the cardinality of capital A is equinumerous with capital A

Means: the cardinality of capital A is equinumerous with capital A

1 occurrence in this chapter

Equation form expr-5d6cde2ac5d5e460

ϵxF(x)\fregeext{x}{F(x)}

Read as: the Frege extension of x capital F of x

Means: the Frege extension of x capital F of x

2 occurrences in this chapter

Equation form expr-5f7bdbc684866ab0

C={a:a is a cardinal}C = \Setabs{\cardfont{a}}{\cardfont{a} \text{ is a cardinal}}

Read as: capital C equals the set of the cardinal a such that the cardinal a is a cardinal

Means: capital C equals the set of the cardinal a such that the cardinal a is a cardinal

1 occurrence in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

3 occurrences in this chapter

Equation form expr-62c66a7a5dd70c31

mm

Read as: m

Means: m

1 occurrence in this chapter

Equation form expr-63447c603b04ac5b

δ≺γ\cardless{\delta}{\gamma}

Read as: delta has smaller cardinality than gamma

Means: delta has smaller cardinality than gamma

1 occurrence in this chapter

Equation form expr-669ac118248289e5

|A|\card{A}

Read as: the cardinality of capital A

Means: the cardinality of capital A

8 occurrences in this chapter

Equation form expr-67610b0632683369

γ\gamma

Read as: gamma

Means: gamma

5 occurrences in this chapter

Equation form expr-67986c51dd3a0875

A≈β\cardeq{A}{\beta}

Read as: capital A is equinumerous with beta

Means: capital A is equinumerous with beta

1 occurrence in this chapter

Equation form expr-68f534f2e97b908a

α≺b\cardless{\alpha}{\cardfont{b}}

Read as: alpha has smaller cardinality than the cardinal b

Means: alpha has smaller cardinality than the cardinal b

1 occurrence in this chapter

Equation form expr-69294beb8a4afbd0

α∈b∈X\alpha \in \cardfont{b} \in X

Read as: alpha belongs to the cardinal b belongs to capital X

Means: alpha belongs to the cardinal b belongs to capital X

1 occurrence in this chapter

Equation form expr-6a751984c342f99a

b>⋃C\cardfont{b} > \bigcup C

Read as: the cardinal b is greater than the union of capital C

Means: the cardinal b is greater than the union of capital C

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

2 occurrences in this chapter

Equation form expr-7046857daf500937

ZF\ZF

Read as: set theory Z F

Means: set theory Z F

3 occurrences in this chapter

Equation form expr-7b506dafe257cd66

a=ω\cardfont{a} = \omega

Read as: the cardinal a equals omega

Means: the cardinal a equals omega

1 occurrence in this chapter

Equation form expr-80e2db86aafbf42f

|A|≤ω\card{A} \leq \omega

Read as: the cardinality of capital A is less than or equal to omega

Means: the cardinality of capital A is less than or equal to omega

4 occurrences in this chapter

Equation form expr-865651a63451e07a

#(x=0∨x=1)\#x(x = 0 \lor x = 1)

Read as: the number of x such that x equals zero or x equals one

Means: the number of x such that x equals zero or x equals one

1 occurrence in this chapter

Equation form expr-88948c390bb8007c

#(x=0)\#x (x = 0)

Read as: the number of x such that x equals zero

Means: the number of x such that x equals zero

1 occurrence in this chapter

Equation form expr-8966968b2da9eb2d

n∈mn \in m

Read as: n belongs to m

Means: n belongs to m

2 occurrences in this chapter

Equation form expr-8a02145016ad595b

m∈nm \in n

Read as: m belongs to n

Means: m belongs to n

1 occurrence in this chapter

Equation form expr-8c2574892063f995

RR

Read as: capital R

Means: capital R

1 occurrence in this chapter

Equation form expr-8cbd4e98d6b0299c

f:A→Bf \colon A \to B

Read as: f colon capital A maps to capital B

Means: f colon capital A maps to capital B

1 occurrence in this chapter

Equation form expr-8ef757b30ec4c33c

δ≺A\cardless{\delta}{A}

Read as: delta has smaller cardinality than capital A

Means: delta has smaller cardinality than capital A

1 occurrence in this chapter

Equation form expr-8f1e69acbf519151

|A|∈ω\card{A} \in \omega

Read as: the cardinality of capital A belongs to omega

Means: the cardinality of capital A belongs to omega

1 occurrence in this chapter

Equation form expr-9061fa7edf06712e

#xF(x)={G:F∼G}\fregenum{x}{F(x)} = \Setabs{G}{F \sim G}

Read as: the Frege number of x capital F of x equals the set of capital G such that capital F is equivalent to capital G

Means: the Frege number of x capital F of x equals the set of capital G such that capital F is equivalent to capital G

1 occurrence in this chapter

Equation form expr-9e5a15c483dc462b

|A|∉ω\card{A} \notin \omega

Read as: the cardinality of capital A does not belong to omega

Means: the cardinality of capital A does not belong to omega

1 occurrence in this chapter

Equation form expr-9f4d7ba9e3683661

n≈A\cardeq{n}{A}

Read as: n is equinumerous with capital A

Means: n is equinumerous with capital A

1 occurrence in this chapter

Equation form expr-9f7455f7ac427be2

A→BA\to B

Read as: capital A maps to capital B

Means: capital A maps to capital B

1 occurrence in this chapter

Equation form expr-a7ec629e5fbab783

n⊊mn \subsetneq m

Read as: n is a proper subset of m

Means: n is a proper subset of m

1 occurrence in this chapter

Equation form expr-aa79c7b00d029717

|A|=|B| iff A≈B.\card{A} = \card{B} \text{ iff } \cardeq{A}{B}.

Read as: the cardinality of capital A equals the cardinality of capital B if and only if capital A is equinumerous with capital B

Means: the cardinality of capital A equals the cardinality of capital B if and only if capital A is equinumerous with capital B

1 occurrence in this chapter

Equation form expr-aba53199c3c0a4dd

|B|≤|A|\card{B} \leq \card{A}

Read as: the cardinality of capital B is less than or equal to the cardinality of capital A

Means: the cardinality of capital B is less than or equal to the cardinality of capital A

1 occurrence in this chapter

Equation form expr-aec0c9d63d542980

|A|=0∈ω\card{A} = 0 \in \omega

Read as: the cardinality of capital A equals zero belongs to omega

Means: the cardinality of capital A equals zero belongs to omega

1 occurrence in this chapter

Equation form expr-b27e3e39b7a6d678

n,m∈ωn, m \in \omega

Read as: n, then m belongs to omega

Means: n, then m belongs to omega

1 occurrence in this chapter

Equation form expr-b4668eacf7fe6b79

A≈B\cardeq{A}{B}

Read as: capital A is equinumerous with capital B

Means: capital A is equinumerous with capital B

3 occurrences in this chapter

Equation form expr-b5c8ccdf1d6b1395

#xF(x)\fregenum{x}{F(x)}

Read as: the Frege number of x capital F of x

Means: the Frege number of x capital F of x

3 occurrences in this chapter

Equation form expr-b6303c4a80c164db

a<b\cardfont{a} < \cardfont{b}

Read as: the cardinal a is less than the cardinal b

Means: the cardinal a is less than the cardinal b

1 occurrence in this chapter

Equation form expr-b87140b8e470dd31

|A|=||A||\card{A} = \card{\card{A}}

Read as: the cardinality of capital A equals the cardinality of the cardinality of capital A

Means: the cardinality of capital A equals the cardinality of the cardinality of capital A

1 occurrence in this chapter

Equation form expr-b8f9f0da7598215a

|A|=γ\card{A} = \gamma

Read as: the cardinality of capital A equals gamma

Means: the cardinality of capital A equals gamma

1 occurrence in this chapter

Equation form expr-b907117d6e98daa0

A≈B iff |A|=|B|A≼B iff |A|≤|B|A≺B iff |A|<|B|\cardeq{A}{B} &\text{ iff } \card{A} = \card{B}\\ \cardle{A}{B} &\text{ iff } \card{A} \leq \card{B}\\ \cardless{A}{B}&\text{ iff } \card{A} < \card{B}

Read as: Source-ordered display. capital A is equinumerous with capital B if and only if the cardinality of capital A equals the cardinality of capital B. Then, capital A has cardinality at most that of capital B if and only if the cardinality of capital A is less than or equal to the cardinality of capital B. Then, capital A has smaller cardinality than capital B if and only if the cardinality of capital A is less than the cardinality of capital B. End display

Means: Source-ordered display. capital A is equinumerous with capital B if and only if the cardinality of capital A equals the cardinality of capital B. Then, capital A has cardinality at most that of capital B if and only if the cardinality of capital A is less than or equal to the cardinality of capital B. Then, capital A has smaller cardinality than capital B if and only if the cardinality of capital A is less than the cardinality of capital B. End display

1 occurrence in this chapter

Equation form expr-bc3b5864a25454f5

x=0x = 0

Read as: x equals zero

Means: x equals zero

1 occurrence in this chapter

Equation form expr-bec65ba5ed754f42

0≠20\neq 2

Read as: zero is not equal to two

Means: zero is not equal to two

1 occurrence in this chapter

Equation form expr-c00f6fe4816249e7

α=β+1\alpha = \beta \ordplus 1

Read as: alpha equals beta ordinal plus one

Means: alpha equals beta ordinal plus one

1 occurrence in this chapter

Equation form expr-c0660778db5dafb5

|A|=ω\card{A} = \omega

Read as: the cardinality of capital A equals omega

Means: the cardinality of capital A equals omega

1 occurrence in this chapter

Equation form expr-cd784b071bc61fe4

|A|≤|B|\card{A} \leq \card{B}

Read as: the cardinality of capital A is less than or equal to the cardinality of capital B

Means: the cardinality of capital A is less than or equal to the cardinality of capital B

1 occurrence in this chapter

Equation form expr-d00767c1ff333812

κ,λ\kappa, \lambda

Read as: kappa, then lambda

Means: kappa, then lambda

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: two

1 occurrence in this chapter

Equation form expr-d4a6ee4f6340ba04

b≤⋃C\cardfont{b} \leq \bigcup C

Read as: the cardinal b is less than or equal to the union of capital C

Means: the cardinal b is less than or equal to the union of capital C

1 occurrence in this chapter

Equation form expr-d4d031a7b4667b96

α∈⋃X\alpha \in \bigcup X

Read as: alpha belongs to the union of capital X

Means: alpha belongs to the union of capital X

1 occurrence in this chapter

Equation form expr-d65e5a7de9994536

ω≈A\cardeq{\omega}{A}

Read as: omega is equinumerous with capital A

Means: omega is equinumerous with capital A

1 occurrence in this chapter

Equation form expr-da0ee11ec8d2fe33

A≼ω\cardle{A}{\omega}

Read as: capital A has cardinality at most that of omega

Means: capital A has cardinality at most that of omega

1 occurrence in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: capital B

Means: capital B

4 occurrences in this chapter

Equation form expr-e2faa02826c74c82

ω<|A|\omega < \card{A}

Read as: omega is less than the cardinality of capital A

Means: omega is less than the cardinality of capital A

1 occurrence in this chapter

Equation form expr-e349851823caf578

ZFC\ZFC

Read as: set theory Z F C

Means: set theory Z F C

7 occurrences in this chapter

Equation form expr-eb18232dc3c96a91

b⊆⋃C\cardfont{b} \subseteq \bigcup{C}

Read as: the cardinal b is a subset of the union of capital C

Means: the cardinal b is a subset of the union of capital C

1 occurrence in this chapter

Equation form expr-f11aed8b3c261db5

B≼A\cardle{B}{A}

Read as: capital B has cardinality at most that of capital A

Means: capital B has cardinality at most that of capital A

2 occurrences in this chapter

Equation form expr-f3b9fd9c94e230f5

x≠xx \neq x

Read as: x is not equal to x

Means: x is not equal to x

1 occurrence in this chapter

Equation form expr-f3c5ce8d27fb8bb6

|A|=n∈ω\card{A} = n \in \omega

Read as: the cardinality of capital A equals n belongs to omega

Means: the cardinality of capital A equals n belongs to omega

1 occurrence in this chapter

Equation form expr-f3f3804480e8551a

β\beta

Read as: beta

Means: beta

3 occurrences in this chapter

Equation form expr-f497d36773855a61

α≉⋃X\cardneq{\alpha}{\bigcup X}

Read as: alpha is not equinumerous with the union of capital X

Means: alpha is not equinumerous with the union of capital X

1 occurrence in this chapter

Equation form expr-f67ab10ad4e4c531

FF

Read as: capital F

Means: capital F

7 occurrences in this chapter

Equation form expr-f804a26ff13002e9

a=|a|≤ω\cardfont{a} = \card{\cardfont{a}} \leq \omega

Read as: the cardinal a equals the cardinality of the cardinal a is less than or equal to omega

Means: the cardinal a equals the cardinality of the cardinal a is less than or equal to omega

1 occurrence in this chapter

Equation form expr-fa45014f67cf2d9b

(x=0∨x=1)(x = 0 \lor x = 1)

Read as: open scope, x equals zero or x equals one, close scope

Means: open scope, x equals zero or x equals one, close scope

1 occurrence in this chapter

Equation form expr-fa54f43189336117

f:m→nf \colon m \to n

Read as: f colon m maps to n

Means: f colon m maps to n

1 occurrence in this chapter

Equation form expr-fc6d010716d8b04d

α≠|α|\alpha \neq \card{\alpha}

Read as: alpha is not equal to the cardinality of alpha

Means: alpha is not equal to the cardinality of alpha

1 occurrence in this chapter

Equation form expr-fc70109744fc3a31

F∼GF \sim G

Read as: capital F is equivalent to capital G

Means: capital F is equivalent to capital G

1 occurrence in this chapter

Equation form expr-fd0ecad03de6a9fb

|β|=|β+1|=|α|\card{\beta} = \card{\beta\ordplus 1} = \card{\alpha}

Read as: the cardinality of beta equals the cardinality of beta ordinal plus one

Means: the cardinality of beta equals the cardinality of beta ordinal plus one

1 occurrence in this chapter

Equation form expr-fe21948b09464fd0

⋃X\bigcup X

Read as: the union of capital X

Means: the union of capital X

3 occurrences in this chapter

Definition one in this chapter

This source definition contains, in source order: capital A; then the cardinality of capital A; then gamma; then capital A is equinumerous with gamma; then gamma; then gamma; then gamma equals the cardinality of gamma. The complete surrounding source prose remains in the continuous listener stream.

Source

Axiom: Well-Ordering

This source axiom contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Lemma one in this chapter

This source lemma contains, in source order: capital A; then the cardinality of capital A; then the cardinality of capital A is equinumerous with capital A; then the cardinality of capital A; then the cardinality of capital A equals the cardinality of the cardinality of capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma two in this chapter

This source lemma contains, in source order: capital A; then capital B; then Source-ordered display. capital A is equinumerous with capital B if and only if the cardinality of capital A equals the cardinality of capital B. Then, capital A has cardinality at most that of capital B if and only if the cardinality of capital A is less than or equal to the cardinality of capital B. Then, capital A has smaller cardinality than capital B if and only if the cardinality of capital A is less than the cardinality of capital B. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math one in this chapter

This source display math contains, in source order: Source-ordered display. capital A is equinumerous with capital B if and only if the cardinality of capital A equals the cardinality of capital B. Then, capital A has cardinality at most that of capital B if and only if the cardinality of capital A is less than or equal to the cardinality of capital B. Then, capital A has smaller cardinality than capital B if and only if the cardinality of capital A is less than the cardinality of capital B. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Cardinality comparison diagram

Commutative comparison diagram. On the top row, capital A maps by an injection to capital B. Vertical bijections connect those two sets, respectively, with the cardinality of capital A and the cardinality of capital B. A dashed bottom arrow is the resulting injection between the two cardinalities. End of diagram.

Source

Definition two in this chapter

This source definition contains, in source order: set theory Z F C; then set theory Z F C; then set theory Z F. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition one in this chapter

This source proposition contains, in source order: n, then m belongs to omega; then n equals m; then n is equinumerous with m. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary one in this chapter

This source corollary contains, in source order: n belongs to omega; then n. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem one in this chapter

This source theorem contains, in source order: capital A; then the cardinality of capital A does not belong to omega; then capital A; then omega is less than or equal to the cardinality of capital A; then capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition three in this chapter

This source definition contains, in source order: capital A; then the cardinality of capital A; then the cardinality of capital A belongs to omega; then capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary two in this chapter

This source corollary contains, in source order: omega. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary three in this chapter

This source corollary contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Proposition two in this chapter

This source proposition contains, in source order: capital A; then the cardinality of capital A is less than or equal to omega; then capital A; then omega is less than the cardinality of capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary four in this chapter

This source corollary contains, in source order: omega. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition three in this chapter

This source proposition contains, in source order: capital X; then the union of capital X. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem two in this chapter

This source theorem contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Theorem three in this chapter

This source theorem contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream.

Source

Display math two in this chapter

This source display math contains, in source order: Source-ordered display. the cardinality of capital A equals the cardinality of capital B if and only if capital A is approximately equal to capital B. Then, This is very similar to what is now called Hume's Principle , which says:. Then, the Frege number of x capital F of x equals the Frege number of x capital G of x if and only if capital F is equivalent to capital G. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math three in this chapter

This source display math contains, in source order: there exists a relation capital R such that three conditions hold. First, whenever capital R relates v to y, capital F holds of v and capital G holds of y. Second, every v satisfying capital F is related by capital R to exactly one y. Third, every y satisfying capital G is related by capital R to exactly one v. End conditions. The complete surrounding source prose remains in the continuous listener stream.

Source

Cross-reference reference-001459

section “Von Neumann's Construction of the Ordinals” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001460

corollary six in chapter “Ordinals”

Source occurrence

Cross-reference reference-001461

section “Equinumerosity” in chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001462

section “Appendix: Hume's Principle” in chapter “Cardinals”

Source occurrence

Cross-reference reference-001463

definition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001464

definition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001465

definition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001466

chapter “Choice”

Source occurrence

Cross-reference reference-001467

definition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001468

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001469

item 1 of lemma one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001470

item 2 of lemma one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001471

item 3 of lemma one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001472

proposition “Equinumerosity as an equivalence relation” in chapter “The Size of Sets”

Source occurrence

Cross-reference reference-001473

the lemma on Cardinals Exist

Source occurrence

Cross-reference reference-001474

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001475

the theorem on schroder bernstein

Source occurrence

Cross-reference reference-001476

section “Appendix: Proving Schröder-Bernstein” in chapter “Infinite Sets”

Source occurrence

Cross-reference reference-001477

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001478

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001479

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001480

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001481

section “Appendix: Proving Schröder-Bernstein” in chapter “Infinite Sets”

Source occurrence

Cross-reference reference-001482

theorem “in set theory Z F” in chapter “Choice”

Source occurrence

Cross-reference reference-001483

definition of the natural numbers and omega in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001484

proposition that natural numbers are not Dedekind infinite in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001485

lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001486

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001487

corollary one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001488

part “Naïve Set Theory”

Source occurrence

Cross-reference reference-001489

chapter “Choice”

Source occurrence

Cross-reference reference-001490

proposition that natural numbers are not Dedekind infinite in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001491

proposition one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001492

lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001493

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001494

the definition on enumerable

Source occurrence

Cross-reference reference-001495

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001496

corollary two in chapter “Cardinals”

Source occurrence

Cross-reference reference-001497

the definition on enumerable

Source occurrence

Cross-reference reference-001498

corollary one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001499

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001500

corollary one in chapter “Cardinals”

Source occurrence

Cross-reference reference-001501

the lemma on Cardinals Behave Right

Source occurrence

Cross-reference reference-001502

corollary two in chapter “Cardinals”

Source occurrence

Cross-reference reference-001503

the theorem on cantor

Source occurrence

Cross-reference reference-001504

the lemma on Cardinals Exist

Source occurrence

Cross-reference reference-001505

proposition three in chapter “Cardinals”

Source occurrence

Cross-reference reference-001506

the lemma on No Largest Cardinal

Source occurrence

Cross-reference reference-001507

section “Cantor's Principle” in chapter “Cardinals”

Source occurrence

Cross-reference reference-001508

(David Hume, 1740, Pt.III Bk.1 §1)

Source occurrence

Cross-reference reference-001509

Gottlob Frege (1884), §63

Source occurrence

Cross-reference reference-001510

section “Appendix: Frege's Basic Law V” in chapter “The Iterative Conception”

Source occurrence

Cross-reference reference-001511

section “Predicative and Impredicative” in chapter “The Iterative Conception”

Source occurrence

Cross-reference reference-001512

section “Appendix: Frege's Basic Law V” in chapter “The Iterative Conception”

Source occurrence

Ordered structures

Cardinality comparison diagram

Structure: diagram tikz.

Commutative comparison diagram. On the top row, capital A maps by an injection to capital B. Vertical bijections connect those two sets, respectively, with the cardinality of capital A and the cardinality of capital B. A dashed bottom arrow is the resulting injection between the two cardinalities. End of diagram.

Read the source-bound structure in context