Set Theory

Ordinals

Equation form expr-00b7e9bac079cbe4

ϕ(∅)\phi(\emptyset)

Read as: phi of the empty set

Means: phi of the empty set

1 occurrence in this chapter

Equation form expr-021d1aa38739acfd

a1<a2 iff b1⋖b2{a_1} < {a_2} \text{ iff }{b_1} \lessdot {b_2}

Read as: a subscript one is less than a subscript two if and only if b subscript one is strictly smaller than b subscript two

Means: a subscript one is less than a subscript two if and only if b subscript one is strictly smaller than b subscript two

1 occurrence in this chapter

Equation form expr-043a2cc0d4d495a0

{τ(x):x∈A}={y:(∃x∈A)y=τ(x)}.\Setabs{\tau(x)}{x \in A} = \Setabs{y}{(\exists x \in A)y = \tau(x)}.

Read as: the set of tau of x such that x belongs to capital A, equals the set of y such that there exists x in capital A, y equals tau of x

Means: the set of tau of x such that x belongs to capital A, equals the set of y such that there exists x in capital A, y equals tau of x

1 occurrence in this chapter

Equation form expr-0758bba7c8612719

(∀x∈A)∃!yϕ(x,y)(\forall x \in A)\lexists![y][\phi(x,y)]

Read as: for every x in capital A, there exists exactly one y, phi of x and y

Means: for every x in capital A, there exists exactly one y, phi of x and y

1 occurrence in this chapter

Equation form expr-07d0e80a6b2c851e

ran(f)\ran{f}

Read as: the range of f

Means: the range of f

2 occurrences in this chapter

Equation form expr-083d3203faaa475e

ord(A,<)=ord(B,⋖) iff ⟨A,<⟩≅⟨B,⋖⟩ord(A,<)<ord(B,⋖) iff ⟨A,<⟩≅⟨Bb,⋖b⟩ for some b∈B\ordtype{A, <} = \ordtype{B, \lessdot} & \text{ iff } \ordeq{\tuple{A, <}}{\tuple{B, \lessdot}}\\ \ordtype{A, <} < \ordtype{B, \lessdot}& \text{ iff }\ordeq{\tuple{A, <}}{\tuple{B_b, \lessdot_b}}\text{ for some }b \in B

Read as: Two proposed order-type principles. First, the order type of capital A under less-than equals the order type of capital B under less-dot if and only if those two well-orderings are order-isomorphic. Second, the first order type is less than the second if and only if the first well-ordering is order-isomorphic to the initial segment of capital B below some b. End principles

Means: Two proposed order-type principles. First, the order type of capital A under less-than equals the order type of capital B under less-dot if and only if those two well-orderings are order-isomorphic. Second, the first order type is less than the second if and only if the first well-ordering is order-isomorphic to the initial segment of capital B below some b. End principles

1 occurrence in this chapter

Equation form expr-089df62f42744dab

¬ϕ(a)\lnot \phi(a)

Read as: not phi of a

Means: not phi of a

1 occurrence in this chapter

Equation form expr-0928828231e90719

a∈Aa \in A

Read as: a belongs to capital A

Means: a belongs to capital A

5 occurrences in this chapter

Equation form expr-0a949794c34081dd

(∀β∈γ)ϕ(β)(\forall \beta \in \gamma)\phi(\beta)

Read as: for every beta in gamma, phi of beta

Means: for every beta in gamma, phi of beta

1 occurrence in this chapter

Equation form expr-0b21fd0c5b88d80e

b<ab < a

Read as: b is less than a

Means: b is less than a

1 occurrence in this chapter

Equation form expr-0dbe18f633a4d9a6

{y:(∃x∈A)τ(x)=y}\Setabs{y}{(\exists x \in A)\tau(x) = y}

Read as: the set of y such that there exists x in capital A, tau of x equals y

Means: the set of y such that there exists x in capital A, tau of x equals y

1 occurrence in this chapter

Equation form expr-0e4f901aae77382f

sup(X)\text{sup}(X)

Read as: the supremum of capital X

Means: the supremum of capital X

2 occurrences in this chapter

Equation form expr-0f1a2d835bf992d1

β+=α\ordsucc{\beta} = \alpha

Read as: the ordinal successor of beta equals alpha

Means: the ordinal successor of beta equals alpha

1 occurrence in this chapter

Equation form expr-10ce0200b450ea95

ϕ\phi

Read as: phi

Means: phi

2 occurrences in this chapter

Equation form expr-11baa595827a4e0f

ω\omega

Read as: omega

Means: omega

5 occurrences in this chapter

Equation form expr-133566a053ef68c4

γ∈dom(f)\gamma \in \dom{f}

Read as: gamma belongs to the domain of f

Means: gamma belongs to the domain of f

1 occurrence in this chapter

Equation form expr-14c5137ad7065181

α∈lsub(X)\alpha \in \supstrict(X)

Read as: alpha belongs to the strict supremum of capital X

Means: alpha belongs to the strict supremum of capital X

2 occurrences in this chapter

Equation form expr-17398d1ee9f4bc45

AaA_a

Read as: capital A subscript a

Means: capital A subscript a

1 occurrence in this chapter

Equation form expr-185ce3b3df1d4897

∀αϕ(α)\forall \alpha \phi(\alpha)

Read as: for every alpha, phi of alpha

Means: for every alpha, phi of alpha

1 occurrence in this chapter

Equation form expr-18ae661dbfa4d037

lsub(X)\supstrict(X)

Read as: the strict supremum of capital X

Means: the strict supremum of capital X

4 occurrences in this chapter

Equation form expr-1e93f11e638811cb

x⋖f(a)x \lessdot f(a)

Read as: x is strictly smaller than f of a

Means: x is strictly smaller than f of a

2 occurrences in this chapter

Equation form expr-223c4d2008755108

a=ba = b

Read as: a equals b

Means: a equals b

1 occurrence in this chapter

Equation form expr-245c2c3b3d0e0c4d

τ[A]\funimage{\tau}{A}

Read as: the image of capital A under tau

Means: the image of capital A under tau

1 occurrence in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: f

12 occurrences in this chapter

Equation form expr-28ffed969bb6ba98

α∉α\alpha \notin \alpha

Read as: alpha does not belong to alpha

Means: alpha does not belong to alpha

1 occurrence in this chapter

Equation form expr-2d711642b726b044

xx

Read as: x

Means: x

4 occurrences in this chapter

Equation form expr-2da4e8a803e7e0bf

β≅⟨Bb,⋖b⟩\ordeq{\beta}{\tuple{B_b, \lessdot_b}}

Read as: beta is order-isomorphic to the initial segment capital B subscript b under restricted less-dot

Means: beta is order-isomorphic to the initial segment capital B subscript b under restricted less-dot

1 occurrence in this chapter

Equation form expr-2ed15f42dc53d5c3

α0>α1>α2>…\alpha_0 > \alpha_1 > \alpha_2 > \ldots

Read as: alpha subscript zero is greater than alpha subscript one, which is greater than alpha subscript two, and the descending sequence continues

Means: alpha subscript zero is greater than alpha subscript one, which is greater than alpha subscript two, and the descending sequence continues

1 occurrence in this chapter

Equation form expr-2fa32a2552f483a7

f={⟨a,b⟩∈A×B:⟨Aa,<a⟩≅⟨Bb,⋖b⟩}.f = \Setabs{\tuple{a, b} \in A \times B}{ \ordeq{\tuple{A_a, <_a}}{\tuple{B_b, \lessdot_b}}}.

Read as: f equals the set of ordered pairs a and b in the Cartesian product of capital A and capital B such that the initial segment of capital A below a is order-isomorphic to the initial segment of capital B below b

Means: f equals the set of ordered pairs a and b in the Cartesian product of capital A and capital B such that the initial segment of capital A below a is order-isomorphic to the initial segment of capital B below b

1 occurrence in this chapter

Equation form expr-305fe666789b0f84

γ=α+\gamma = \ordsucc{\alpha}

Read as: gamma equals the ordinal successor of alpha

Means: gamma equals the ordinal successor of alpha

1 occurrence in this chapter

Equation form expr-307136f10de6ef37

⋖\lessdot

Read as: is strictly smaller than

Means: is strictly smaller than

1 occurrence in this chapter

Equation form expr-31440f54fb4f03d2

α∈β∨α=β∨β∈α\alpha \in \beta \lor \alpha = \beta \lor \beta \in \alpha

Read as: alpha belongs to beta or alpha equals beta or beta belongs to alpha

Means: alpha belongs to beta or alpha equals beta or beta belongs to alpha

1 occurrence in this chapter

Equation form expr-316a5728201e9b4b

ran(f)=B\ran{f} = B

Read as: the range of f equals capital B

Means: the range of f equals capital B

1 occurrence in this chapter

Equation form expr-32a129330a8e13f1

a1∈dom(f)a_1 \in \dom{f}

Read as: a subscript one belongs to the domain of f

Means: a subscript one belongs to the domain of f

1 occurrence in this chapter

Equation form expr-3333d924e1a68074

(∀n∈ω)((∀m<n)ϕ(m)→ϕ(n))(\forall n \in \omega)((\forall m < n)\phi(m) \lif \phi(n))

Read as: for every n in omega, if phi holds for every m less than n, then phi holds for n

Means: for every n in omega, if phi holds for every m less than n, then phi holds for n

1 occurrence in this chapter

Equation form expr-34523ac6703a1a88

⋖=<a\mathord{\lessdot} = \mathord{<_a}

Read as: the less-dot relation equals the less-than relation restricted below a

Means: the less-dot relation equals the less-than relation restricted below a

1 occurrence in this chapter

Equation form expr-3593a87feec74cf7

α∈β+∈Y\alpha \in \ordsucc{\beta} \in Y

Read as: alpha belongs to the ordinal successor of beta, and that successor belongs to capital Y

Means: alpha belongs to the ordinal successor of beta, and that successor belongs to capital Y

1 occurrence in this chapter

Equation form expr-363435b5ce5f64a2

β⊆α\beta \subseteq \alpha

Read as: beta is a subset of alpha

Means: beta is a subset of alpha

3 occurrences in this chapter

Equation form expr-38c815663bfb7ec0

⟨B,⋖⟩\tuple{B, \lessdot}

Read as: the ordered pair consisting of capital B and the less-dot relation

Means: the ordered pair consisting of capital B and the less-dot relation

6 occurrences in this chapter

Equation form expr-3958848973f252eb

if ∃αϕ(α), then ∃α(ϕ(α)∧(∀β∈α)¬ϕ(β))\text{if }\exists \alpha \phi(\alpha)\text{, then }\exists \alpha(\phi(\alpha) \land (\forall \beta \in \alpha) \lnot \phi(\beta))

Read as: if some alpha satisfies phi, then there is an alpha satisfying phi such that no beta belonging to alpha satisfies phi

Means: if some alpha satisfies phi, then there is an alpha satisfying phi such that no beta belonging to alpha satisfies phi

1 occurrence in this chapter

Equation form expr-3a4a4c1a7b43aea5

f[A]={f(x):x∈A}\funimage{f}{A} = \Setabs{f(x)}{x \in A}

Read as: the image of capital A under f equals the set of f of x such that x belongs to capital A

Means: the image of capital A under f equals the set of f of x such that x belongs to capital A

1 occurrence in this chapter

Equation form expr-3b10bc53049617cb

f:β→⟨B,⋖⟩f \colon \beta \to \tuple {B, \lessdot}

Read as: f maps beta isomorphically to capital B under the less-dot relation

Means: f maps beta isomorphically to capital B under the less-dot relation

1 occurrence in this chapter

Equation form expr-3b2591dd01df745a

x∈Bx \in B

Read as: x belongs to capital B

Means: x belongs to capital B

1 occurrence in this chapter

Equation form expr-3d830583ad97e207

f(x)⋖f(y)f(x) \lessdot f(y)

Read as: f of x is strictly smaller than f of y

Means: f of x is strictly smaller than f of y

1 occurrence in this chapter

Equation form expr-3e23e8160039594a

bb

Read as: b

Means: b

6 occurrences in this chapter

Equation form expr-3e3e36bfd1feb5a0

g:B→Cg \colon B \to C

Read as: g maps capital B to capital C

Means: g maps capital B to capital C

1 occurrence in this chapter

Equation form expr-3ebf1bb63c2c8868

O∈OO \in O

Read as: capital O belongs to capital O

Means: capital O belongs to capital O

1 occurrence in this chapter

Equation form expr-403b41511aaabb24

α∈β∨α=β\alpha \in \beta \lor \alpha = \beta

Read as: alpha belongs to beta or alpha equals beta

Means: alpha belongs to beta or alpha equals beta

1 occurrence in this chapter

Equation form expr-41a41ea281c260a8

α≤β∈X\alpha \leq \beta \in X

Read as: alpha is less than or equal to beta, and beta belongs to capital X

Means: alpha is less than or equal to beta, and beta belongs to capital X

1 occurrence in this chapter

Equation form expr-46e367ad911fd441

b1⋖b2{b_1} \lessdot {b_2}

Read as: b subscript one is strictly smaller than b subscript two

Means: b subscript one is strictly smaller than b subscript two

1 occurrence in this chapter

Equation form expr-47caee382a6a8253

⟨A,<⟩≅α\ordeq{\tuple{A, < }}{\alpha}

Read as: capital A under less-than is order-isomorphic to alpha

Means: capital A under less-than is order-isomorphic to alpha

1 occurrence in this chapter

Equation form expr-4893e9df8b5496eb

α\alpha

Read as: alpha

Means: alpha

32 occurrences in this chapter

Equation form expr-496ff862ff24757d

α∈β iff f↾α:α→Bf(α) is an isomorphism iff ⟨A,<⟩≅⟨Bf(α),⋖f(α)⟩ iff ⟨A,<⟩≅⟨Bb,⋖b⟩ for some b∈B\alpha \in \beta&\text{ iff }\funrestrictionto{f}{\alpha} \colon \alpha \to B_{f(\alpha)}\text{ is an isomorphism}\\ &\text{ iff }\ordeq{\tuple{A, <}}{\tuple{B_{f(\alpha)}, \lessdot_{f(\alpha)}}}\\ &\text{ iff }\ordeq{\tuple{A, <}}{\tuple{B_b, \lessdot_b}}\text{ for some $b \in B$}

Read as: Three equivalent conditions. Alpha belongs to beta if and only if the restriction of f to alpha is an isomorphism from alpha to the initial segment of capital B below f of alpha. This holds if and only if capital A under less-than is order-isomorphic to that initial segment, and if and only if it is order-isomorphic to the initial segment of capital B below some b. End equivalences

Means: Three equivalent conditions. Alpha belongs to beta if and only if the restriction of f to alpha is an isomorphism from alpha to the initial segment of capital B below f of alpha. This holds if and only if capital A under less-than is order-isomorphic to that initial segment, and if and only if it is order-isomorphic to the initial segment of capital B below some b. End equivalences

1 occurrence in this chapter

Equation form expr-49c95203d0c3c99f

(∀b<a)f(b)=g(b)(\forall b < a)f(b) = g(b)

Read as: for every b less than a, f of b equals g of b

Means: for every b less than a, f of b equals g of b

1 occurrence in this chapter

Equation form expr-4b227777d4dd1fc6

44

Read as: four

Means: four

1 occurrence in this chapter

Equation form expr-4b68ab3847feda7d

XX

Read as: capital X

Means: capital X

10 occurrences in this chapter

Equation form expr-4b7ce6b75f0ec36f

ϕ(x)\phi(x)

Read as: phi of x

Means: phi of x

3 occurrences in this chapter

Equation form expr-4b9193d12c947630

α→B\alpha \to B

Read as: functions from alpha to capital B

Means: functions from alpha to capital B

1 occurrence in this chapter

Equation form expr-4c9bc29932ade64f

f:Aa2→Bb2f \colon A_{{a_2}} \to B_{{b_2}}

Read as: f maps capital A subscript a subscript two to capital B subscript b subscript two

Means: f maps capital A subscript a subscript two to capital B subscript b subscript two

1 occurrence in this chapter

Equation form expr-4db9fec3bb2f4f77

α⊆β∨β⊆α\alpha \subseteq \beta \lor \beta \subseteq \alpha

Read as: alpha is a subset of beta or beta is a subset of alpha

Means: alpha is a subset of beta or beta is a subset of alpha

1 occurrence in this chapter

Equation form expr-4de611f5e3b7475a

ϕ(α+)\phi(\ordsucc{\alpha})

Read as: phi of the ordinal successor of alpha

Means: phi of the ordinal successor of alpha

1 occurrence in this chapter

Equation form expr-4de8c0e6ad0a36eb

α≤β\alpha \leq \beta

Read as: alpha is less than or equal to beta

Means: alpha is less than or equal to beta

1 occurrence in this chapter

Equation form expr-4e07408562bedb8b

33

Read as: three

Means: three

1 occurrence in this chapter

Equation form expr-4fb37c575281da58

α+∈Y\ordsucc{\alpha} \in Y

Read as: the ordinal successor of alpha belongs to capital Y

Means: the ordinal successor of alpha belongs to capital Y

1 occurrence in this chapter

Equation form expr-522df6ad7d1bb5ca

Z−\Zminus

Read as: set theory Z minus

Means: set theory Z minus

2 occurrences in this chapter

Equation form expr-527168bf4f68bf9a

α∈β∈O\alpha \in \beta \in O

Read as: alpha belongs to beta, and beta belongs to capital O

Means: alpha belongs to beta, and beta belongs to capital O

1 occurrence in this chapter

Equation form expr-530e566d05205c99

∀A[(∀x∈A)∃!yϕ(x,y)→∃B∀y(y∈B↔(∃x∈A)ϕ(x,y))]\forall A[(\forall x \in A)\lexists![y][\phi(x,y)] \lif \exists B\forall y (y \in B \liff (\exists x \in A)\phi(x,y))]

Read as: for every capital A, if every x in capital A has exactly one y such that phi holds of x and y, then there exists a capital B such that, for every y, y belongs to capital B if and only if phi holds of x and y for some x in capital A

Means: for every capital A, if every x in capital A has exactly one y such that phi holds of x and y, then there exists a capital B such that, for every y, y belongs to capital B if and only if phi holds of x and y for some x in capital A

1 occurrence in this chapter

Equation form expr-53993ffa4fd6146e

∃α(ϕ(α)∧∀β(ϕ(β)→α≤β))\exists \alpha(\phi(\alpha) \land \forall \beta(\phi(\beta) \lif \alpha \leq \beta))

Read as: there exists an alpha satisfying phi such that every beta satisfying phi is greater than or equal to alpha

Means: there exists an alpha satisfying phi such that every beta satisfying phi is greater than or equal to alpha

1 occurrence in this chapter

Equation form expr-5457674e632558b5

α⊆β\alpha \subseteq \beta

Read as: alpha is a subset of beta

Means: alpha is a subset of beta

4 occurrences in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

23 occurrences in this chapter

Equation form expr-559bd91e2be0ade0

<a<_a

Read as: the less-than relation restricted below a

Means: the less-than relation restricted below a

1 occurrence in this chapter

Equation form expr-55d1823545b5b4c3

τ\tau

Read as: tau

Means: tau

3 occurrences in this chapter

Equation form expr-57a882ce825ae828

(∀a∈A)f(a)=g(a)(\forall a \in A)f(a) = g(a)

Read as: for every a in capital A, f of a equals g of a

Means: for every a in capital A, f of a equals g of a

1 occurrence in this chapter

Equation form expr-5ac9047330d456ed

α∈β∈α+\alpha \in \beta \in \ordsucc{\alpha}

Read as: alpha belongs to beta, and beta belongs to the ordinal successor of alpha

Means: alpha belongs to beta, and beta belongs to the ordinal successor of alpha

2 occurrences in this chapter

Equation form expr-5b7e81fd3330f7e3

b≠f(b)b \neq f(b)

Read as: b is not equal to f of b

Means: b is not equal to f of b

1 occurrence in this chapter

Equation form expr-5bc5e9a6bcde9ba8

ord(A,<)\ordtype{A, <}

Read as: the order type of capital A under the less-than relation

Means: the order type of capital A under the less-than relation

3 occurrences in this chapter

Equation form expr-5c7cd7a3181b8596

<↾Aa2\funrestrictionto{\mathord{<}}{A_a^2}

Read as: the restriction of the less-than relation to capital A subscript a squared

Means: the restriction of the less-than relation to capital A subscript a squared

1 occurrence in this chapter

Equation form expr-5ce5fbbc12221457

f(γ)⋖f(β)f(\gamma) \lessdot f(\beta)

Read as: f of gamma is strictly smaller than f of beta

Means: f of gamma is strictly smaller than f of beta

1 occurrence in this chapter

Equation form expr-5ebae97925e93364

⟨Bb1,⋖b1⟩≅⟨Bf(a1),⋖f(a1)⟩\ordeq{\tuple{B_{b_1}, \lessdot_{b_1}}}{\tuple{B_{f({a_1})}, \lessdot_{f({a_1})}}}

Read as: the initial segment capital B subscript b subscript one is order-isomorphic to the initial segment capital B subscript f of a subscript one, using their restricted less-dot relations

Means: the initial segment capital B subscript b subscript one is order-isomorphic to the initial segment capital B subscript f of a subscript one, using their restricted less-dot relations

1 occurrence in this chapter

Equation form expr-5f5e6e77d0165ddc

x,c,b∈αx, c, b \in \alpha

Read as: x, c, and b belong to alpha

Means: x, c, and b belong to alpha

1 occurrence in this chapter

Equation form expr-5f67a793a81728e3

α,β\alpha, \beta

Read as: alpha, then beta

Means: alpha, then beta

2 occurrences in this chapter

Equation form expr-5fe8b01d3b1ad332

α∈¯β\alpha \mathrel{\underline{\in}} \beta

Read as: alpha bears the underlined membership relation to beta

Means: alpha bears the underlined membership relation to beta

1 occurrence in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

3 occurrences in this chapter

Equation form expr-61b55354f7616dbb

β∉α\beta \notin \alpha

Read as: beta does not belong to alpha

Means: beta does not belong to alpha

1 occurrence in this chapter

Equation form expr-62c66a7a5dd70c31

mm

Read as: m

Means: m

2 occurrences in this chapter

Equation form expr-657e514199c62b2d

ord(B,⋖)=β\ordtype{B, \lessdot} = \beta

Read as: the order type of capital B under less-dot equals beta

Means: the order type of capital B under less-dot equals beta

1 occurrence in this chapter

Equation form expr-66a27a4f24465fcc

α∈α∪{α}=α+\alpha \in \alpha \cup \{\alpha\} = \ordsucc{\alpha}

Read as: alpha belongs to alpha union the set containing alpha equals the ordinal successor of alpha

Means: alpha belongs to alpha union the set containing alpha equals the ordinal successor of alpha

1 occurrence in this chapter

Equation form expr-66e45ee22dbf84b0

α={β∈α:β is an ordinal}\alpha = \Setabs{\beta \in \alpha}{\beta \text{ is an ordinal}}

Read as: alpha equals the set of ordinals beta that belong to alpha

Means: alpha equals the set of ordinals beta that belong to alpha

1 occurrence in this chapter

Equation form expr-67610b0632683369

γ\gamma

Read as: gamma

Means: gamma

3 occurrences in this chapter

Equation form expr-67f472aef5ca147a

{y:(∃x∈A)ϕ(x,y)}\Setabs{y}{(\exists x \in A)\phi(x,y)}

Read as: the set of y such that there exists x in capital A, phi of x and y

Means: the set of y such that there exists x in capital A, phi of x and y

1 occurrence in this chapter

Equation form expr-69325f81a8de9471

f−1(x)=xf^{-1}(x) = x

Read as: the inverse of f at x equals x

Means: the inverse of f at x equals x

1 occurrence in this chapter

Equation form expr-6a6680d0205c89b2

(∀x∈B)(x⋖f(a)↔x⋖g(a))(\forall x \in B)(x \lessdot f(a) \liff x \lessdot g(a))

Read as: for every x in capital B, x is less-dot below f of a if and only if x is less-dot below g of a

Means: for every x in capital B, x is less-dot below f of a if and only if x is less-dot below g of a

1 occurrence in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

1 occurrence in this chapter

Equation form expr-6f8137fe28207fa0

1,2,3,4,5,…1, 2, 3, 4, 5, \ldots

Read as: one, then two, then three, then four, then five, and so on

Means: one, then two, then three, then four, then five, and so on

1 occurrence in this chapter

Equation form expr-6ffc8155d28597c6

β∈α\beta \in \alpha

Read as: beta belongs to alpha

Means: beta belongs to alpha

3 occurrences in this chapter

Equation form expr-729f2cd8398e9960

∈\in

Read as: the membership relation

Means: the membership relation

9 occurrences in this chapter

Equation form expr-72d532267cd81210

x∈c∈bx \in c \in b

Read as: x belongs to c, and c belongs to b

Means: x belongs to c, and c belongs to b

2 occurrences in this chapter

Equation form expr-75f5867578347b4a

∃αϕ(α)\exists \alpha\phi(\alpha)

Read as: there exists alpha, phi of alpha

Means: there exists alpha, phi of alpha

1 occurrence in this chapter

Equation form expr-762675b8339568ec

(∀β∈α)ϕ(β)(\forall \beta \in \alpha)\phi(\beta)

Read as: for every beta in alpha, phi of beta

Means: for every beta in alpha, phi of beta

1 occurrence in this chapter

Equation form expr-7a0515fba5adacd5

X≅Y\ordeq{X}{Y}

Read as: capital X is order-isomorphic to capital Y

Means: capital X is order-isomorphic to capital Y

1 occurrence in this chapter

Equation form expr-7a3734008aa0cf39

(∀β∈α)¬ϕ(β)(\forall \beta \in \alpha) \lnot \phi(\beta)

Read as: for every beta in alpha, not phi of beta

Means: for every beta in alpha, not phi of beta

1 occurrence in this chapter

Equation form expr-7d558d06bb01f5fe

x⋖g(a)x \lessdot g(a)

Read as: x is strictly smaller than g of a

Means: x is strictly smaller than g of a

2 occurrences in this chapter

Equation form expr-7e0c48e2761c88b9

ran(f)=Bb\ran{f} = B_b

Read as: the range of f equals capital B subscript b

Means: the range of f equals capital B subscript b

1 occurrence in this chapter

Equation form expr-7e1e9c9fdb832df9

(∀b<a)ϕ(b)(\forall b < a)\phi(b)

Read as: for every b less than a, phi of b

Means: for every b less than a, phi of b

1 occurrence in this chapter

Equation form expr-7e3fb797657350ce

γ∈α\gamma \in \alpha

Read as: gamma belongs to alpha

Means: gamma belongs to alpha

1 occurrence in this chapter

Equation form expr-7fdd084ac46a4001

x=f(x)x = f(x)

Read as: x equals f of x

Means: x equals f of x

1 occurrence in this chapter

Equation form expr-808f065c73781a5d

ord(A,<)=α\ordtype{A, <} = \alpha

Read as: the order type of capital A under less-than equals alpha

Means: the order type of capital A under less-than equals alpha

1 occurrence in this chapter

Equation form expr-80fc781c4e6c99a5

f:A→Aaf \colon A \to A_a

Read as: f maps capital A to capital A subscript a

Means: f maps capital A to capital A subscript a

1 occurrence in this chapter

Equation form expr-8239088d836daffd

β∈γ\beta \in \gamma

Read as: beta belongs to gamma

Means: beta belongs to gamma

1 occurrence in this chapter

Equation form expr-837a81f8b9a4eedd

dom(f)\dom{f}

Read as: the domain of f

Means: the domain of f

3 occurrences in this chapter

Equation form expr-83fd113947a10f3b

⟨a,b⟩∈f\tuple{a, b} \in f

Read as: the ordered pair a, then b belongs to f

Means: the ordered pair a, then b belongs to f

1 occurrence in this chapter

Equation form expr-84992cde4f19f4a1

b1⋖b2b_1 \lessdot b_2

Read as: b subscript one is strictly smaller than b subscript two

Means: b subscript one is strictly smaller than b subscript two

1 occurrence in this chapter

Equation form expr-876ef64f72817f36

α∈α+\alpha \in \ordsucc{\alpha}

Read as: alpha belongs to the ordinal successor of alpha

Means: alpha belongs to the ordinal successor of alpha

1 occurrence in this chapter

Equation form expr-87fd2e0563cf5695

a1<a2a_1 < a_2

Read as: a subscript one is less than a subscript two

Means: a subscript one is less than a subscript two

2 occurrences in this chapter

Equation form expr-88049d5f38b12945

⟨A,<⟩≇⟨Aa,<a⟩\ordneq{\tuple{A, <}}{\tuple{A_a, <_a}}

Read as: capital A under less-than is not order-isomorphic to its initial segment capital A subscript a under the restricted relation

Means: capital A under less-than is not order-isomorphic to its initial segment capital A subscript a under the restricted relation

1 occurrence in this chapter

Equation form expr-88c151848a108244

β=α\beta = \alpha

Read as: beta equals alpha

Means: beta equals alpha

1 occurrence in this chapter

Equation form expr-8951b18aca64248c

ω+ω\omega+\omega

Read as: omega plus omega

Means: omega plus omega

1 occurrence in this chapter

Equation form expr-8af4c856eab5b503

{y∈⋃⋃f:(∃x∈A)y=f(x)}\Setabs{y \in \bigcup \bigcup f}{(\exists x \in A)y = f(x)}

Read as: the set of y in the double union of f such that y equals f of x for some x in capital A

Means: the set of y in the double union of f such that y equals f of x for some x in capital A

1 occurrence in this chapter

Equation form expr-8b5c234d9d72eadb

(∀x∈A)(x<b↔x<f(b))(\forall x \in A)(x<b \liff x < f(b))

Read as: for every x in capital A, x is less than b if and only if x is less than f of b

Means: for every x in capital A, x is less than b if and only if x is less than f of b

1 occurrence in this chapter

Equation form expr-8c49fbee10c92748

⟨A,<⟩\tuple{A, <}

Read as: the ordered pair consisting of capital A and the less-than relation

Means: the ordered pair consisting of capital A and the less-than relation

14 occurrences in this chapter

Equation form expr-8cbd4e98d6b0299c

f:A→Bf \colon A \to B

Read as: f maps capital A to capital B

Means: f maps capital A to capital B

3 occurrences in this chapter

Equation form expr-8d2cacefc75ba038

∅\emptyset

Read as: the empty set

Means: the empty set

1 occurrence in this chapter

Equation form expr-8eb03a9f4f33b3ac

f(x)<f(b)f(x) < f(b)

Read as: f of x is less than f of b

Means: f of x is less than f of b

1 occurrence in this chapter

Equation form expr-8f78f8a6ac22f4a8

⟨Aa1,<a1⟩≅⟨Bb1,⋖b1⟩\ordeq{\tuple{A_{a_1}, <_{a_1}}}{\tuple{B_{b_1}, \lessdot_{b_1}}}

Read as: the initial segment capital A subscript a subscript one under restricted less-than is order-isomorphic to the initial segment capital B subscript b subscript one under restricted less-dot

Means: the initial segment capital A subscript a subscript one under restricted less-than is order-isomorphic to the initial segment capital B subscript b subscript one under restricted less-dot

2 occurrences in this chapter

Equation form expr-8fe5c714244928bc

α,β,γ\alpha, \beta, \gamma

Read as: alpha, then beta, then gamma

Means: alpha, then beta, then gamma

1 occurrence in this chapter

Equation form expr-916c9823c95a16f9

a1<a2{a_1} < {a_2}

Read as: a subscript one is less than a subscript two

Means: a subscript one is less than a subscript two

1 occurrence in this chapter

Equation form expr-9206214346057a8f

α≅β\ordeq{\alpha}{\beta}

Read as: alpha is order-isomorphic to beta

Means: alpha is order-isomorphic to beta

1 occurrence in this chapter

Equation form expr-9229896a74117a83

{a,b,c}\{a, b, c\}

Read as: the set containing a, then b, then c

Means: the set containing a, then b, then c

1 occurrence in this chapter

Equation form expr-9966082dffa0c4c7

⋖=<\mathord{\lessdot} = \mathord{<}

Read as: the less-dot relation equals the less-than relation

Means: the less-dot relation equals the less-than relation

1 occurrence in this chapter

Equation form expr-9b76f4ba3a0e618c

a∈Aa\in A

Read as: a belongs to capital A

Means: a belongs to capital A

1 occurrence in this chapter

Equation form expr-9c6859ca99e0a328

{τ(x):x∈A}\Setabs{\tau(x)}{x \in A}

Read as: the set of tau of x such that x belongs to capital A

Means: the set of tau of x such that x belongs to capital A

2 occurrences in this chapter

Equation form expr-9e2782833fb8ffa6

α+⊆⋃Y=lsub(X)\ordsucc{\alpha} \subseteq \bigcup Y = \supstrict(X)

Read as: the ordinal successor of alpha is a subset of the union of capital Y equals the strict supremum of capital X

Means: the ordinal successor of alpha is a subset of the union of capital Y equals the strict supremum of capital X

1 occurrence in this chapter

Equation form expr-9ef714289deedcf5

γ∈β\gamma \in \beta

Read as: gamma belongs to beta

Means: gamma belongs to beta

3 occurrences in this chapter

Equation form expr-a10c8fc3a5760311

α=β\alpha = \beta

Read as: alpha equals beta

Means: alpha equals beta

4 occurrences in this chapter

Equation form expr-a1fce4363854ff88

yy

Read as: y

Means: y

2 occurrences in this chapter

Equation form expr-a23b204872928a62

¬(∀a∈A)ϕ(a)\lnot(\forall a \in A)\phi(a)

Read as: not for every a in capital A, phi of a

Means: not for every a in capital A, phi of a

1 occurrence in this chapter

Equation form expr-a3e32bbe007482fc

α∈β∈γ→α∈γ\alpha \in \beta \in \gamma \lif \alpha \in \gamma

Read as: if alpha belongs to beta and beta belongs to gamma, then alpha belongs to gamma

Means: if alpha belongs to beta and beta belongs to gamma, then alpha belongs to gamma

1 occurrence in this chapter

Equation form expr-a6a47b7c667d30a7

X={x∈A:¬ϕ(x)}≠∅X = \Setabs{x \in A}{\lnot\phi(x)} \neq \emptyset

Read as: capital X is the nonempty set of x in capital A for which phi does not hold

Means: capital X is the nonempty set of x in capital A for which phi does not hold

1 occurrence in this chapter

Equation form expr-a6b4751a285107e1

ord(A,<)=ord(B,⋖) iff ⟨A,<⟩≅⟨B,⋖⟩ord(A,<)∈ord(B,⋖) iff ⟨A,<⟩≅⟨Bb,⋖b⟩ for some b∈B\ordtype{A, <} = \ordtype{B, \lessdot}&\text{ iff }\ordeq{\tuple{A, <}}{\tuple{B, \lessdot}}\\ \ordtype{A, <} \in \ordtype{B, \lessdot}&\text{ iff }\ordeq{\tuple{A, <}}{\tuple{B_b, \lessdot_b}}\text{ for some }b \in B

Read as: Two order-type principles. First, the order types are equal if and only if capital A under less-than is order-isomorphic to capital B under less-dot. Second, the first order type belongs to the second if and only if capital A under less-than is order-isomorphic to the initial segment of capital B below some b. End principles

Means: Two order-type principles. First, the order types are equal if and only if capital A under less-than is order-isomorphic to capital B under less-dot. Second, the first order type belongs to the second if and only if capital A under less-than is order-isomorphic to the initial segment of capital B below some b. End principles

1 occurrence in this chapter

Equation form expr-a849bfb6754fe653

f−1(x)<af^{-1}(x) < a

Read as: the inverse of f at x is less than a

Means: the inverse of f at x is less than a

2 occurrences in this chapter

Equation form expr-a86d6dedf5bd962e

0,1,2,3,4,5\x

Read as: the current loop value

Means: the current loop value

2 occurrences in this chapter

Equation form expr-a9d50f6f3254e167

⟨A,<⟩≅α\tuple{A, <} \isomorphic \alpha

Read as: capital A under less-than is isomorphic to alpha

Means: capital A under less-than is isomorphic to alpha

1 occurrence in this chapter

Equation form expr-aa0adc5fdc1f35d2

β⊆dom(f)\beta \subseteq \dom{f}

Read as: beta is a subset of the domain of f

Means: beta is a subset of the domain of f

1 occurrence in this chapter

Equation form expr-aa380901b9c8defa

a<ca < c

Read as: a is less than c

Means: a is less than c

1 occurrence in this chapter

Equation form expr-aa59900449c1df16

α∈O\alpha \in O

Read as: alpha belongs to capital O

Means: alpha belongs to capital O

1 occurrence in this chapter

Equation form expr-ab6ac285f31b1747

∈α={⟨x,y⟩∈α2:x∈y}\in_\alpha = \Setabs{\tuple{x, y} \in \alpha^2}{x \in y}

Read as: membership restricted to alpha equals the set of ordered pairs x and y in alpha squared such that x belongs to y

Means: membership restricted to alpha equals the set of ordered pairs x and y in alpha squared such that x belongs to y

1 occurrence in this chapter

Equation form expr-ac331c69a4afaf5f

g(f−1(x))⋖g(a)g(f^{-1}(x)) \lessdot g(a)

Read as: g of the inverse of f at x is less-dot below g of a

Means: g of the inverse of f at x is less-dot below g of a

1 occurrence in this chapter

Equation form expr-ad22e315da8ebf34

⟨Aa1,<a1⟩≅⟨Bf(a1),⋖f(a1)⟩\ordeq{\tuple{A_{a_1}, <_{a_1}}}{\tuple{B_{f({a_1})}, \lessdot_{f({a_1})}}}

Read as: the initial segment capital A subscript a subscript one under restricted less-than is order-isomorphic to the initial segment capital B subscript f of a subscript one under restricted less-dot

Means: the initial segment capital A subscript a subscript one under restricted less-than is order-isomorphic to the initial segment capital B subscript f of a subscript one under restricted less-dot

1 occurrence in this chapter

Equation form expr-ade77833ad127348

A∖dom(f)A \setminus \dom{f}

Read as: capital A set minus the domain of f

Means: capital A set minus the domain of f

1 occurrence in this chapter

Equation form expr-ae81696353633c0e

a≮aa \nless a

Read as: a is not less than a

Means: a is not less than a

1 occurrence in this chapter

Equation form expr-aea71cbfc818e6c4

{x}\{x\}

Read as: the set containing x

Means: the set containing x

1 occurrence in this chapter

Equation form expr-b1196a9f9c627cfa

if (∀a∈A)((∀b<a)ϕ(b)→ϕ(a)), then (∀a∈A)ϕ(a).\text{if }(\forall a \in A)((\forall b < a)\phi(b) \lif \phi(a))\text{, then }(\forall a \in A)\phi(a).

Read as: if phi holds of each a in capital A whenever it holds of every b less than a, then phi holds of every a in capital A

Means: if phi holds of each a in capital A whenever it holds of every b less than a, then phi holds of every a in capital A

1 occurrence in this chapter

Equation form expr-b138b2ad5cb401ac

b=f(b)b = f(b)

Read as: b equals f of b

Means: b equals f of b

1 occurrence in this chapter

Equation form expr-b1ded1d37edfd743

(∀x∈A)x⊆A(\forall x \in A)x \subseteq A

Read as: for every x in capital A, x is a subset of capital A

Means: for every x in capital A, x is a subset of capital A

1 occurrence in this chapter

Equation form expr-b59be578947fddef

a,b∈Aa, b \in A

Read as: a and b belong to capital A

Means: a and b belong to capital A

1 occurrence in this chapter

Equation form expr-b6c9f7eeb5b5c2bf

α∉β\alpha \notin \beta

Read as: alpha does not belong to beta

Means: alpha does not belong to beta

1 occurrence in this chapter

Equation form expr-b88f2c670760bac5

f↾Aa\funrestrictionto{f}{A_a}

Read as: the restriction of f to capital A subscript a

Means: the restriction of f to capital A subscript a

1 occurrence in this chapter

Equation form expr-bba2711758449d10

Bb2B_{{b_2}}

Read as: capital B subscript b subscript two

Means: capital B subscript b subscript two

1 occurrence in this chapter

Equation form expr-bc57724b0c8fb691

τ(x)\tau(x)

Read as: tau of x

Means: tau of x

1 occurrence in this chapter

Equation form expr-bc98d88a67d6b61a

f↾Aa:Aa→Bf(a)\funrestrictionto{f}{A_{a}} : A_a \to B_{f(a)}

Read as: the restriction of f to capital A subscript a maps capital A subscript a to capital B subscript f of a

Means: the restriction of f to capital A subscript a maps capital A subscript a to capital B subscript f of a

1 occurrence in this chapter

Equation form expr-bf2a2f65d4938eb6

γ=β\gamma = \beta

Read as: gamma equals beta

Means: gamma equals beta

1 occurrence in this chapter

Equation form expr-c1164388ed8aa905

⟨A,<⟩≅⟨B,⋖⟩\ordeq{\tuple{A, <}}{\tuple{B, \lessdot}}

Read as: capital A under less-than is order-isomorphic to capital B under less-dot

Means: capital A under less-than is order-isomorphic to capital B under less-dot

1 occurrence in this chapter

Equation form expr-c18236f4e2b21722

γ=∅\gamma = \emptyset

Read as: gamma equals the empty set

Means: gamma equals the empty set

1 occurrence in this chapter

Equation form expr-c2a723c186d0f984

b1=f(a1){b_1} = f({a_1})

Read as: b subscript one equals f of a subscript one

Means: b subscript one equals f of a subscript one

1 occurrence in this chapter

Equation form expr-c3f8d9046b9b4f08

(∀n∈ω)ϕ(n)(\forall n \in \omega)\phi(n)

Read as: for every n in omega, phi of n

Means: for every n in omega, phi of n

1 occurrence in this chapter

Equation form expr-c3f942284c39de16

⟨a1,b1⟩,⟨a2,b2⟩∈f\tuple{a_1, b_1}, \tuple{a_2, b_2} \in f

Read as: the ordered pairs a subscript one with b subscript one, and a subscript two with b subscript two, belong to f

Means: the ordered pairs a subscript one with b subscript one, and a subscript two with b subscript two, belong to f

1 occurrence in this chapter

Equation form expr-c4694f2e93d5c4e7

OO

Read as: capital O

Means: capital O

3 occurrences in this chapter

Equation form expr-c73254e6088d1bc4

α∈β\alpha \in \beta

Read as: alpha belongs to beta

Means: alpha belongs to beta

2 occurrences in this chapter

Equation form expr-c750645c650e47fb

f:dom(f)→ran(f)f \colon \dom{f} \to \ran{f}

Read as: f maps its domain to its range

Means: f maps its domain to its range

1 occurrence in this chapter

Equation form expr-c7934d1a2378b2fc

⟨Aa,<a⟩\tuple{A_a, <_a}

Read as: the ordered pair consisting of capital A subscript a and its restricted less-than relation

Means: the ordered pair consisting of capital A subscript a and its restricted less-than relation

1 occurrence in this chapter

Equation form expr-c83e85f0d76191fa

c⊆bc \subseteq b

Read as: c is a subset of b

Means: c is a subset of b

1 occurrence in this chapter

Equation form expr-c9d42c8a545dff20

x<f(b)x < f(b)

Read as: x is less than f of b

Means: x is less than f of b

2 occurrences in this chapter

Equation form expr-ca978112ca1bbdca

aa

Read as: a

Means: a

3 occurrences in this chapter

Equation form expr-cc15cb2971a5fdbd

b⊆αb \subseteq \alpha

Read as: b is a subset of alpha

Means: b is a subset of alpha

2 occurrences in this chapter

Equation form expr-cd0aa9856147b6c5

gg

Read as: g

Means: g

2 occurrences in this chapter

Equation form expr-cd9ca3957367f4bb

¬ϕ\lnot\phi

Read as: not phi

Means: not phi

1 occurrence in this chapter

Equation form expr-cebc72898cc3a1f9

b∈αb \in \alpha

Read as: b belongs to alpha

Means: b belongs to alpha

1 occurrence in this chapter

Equation form expr-cef3efcc20756bf0

α+=α∪{α}\ordsucc{\alpha} =\alpha \cup \{\alpha\}

Read as: the ordinal successor of alpha equals alpha union the set containing alpha

Means: the ordinal successor of alpha equals alpha union the set containing alpha

1 occurrence in this chapter

Equation form expr-cfc51773bd87a244

f={⟨β,b⟩:b∈B and β≅⟨Bb,⋖b⟩}f = \Setabs{\tuple{\beta, b}}{b \in B\text{ and }\ordeq{\beta}{\tuple{B_b, \lessdot_b}}}

Read as: f equals the set of ordered pairs beta and b such that b belongs to capital B and beta is order-isomorphic to the initial segment capital B subscript b under restricted less-dot

Means: f equals the set of ordered pairs beta and b such that b belongs to capital B and beta is order-isomorphic to the initial segment capital B subscript b under restricted less-dot

1 occurrence in this chapter

Equation form expr-d01032491560ec4e

A→BA \to B

Read as: functions from capital A to capital B

Means: functions from capital A to capital B

1 occurrence in this chapter

Equation form expr-d057b5baa71abc46

c⊆αc \subseteq \alpha

Read as: c is a subset of alpha

Means: c is a subset of alpha

1 occurrence in this chapter

Equation form expr-d0fc5eaf2ee958fe

B∖ran(f)B \setminus \ran{f}

Read as: capital B set minus the range of f

Means: capital B set minus the range of f

1 occurrence in this chapter

Equation form expr-d221f5671d9ec45b

ZF−\ZFminus

Read as: set theory Z F minus

Means: set theory Z F minus

5 occurrences in this chapter

Equation form expr-d438a18a7da07bf0

x<yx < y

Read as: x is less than y

Means: x is less than y

1 occurrence in this chapter

Equation form expr-d4735e3a265e16ee

22

Read as: two

Means: two

1 occurrence in this chapter

Equation form expr-d482e9c278d41825

¬ϕ(α)\lnot\phi(\alpha)

Read as: not phi of alpha

Means: not phi of alpha

1 occurrence in this chapter

Equation form expr-d51246042fbc0a3f

B=AB = A

Read as: capital B equals capital A

Means: capital B equals capital A

1 occurrence in this chapter

Equation form expr-d5245ea5fc47e082

dom(f)=Aa\dom{f} = A_a

Read as: the domain of f equals capital A subscript a

Means: the domain of f equals capital A subscript a

1 occurrence in this chapter

Equation form expr-d6fbd5a225bd83fa

Aa⊊AA_a \subsetneq A

Read as: capital A subscript a is a proper subset of capital A

Means: capital A subscript a is a proper subset of capital A

1 occurrence in this chapter

Equation form expr-d818c9808f327956

α+\ordsucc{\alpha}

Read as: the ordinal successor of alpha

Means: the ordinal successor of alpha

3 occurrences in this chapter

Equation form expr-d880f68152806d22

x=f(f−1(x))=g(f−1(x))x =f(f^{-1}(x)) = g(f^{-1}(x))

Read as: x equals f of the inverse of f at x, which equals g of the inverse of f at x

Means: x equals f of the inverse of f at x, which equals g of the inverse of f at x

1 occurrence in this chapter

Equation form expr-d91df0d8bb09bc22

⟨Aa2,<a2⟩≅⟨Bb2,⋖b2⟩\ordeq{\tuple{A_{{a_2}}, <_{a_2}}}{\tuple{B_{{b_2}}, \lessdot_{b_2}}}

Read as: the initial segment capital A subscript a subscript two under restricted less-than is order-isomorphic to the initial segment capital B subscript b subscript two under restricted less-dot

Means: the initial segment capital A subscript a subscript two under restricted less-than is order-isomorphic to the initial segment capital B subscript b subscript two under restricted less-dot

2 occurrences in this chapter

Equation form expr-d95a2ca6953f247a

B=AaB = A_a

Read as: capital B equals capital A subscript a

Means: capital B equals capital A subscript a

1 occurrence in this chapter

Equation form expr-da919cb2feadb2bc

β∈dom(f)\beta \in \dom{f}

Read as: beta belongs to the domain of f

Means: beta belongs to the domain of f

1 occurrence in this chapter

Equation form expr-dabd3aff769f07eb

<<

Read as: is less than

Means: is less than

27 occurrences in this chapter

Equation form expr-db814bf42ffdbbd1

b∈Ab \in A

Read as: b belongs to capital A

Means: b belongs to capital A

1 occurrence in this chapter

Equation form expr-dbd4c76b25c1bf05

x∈αx \in \alpha

Read as: x belongs to alpha

Means: x belongs to alpha

1 occurrence in this chapter

Equation form expr-dcc3a945dbc390de

(∀z∈X)m≤z(\forall z \in X)m \leq z

Read as: for every z in capital X, m is less than or equal to z

Means: for every z in capital X, m is less than or equal to z

1 occurrence in this chapter

Equation form expr-dd5a7596b02bd0b7

lsub(X)=⋃Y\supstrict(X) = \bigcup Y

Read as: the strict supremum of capital X equals the union of capital Y

Means: the strict supremum of capital X equals the union of capital Y

1 occurrence in this chapter

Equation form expr-deb7ee55dd7f1328

ϕ(x,y)\phi(x,y)

Read as: phi of x and y

Means: phi of x and y

1 occurrence in this chapter

Equation form expr-df7e70e5021544f4

BB

Read as: capital B

Means: capital B

3 occurrences in this chapter

Equation form expr-df8bc276256112d2

Aa={x∈A:x<a}A_a = \Setabs{x \in A}{x < a}

Read as: capital A subscript a equals the set of x in capital A that are less than a

Means: capital A subscript a equals the set of x in capital A that are less than a

1 occurrence in this chapter

Equation form expr-dff9e18c96575495

β∈X\beta \in X

Read as: beta belongs to capital X

Means: beta belongs to capital X

1 occurrence in this chapter

Equation form expr-e0a1d8a7dc5f9e9c

α<β\alpha <\beta

Read as: alpha is less than beta

Means: alpha is less than beta

1 occurrence in this chapter

Equation form expr-e0dba47e4a8ce57c

f[Aa]=f[{x∈A:x<a}]=f[{f−1(y)∈A:f−1(y)<a}]={y∈B:y⋖f(a)}=Bf(a)\funimage{f}{A_a} &= \funimage{f}{\Setabs{x \in A}{x < a}}\\ &= \funimage{f}{\Setabs{f^{-1}(y) \in A}{f^{-1}(y) < a}} \\ &= \Setabs{y \in B}{y \lessdot f(a)} \\ &=B_{f(a)}

Read as: Source-ordered equality chain. The image of capital A subscript a under f equals the image under f of all x in capital A below a. This equals the image under f of all inverse images of y that lie below a. It equals the set of y in capital B that are less-dot below f of a, and therefore equals capital B subscript f of a. End equality chain

Means: Source-ordered equality chain. The image of capital A subscript a under f equals the image under f of all x in capital A below a. This equals the image under f of all inverse images of y that lie below a. It equals the set of y in capital B that are less-dot below f of a, and therefore equals capital B subscript f of a. End equality chain

1 occurrence in this chapter

Equation form expr-e2e5e9c0b995314e

Y={α+:α∈X}Y = \Setabs{\ordsucc{\alpha}}{\alpha \in X}

Read as: capital Y equals the set of ordinal successors of alpha as alpha ranges over capital X

Means: capital Y equals the set of ordinal successors of alpha as alpha ranges over capital X

1 occurrence in this chapter

Equation form expr-e325514196af49a9

{a}\{a\}

Read as: the set containing a

Means: the set containing a

1 occurrence in this chapter

Equation form expr-e4a7561b1e2ae3f9

lsub(X)=⋃α∈Xα+\supstrict(X) = \bigcup_{\alpha \in X} \ordsucc{\alpha}

Read as: the strict supremum of capital X equals the union, over alpha in capital X, of the ordinal successors of alpha

Means: the strict supremum of capital X equals the union, over alpha in capital X, of the ordinal successors of alpha

1 occurrence in this chapter

Equation form expr-e567ac37e8fc3c30

α∈X\alpha \in X

Read as: alpha belongs to capital X

Means: alpha belongs to capital X

1 occurrence in this chapter

Equation form expr-e6e6dd53a669cfab

ϕ(x,y)\phi(x, y)

Read as: phi of x and y

Means: phi of x and y

1 occurrence in this chapter

Equation form expr-e7389bc1a350f461

⟨α,∈α⟩\tuple{\alpha, \in_\alpha}

Read as: the ordered pair consisting of alpha and membership restricted to alpha

Means: the ordered pair consisting of alpha and membership restricted to alpha

1 occurrence in this chapter

Equation form expr-e9437ab33fd66b11

f:Aa→Bbf \colon A_a \to B_b

Read as: f maps capital A subscript a to capital B subscript b

Means: f maps capital A subscript a to capital B subscript b

1 occurrence in this chapter

Equation form expr-ea0b0a274a95af0c

∀x∃!yτ(x)=y\forall x \lexists![y][\tau(x) = y]

Read as: for every x, there exists exactly one y, tau of x equals y

Means: for every x, there exists exactly one y, tau of x equals y

1 occurrence in this chapter

Equation form expr-ea0b97f74358ec16

(g∘f):A→C(g \circ f) \colon A \to C

Read as: g composed with f maps capital A to capital C

Means: g composed with f maps capital A to capital C

1 occurrence in this chapter

Equation form expr-ec9672dcbdd3d677

(∀x∈A)∃!yτ(x)=y(\forall x \in A)\lexists![y][\tau(x) = y]

Read as: for every x in capital A, there exists exactly one y, tau of x equals y

Means: for every x in capital A, there exists exactly one y, tau of x equals y

1 occurrence in this chapter

Equation form expr-efe1e5c7656bf0ba

a<ba < b

Read as: a is less than b

Means: a is less than b

1 occurrence in this chapter

Equation form expr-f0547f283aaf5d8b

a2∈dom(f)a_2 \in \dom{f}

Read as: a subscript two belongs to the domain of f

Means: a subscript two belongs to the domain of f

1 occurrence in this chapter

Equation form expr-f0dc60cd5b6e9148

…\ldots

Read as: and so on

Means: and so on

4 occurrences in this chapter

Equation form expr-f17057058ab950bf

x∈bx \in b

Read as: x belongs to b

Means: x belongs to b

1 occurrence in this chapter

Equation form expr-f27007fd6d9237af

ϕ(α)\phi(\alpha)

Read as: phi of alpha

Means: phi of alpha

4 occurrences in this chapter

Equation form expr-f28a2757bfdff993

∅≠X⊆A\emptyset \neq X \subseteq A

Read as: capital X is nonempty and is a subset of capital A

Means: capital X is nonempty and is a subset of capital A

1 occurrence in this chapter

Equation form expr-f29aedeeddad1283

x∈y∨x=y∨y∈xx \in y \lor x = y \lor y \in x

Read as: x belongs to y or x equals y or y belongs to x

Means: x belongs to y or x equals y or y belongs to x

1 occurrence in this chapter

Equation form expr-f31c89d9fd163089

c∈αc \in \alpha

Read as: c belongs to alpha

Means: c belongs to alpha

1 occurrence in this chapter

Equation form expr-f34dc994cc9b0501

x∪yx \cup y

Read as: x union y

Means: x union y

1 occurrence in this chapter

Equation form expr-f3d3a0f30115de54

a<b<ca < b < c

Read as: a is less than b is less than c

Means: a is less than b is less than c

1 occurrence in this chapter

Equation form expr-f3f3804480e8551a

β\beta

Read as: beta

Means: beta

8 occurrences in this chapter

Equation form expr-f4b6980a038b6ca9

if ∀α((∀β∈α)ϕ(β)→ϕ(α)), then ∀αϕ(α)\text{if }\forall \alpha((\forall \beta \in \alpha)\phi(\beta) \lif \phi(\alpha))\text{, then }\forall \alpha\phi(\alpha)

Read as: if phi holds of every alpha whenever it holds of every beta belonging to alpha, then phi holds of every alpha

Means: if phi holds of every alpha whenever it holds of every beta belonging to alpha, then phi holds of every alpha

1 occurrence in this chapter

Equation form expr-f6903cbb1391d070

x<bx < b

Read as: x is less than b

Means: x is less than b

2 occurrences in this chapter

Equation form expr-f837b0d8707430f0

(∀z∈X)z≮m(\forall z \in X)z \nless m

Read as: for every z in capital X, z is not less than m

Means: for every z in capital X, z is not less than m

1 occurrence in this chapter

Equation form expr-f854ba88a4b9e5c9

ω+1\omega+1

Read as: omega plus one

Means: omega plus one

1 occurrence in this chapter

Equation form expr-fcad1fc8862e942e

f(a)=g(a)f(a) = g(a)

Read as: f of a equals g of a

Means: f of a equals g of a

1 occurrence in this chapter

Equation form expr-fd39650a8fc246c4

∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}},…\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}, \ldots

Read as: the empty set; then the singleton containing the empty set; then the set containing the empty set and that singleton; then the set containing the empty set, that singleton, and the preceding two-element set; and so on

Means: the empty set; then the singleton containing the empty set; then the set containing the empty set and that singleton; then the set containing the empty set, that singleton, and the preceding two-element set; and so on

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

2 occurrences in this chapter

Equation form expr-ff08e946ad2d6732

f−1(x)<bf^{-1}(x) < b

Read as: the inverse of f at x is less than b

Means: the inverse of f at x is less than b

1 occurrence in this chapter

Equation form expr-ff70c770816a0a01

(∃m∈X)(∀z∈X)z≮m(\exists m \in X)(\forall z \in X)z \nless m

Read as: there exists m in capital X, for every z in capital X, z is not less than m

Means: there exists m in capital X, for every z in capital X, z is not less than m

1 occurrence in this chapter

Natural numbers in their usual order

The diagram expands a source loop. In that loop, the current loop value successively takes the values zero through five. Each displayed value is less than the next. After five the diagram continues, and so on. Thus, from left to right: zero is less than one, one is less than two, two is less than three, three is less than four, four is less than five, and the sequence continues. End diagram.

Source

Positive natural numbers followed by zero

In the source loop, the current loop value successively takes the values one through five. Each loop value is ordered before the next. After five the positive-number sequence continues, and so on. The entire continuing sequence is ordered before the final zero. The resulting order has all positive natural numbers first and zero last. End diagram.

Source

Even natural numbers followed by odd natural numbers

The diagram reads left to right. The first displayed value is zero; it is ordered before two, which is ordered before four. Four is ordered before the later even natural numbers, and so on. The entire continuing even sequence is ordered before one, which is ordered before three. Three is ordered before the later odd natural numbers, and so on. It places every even natural number in increasing order before every odd natural number in increasing order. End diagram.

Source

Definition one in this chapter

This source definition contains, in source order: is less than; then capital A; then is less than; then a and b belong to capital A; then a is less than b; then a equals b; then b is less than a; then capital A; then is less than; then capital X is nonempty and is a subset of capital A; then there exists m in capital X, for every z in capital X, z is not less than m. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise one in this chapter

This source exercise contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Proposition one in this chapter

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

Source

Proposition two in this chapter

This source proposition contains, in source order: is less than; then capital A; then phi of x; then if phi holds of each a in capital A whenever it holds of every b less than a, then phi holds of every a in capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition two in this chapter

This source definition contains, in source order: the ordered pair consisting of capital A and the less-than relation; then is less than; then capital A; then the ordered pair consisting of capital A and the less-than relation; then the ordered pair consisting of capital B and the less-dot relation; then f maps capital A to capital B; then x is less than y; then f of x is strictly smaller than f of y; then capital A under less-than is order-isomorphic to capital B under less-dot; then f. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma one in this chapter

This source lemma contains, in source order: f maps capital A to capital B; then g maps capital B to capital C; then g composed with f maps capital A to capital C. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise two in this chapter

This source exercise contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Corollary one in this chapter

This source corollary contains, in source order: capital X is order-isomorphic to capital Y. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition three in this chapter

This source proposition contains, in source order: the ordered pair consisting of capital A and the less-than relation; then the ordered pair consisting of capital B and the less-dot relation. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition three in this chapter

This source definition contains, in source order: the ordered pair consisting of capital A and the less-than relation; then a belongs to capital A; then capital A subscript a equals the set of x in capital A that are less than a; then capital A subscript a; then capital A; then capital A; then capital A; then the less-than relation restricted below a; then is less than; then the restriction of the less-than relation to capital A subscript a squared. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma two in this chapter

This source lemma contains, in source order: the ordered pair consisting of capital A and the less-than relation; then a belongs to capital A; then capital A under less-than is not order-isomorphic to its initial segment capital A subscript a under the restricted relation. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma three in this chapter

This source lemma contains, in source order: the ordered pair consisting of capital A and the less-than relation; then the ordered pair consisting of capital B and the less-dot relation; then f maps capital A to capital B; then a belongs to capital A; then the restriction of f to capital A subscript a maps capital A subscript a to capital B subscript f of a. 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 equality chain. The image of capital A subscript a under f equals the image under f of all x in capital A below a. This equals the image under f of all inverse images of y that lie below a. It equals the set of y in capital B that are less-dot below f of a, and therefore equals capital B subscript f of a. End equality chain. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma four in this chapter

This source lemma contains, in source order: the ordered pair consisting of capital A and the less-than relation; then the ordered pair consisting of capital B and the less-dot relation; then the initial segment capital A subscript a subscript one under restricted less-than is order-isomorphic to the initial segment capital B subscript b subscript one under restricted less-dot; then the initial segment capital A subscript a subscript two under restricted less-than is order-isomorphic to the initial segment capital B subscript b subscript two under restricted less-dot; then a subscript one is less than a subscript two if and only if b subscript one is strictly smaller than b subscript two. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem one 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: Two proposed order-type principles. First, the order type of capital A under less-than equals the order type of capital B under less-dot if and only if those two well-orderings are order-isomorphic. Second, the first order type is less than the second if and only if the first well-ordering is order-isomorphic to the initial segment of capital B below some b. End principles. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition four in this chapter

This source definition contains, in source order: capital A; then for every x in capital A, x is a subset of capital A; then capital A; then capital A; then the membership relation. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma five in this chapter

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

Source

Corollary two in this chapter

This source corollary contains, in source order: alpha equals the set of ordinals beta that belong to alpha; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Transfinite Induction

This source theorem contains, in source order: phi of x; then if some alpha satisfies phi, then there is an alpha satisfying phi such that no beta belonging to alpha satisfies phi. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Trichotomy

This source theorem contains, in source order: alpha belongs to beta or alpha equals beta or beta belongs to alpha; then alpha; then beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary three in this chapter

This source corollary contains, in source order: there exists alpha, phi of alpha; then there exists an alpha satisfying phi such that every beta satisfying phi is greater than or equal to alpha; then alpha, then beta, then gamma; then alpha does not belong to alpha; then if alpha belongs to beta and beta belongs to gamma, then alpha belongs to gamma. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise three in this chapter

This source exercise contains no delimiter-counted formula. Its complete source wording remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Corollary four in this chapter

This source corollary contains, in source order: capital A; then capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Burali-Forti Paradox

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

Source

Proposition four in this chapter

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

Source

Proposition five in this chapter

This source proposition contains, in source order: alpha is a subset of beta or beta is a subset of alpha; then alpha, then beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition six in this chapter

This source proposition contains, in source order: alpha equals beta; then alpha is order-isomorphic to beta; then alpha, then beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise four in this chapter

This source exercise contains, in source order: capital X; then the union of capital X. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Axiom: Scheme of Replacement

This source axiom contains, in source order: phi of x and y; then capital A; then for every x in capital A, there exists exactly one y, phi of x and y; then the set of y such that there exists x in capital A, phi of x and y. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition five in this chapter

This source definition contains, in source order: phi of x and y; then capital B; then for every capital A, if every x in capital A has exactly one y such that phi holds of x and y, then there exists a capital B such that, for every y, y belongs to capital B if and only if phi holds of x and y for some x in capital A. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary five in this chapter

This source corollary contains, in source order: tau of x; then capital A; then the set of tau of x such that x belongs to capital A, equals the set of y such that there exists x in capital A, y equals tau of x. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition six in this chapter

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

Source

Theorem five in this chapter

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

Source

Definition seven in this chapter

This source definition contains, in source order: the ordered pair consisting of capital A and the less-than relation; then the order type of capital A under the less-than relation; then alpha; then capital A under less-than is order-isomorphic to alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Corollary six in this chapter

This source corollary contains, in source order: the ordered pair consisting of capital A and the less-than relation; then the ordered pair consisting of capital B and the less-dot relation; then Two order-type principles. First, the order types are equal if and only if capital A under less-than is order-isomorphic to capital B under less-dot. Second, the first order type belongs to the second if and only if capital A under less-than is order-isomorphic to the initial segment of capital B below some b. End principles. 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: Two order-type principles. First, the order types are equal if and only if capital A under less-than is order-isomorphic to capital B under less-dot. Second, the first order type belongs to the second if and only if capital A under less-than is order-isomorphic to the initial segment of capital B below some b. End principles. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math four in this chapter

This source display math contains, in source order: Three equivalent conditions. Alpha belongs to beta if and only if the restriction of f to alpha is an isomorphism from alpha to the initial segment of capital B below f of alpha. This holds if and only if capital A under less-than is order-isomorphic to that initial segment, and if and only if it is order-isomorphic to the initial segment of capital B below some b. End equivalences. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition eight in this chapter

This source definition contains, in source order: alpha; then the ordinal successor of alpha equals alpha union the set containing alpha; then alpha; then the ordinal successor of beta equals alpha; then beta; then alpha; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition seven in this chapter

This source proposition contains, in source order: alpha; then alpha belongs to the ordinal successor of alpha; then the ordinal successor of alpha; then beta; then alpha belongs to beta, and beta belongs to the ordinal successor of alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Theorem: Simple Transfinite Induction

This source theorem contains, in source order: phi of x; then phi of the empty set; then alpha; then phi of alpha; then phi of the ordinal successor of alpha; then alpha; then for every beta in alpha, phi of beta; then phi of alpha; then for every alpha, phi of alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition nine in this chapter

This source definition contains, in source order: capital X; then the strict supremum of capital X equals the union, over alpha in capital X, of the ordinal successors of alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition eight in this chapter

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

Source

Cross-reference reference-001237

chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001238

Section “The General Idea of an Ordinal” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001239

lemma one in chapter “Ordinals”

Source occurrence

Cross-reference reference-001240

proposition two in chapter “Ordinals”

Source occurrence

Cross-reference reference-001241

the proposition on extensionality strictlinearorders

Source occurrence

Cross-reference reference-001242

proposition two in chapter “Ordinals”

Source occurrence

Cross-reference reference-001243

proposition one in chapter “Ordinals”

Source occurrence

Cross-reference reference-001244

the proposition on extensionality strictlinearorders

Source occurrence

Cross-reference reference-001245

lemma three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001246

lemma two in chapter “Ordinals”

Source occurrence

Cross-reference reference-001247

lemma four in chapter “Ordinals”

Source occurrence

Cross-reference reference-001248

lemma three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001249

the theorem on woalwayscomparable

Source occurrence

Cross-reference reference-001250

section “Infinity” in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001251

section “Philosophical Reflections” in chapter “Relations”

Source occurrence

Cross-reference reference-001252

section “Some Philosophical Reflections” in chapter “Arithmetization”

Source occurrence

Cross-reference reference-001253

section “Dedekind's “Proof” of the Existence of an Infinite Set” in chapter “Infinite Sets”

Source occurrence

Cross-reference reference-001254

section “Selecting our Natural Numbers” in chapter “Steps towards Z”

Source occurrence

Cross-reference reference-001255

proposition one in chapter “Ordinals”

Source occurrence

Cross-reference reference-001256

lemma five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001257

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001258

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001259

lemma five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001260

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001261

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001262

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001263

lemma five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001264

proposition one in chapter “Ordinals”

Source occurrence

Cross-reference reference-001265

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001266

lemma five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001267

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001268

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001269

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001270

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001271

lemma five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001272

corollary four in chapter “Ordinals”

Source occurrence

Cross-reference reference-001273

corollary three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001274

Cesare Burali-Forti

Source occurrence

Cross-reference reference-001275

(Jean van Heijenoort, 1967, p. 105)

Source occurrence

Cross-reference reference-001276

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001277

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001278

theorem “Transfinite Induction” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001279

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001280

theorem “Trichotomy” in chapter “Ordinals”

Source occurrence

Cross-reference reference-001281

lemma two in chapter “Ordinals”

Source occurrence

Cross-reference reference-001282

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

Source occurrence

Cross-reference reference-001283

section “The Strength of Replacement” in chapter “Replacement”

Source occurrence

Cross-reference reference-001284

section “Functions as Relations” in chapter “Functions”

Source occurrence

Cross-reference reference-001285

chapter “Replacement”

Source occurrence

Cross-reference reference-001286

chapter “Replacement”

Source occurrence

Cross-reference reference-001287

1922

Source occurrence

Cross-reference reference-001288

Thoralf Skolem (1922)

Source occurrence

Cross-reference reference-001289

proposition six in chapter “Ordinals”

Source occurrence

Cross-reference reference-001290

lemma four in chapter “Ordinals”

Source occurrence

Cross-reference reference-001291

corollary four in chapter “Ordinals”

Source occurrence

Cross-reference reference-001292

lemma three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001293

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

Source occurrence

Cross-reference reference-001294

proposition six in chapter “Ordinals”

Source occurrence

Cross-reference reference-001295

proposition three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001296

lemma three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001297

corollary two in chapter “Ordinals”

Source occurrence

Cross-reference reference-001298

corollary four in chapter “Ordinals”

Source occurrence

Cross-reference reference-001299

corollary three in chapter “Ordinals”

Source occurrence

Cross-reference reference-001300

corollary four in chapter “Ordinals”

Source occurrence

Ordered structures

Natural numbers in their usual order

Structure: diagram tikz.

The diagram expands a source loop. In that loop, the current loop value successively takes the values zero through five. Each displayed value is less than the next. After five the diagram continues, and so on. Thus, from left to right: zero is less than one, one is less than two, two is less than three, three is less than four, four is less than five, and the sequence continues. End diagram.

Read the source-bound structure in context

Positive natural numbers followed by zero

Structure: diagram tikz.

In the source loop, the current loop value successively takes the values one through five. Each loop value is ordered before the next. After five the positive-number sequence continues, and so on. The entire continuing sequence is ordered before the final zero. The resulting order has all positive natural numbers first and zero last. End diagram.

Read the source-bound structure in context

Even natural numbers followed by odd natural numbers

Structure: diagram tikz.

The diagram reads left to right. The first displayed value is zero; it is ordered before two, which is ordered before four. Four is ordered before the later even natural numbers, and so on. The entire continuing even sequence is ordered before one, which is ordered before three. Three is ordered before the later odd natural numbers, and so on. It places every even natural number in increasing order before every odd natural number in increasing order. End diagram.

Read the source-bound structure in context