Set Theory

Ordinal Arithmetic

Equation form expr-007d3ba7b1c1b0d3

⋃A\bigcup A

Read as: the union of capital A

Means: the union of capital A

1 occurrence in this chapter

Equation form expr-0650765e41deeb4e

{⟨x,y⟩∈α⊔β:x∢y}\Setabs{\tuple{x,y}\in \alpha\disjointsum\beta}{x \rlexless y}

Read as: the set of the ordered pair x, then y belongs to alpha disjoint sum beta such that x is less than in reverse lexicographic order y

Means: the set of the ordered pair x, then y belongs to alpha disjoint sum beta such that x is less than in reverse lexicographic order y

1 occurrence in this chapter

Equation form expr-078637ab65700a46

α⊔β\alpha \disjointsum \beta

Read as: alpha disjoint sum beta

Means: alpha disjoint sum beta

1 occurrence in this chapter

Equation form expr-0bfaea287a6ff3b5

α+β=ord(α⊔β,∢)\alpha \ordplus \beta = \ordtype{\alpha \disjointsum \beta, \rlexless}

Read as: alpha ordinal plus beta equals the order type of alpha disjoint sum beta is less than in reverse lexicographic order

Means: alpha ordinal plus beta equals the order type of alpha disjoint sum beta is less than in reverse lexicographic order

1 occurrence in this chapter

Equation form expr-0cb3d9a00cbe0d33

α≤γ<α+β\alpha \leq \gamma < \alpha \ordplus \beta

Read as: alpha is less than or equal to gamma is less than alpha ordinal plus beta

Means: alpha is less than or equal to gamma is less than alpha ordinal plus beta

1 occurrence in this chapter

Equation form expr-0e4cf2a37b14a098

γ0=max{γ∈α:f(γ)≠g(γ)}\gamma_0 = \text{max}\Setabs{\gamma \in \alpha}{f(\gamma) \neq g(\gamma)}

Read as: gamma subscript zero equals max, the set of gamma belongs to alpha such that f of gamma is not equal to g of gamma

Means: gamma subscript zero equals max, the set of gamma belongs to alpha such that f of gamma is not equal to g of gamma

1 occurrence 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-102062253e78b96e

2·ω=lsubn<ω(2·n)=ω∈lsubn<ω(ω+n)=ω+ω=ω·22 \ordtimes \omega = \supstrict_{n < \omega}(2\ordtimes n) = \omega \in \supstrict_{n < \omega}(\omega \ordplus n) = \omega \ordplus \omega = \omega \ordtimes 2

Read as: two ordinal times omega equals the strict supremum over n is less than omega of two ordinal times n equals omega belongs to the strict supremum over n is less than omega of open scope, omega ordinal plus n, close scope equals omega ordinal plus omega equals omega ordinal times two

Means: two ordinal times omega equals the strict supremum over n is less than omega of two ordinal times n equals omega belongs to the strict supremum over n is less than omega of open scope, omega ordinal plus n, close scope equals omega ordinal plus omega equals omega ordinal times two

1 occurrence in this chapter

Equation form expr-11baa595827a4e0f

ω\omega

Read as: omega

Means: omega

3 occurrences in this chapter

Equation form expr-11f6440cc92458d2

⟨α×β,∢⟩\tuple{\alpha \times \beta, \rlexless}

Read as: the ordered pair alpha times beta, then is less than in reverse lexicographic order

Means: the ordered pair alpha times beta, then is less than in reverse lexicographic order

1 occurrence in this chapter

Equation form expr-124547e317cd8ebf

c∈Ac \in A

Read as: c belongs to capital A

Means: c belongs to capital A

1 occurrence in this chapter

Equation form expr-12a9b7e1bef1eb61

Y={⟨γ,i⟩∈X:(∀⟨δ,j⟩∈X)i≤j}Y = \Setabs{\tuple{\gamma, i} \in X}{(\forall \tuple{\delta, j} \in X)i \leq j}

Read as: capital Y equals the set of the ordered pair gamma, then i belongs to capital X such that for every the ordered pair delta, then j in capital X, i is less than or equal to j

Means: capital Y equals the set of the ordered pair gamma, then i belongs to capital X such that for every the ordered pair delta, then j in capital X, i is less than or equal to j

1 occurrence in this chapter

Equation form expr-131ff57ab08bc793

α·(β·γ)=(α·β)·γ\alpha \ordtimes (\beta \ordtimes \gamma) = (\alpha \ordtimes \beta) \ordtimes \gamma

Read as: alpha ordinal times open scope, beta ordinal times gamma, close scope equals open scope, alpha ordinal times beta, close scope ordinal times gamma

Means: alpha ordinal times open scope, beta ordinal times gamma, close scope equals open scope, alpha ordinal times beta, close scope ordinal times gamma

1 occurrence in this chapter

Equation form expr-170bdc6ed66e9cb9

x⊆Ax \subseteq A

Read as: x is a subset of capital A

Means: x is a subset of capital A

1 occurrence in this chapter

Equation form expr-18f5384d58bcb1bb

YY

Read as: capital Y

Means: capital Y

2 occurrences in this chapter

Equation form expr-1c2b21ca0ef53d56

ω(2)=ω·ω\ordexpo{\omega}{2} = \omega \ordtimes \omega

Read as: ordinal exponentiation of omega to the power two equals omega ordinal times omega

Means: ordinal exponentiation of omega to the power two equals omega ordinal times omega

1 occurrence in this chapter

Equation form expr-1dbfe792fb165970

{γ∈α:f(γ)≠0}\Setabs{\gamma \in \alpha}{f(\gamma) \neq 0}

Read as: the set of gamma belongs to alpha such that f of gamma is not equal to zero

Means: the set of gamma belongs to alpha such that f of gamma is not equal to zero

1 occurrence in this chapter

Equation form expr-1e756bbccd42b487

α≥ω\alpha \geq \omega

Read as: alpha is greater than or equal to omega

Means: alpha is greater than or equal to omega

1 occurrence in this chapter

Equation form expr-1fe8d17a31455179

rank(℘(A))=α+1\setrank{\Pow{A}} = \alpha \ordplus 1

Read as: the rank of the power set of capital A equals alpha ordinal plus one

Means: the rank of the power set of capital A equals alpha ordinal plus one

1 occurrence in this chapter

Equation form expr-20a3512ed1d530f7

α≈α+1\alpha \approx \alpha \ordplus 1

Read as: alpha is approximately equal to alpha ordinal plus one

Means: alpha is approximately equal to alpha ordinal plus one

1 occurrence in this chapter

Equation form expr-20f49c2ade99ee8d

X={α+δ:δ<β}X = \Setabs{\alpha \ordplus \delta}{\delta < \beta}

Read as: capital X equals the set of alpha ordinal plus delta such that delta is less than beta

Means: capital X equals the set of alpha ordinal plus delta such that delta is less than beta

1 occurrence in this chapter

Equation form expr-252f10c83610ebca

ff

Read as: f

Means: f

1 occurrence in this chapter

Equation form expr-27389467e8e63847

α+1=α+\alpha \ordplus 1 = \ordsucc{\alpha}

Read as: alpha ordinal plus one equals the ordinal successor of alpha

Means: alpha ordinal plus one equals the ordinal successor of alpha

1 occurrence in this chapter

Equation form expr-283d5dfc90b06083

α+δ\alpha \ordplus \delta

Read as: alpha ordinal plus delta

Means: alpha ordinal plus delta

1 occurrence in this chapter

Equation form expr-2d4f7c9397247799

α·β<α·γ\alpha \ordtimes \beta < \alpha \ordtimes \gamma

Read as: alpha ordinal times beta is less than alpha ordinal times gamma

Means: alpha ordinal times beta is less than alpha ordinal times gamma

1 occurrence in this chapter

Equation form expr-351f2b663ca388e1

α⊔1=(α×{0})⊔({0}×{1})\alpha\disjointsum1 = (\alpha \times \{0\}) \disjointsum (\{0\} \times \{1\})

Read as: alpha disjoint sum one equals open scope, alpha times the set containing zero, close scope disjoint sum open scope, the set containing zero times the set containing one, close scope

Means: alpha disjoint sum one equals open scope, alpha times the set containing zero, close scope disjoint sum open scope, the set containing zero times the set containing one, close scope

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

1 occurrence in this chapter

Equation form expr-3bf8261e0d2dfb32

rank(A×B)max(rank(A),rank(B))+2\setrank{A \times B}\max(\setrank{A}, \setrank{B}) \ordplus 2

Read as: the rank of capital A times capital B the maximum of open scope, the rank of capital A the empty expression, then the rank of capital B, close scope ordinal plus two

Means: the rank of capital A times capital B the maximum of open scope, the rank of capital A the empty expression, then the rank of capital B, close scope ordinal plus two

1 occurrence in this chapter

Equation form expr-3da96fd8ab43fab4

α+β<α+γ\alpha \ordplus \beta < \alpha \ordplus \gamma

Read as: alpha ordinal plus beta is less than alpha ordinal plus gamma

Means: alpha ordinal plus beta is less than alpha ordinal plus gamma

1 occurrence in this chapter

Equation form expr-3e1f222d09c5cffd

α(β)\ordexpo{\alpha}{\beta}

Read as: ordinal exponentiation of alpha to the power beta

Means: ordinal exponentiation of alpha to the power beta

1 occurrence in this chapter

Equation form expr-3fb53d927c6bf9ec

α·β=ord(α×β,∢)\alpha \ordtimes \beta = \ordtype{\alpha \times \beta, \rlexless}

Read as: alpha ordinal times beta equals the order type of alpha times beta is less than in reverse lexicographic order

Means: alpha ordinal times beta equals the order type of alpha times beta is less than in reverse lexicographic order

1 occurrence in this chapter

Equation form expr-4026a46e7048ee4a

f(0,1)∈α∖ran(g)f(0,1) \in \alpha \setminus \ran{g}

Read as: f of zero and one belongs to alpha set minus the range of g

Means: f of zero and one belongs to alpha set minus the range of g

1 occurrence in this chapter

Equation form expr-4099dc1518fd7e25

1={0}1 = \{0\}

Read as: one equals the set containing zero

Means: one equals the set containing zero

1 occurrence in this chapter

Equation form expr-46328f2e683dead2

A⊔B=(A×{0})∪(B×{1})A \disjointsum B = (A\times \{0\}) \cup (B \times \{1\})

Read as: capital A disjoint sum capital B equals open scope, capital A times the set containing zero, close scope union open scope, capital B times the set containing one, close scope

Means: capital A disjoint sum capital B equals open scope, capital A times the set containing zero, close scope union open scope, capital B times the set containing one, close scope

1 occurrence in this chapter

Equation form expr-4893e9df8b5496eb

α\alpha

Read as: alpha

Means: alpha

26 occurrences in this chapter

Equation form expr-498141e4330a70e5

α(β)=ord(finfun(α,β),⊏)\ordexpo{\alpha}{\beta} = \ordtype{\text{finfun}(\alpha, \beta), \sqsubset}

Read as: ordinal exponentiation of alpha to the power beta equals the order type of finfun, open scope, alpha, then beta, close scope is a proper initial segment of

Means: ordinal exponentiation of alpha to the power beta equals the order type of finfun, open scope, alpha, then beta, close scope is a proper initial segment of

1 occurrence in this chapter

Equation form expr-4b32d3f1991eab62

1+ω=ω<ω+11 \ordplus \omega = \omega < \omega \ordplus 1

Read as: one ordinal plus omega equals omega is less than omega ordinal plus one

Means: one ordinal plus omega equals omega is less than omega ordinal plus one

1 occurrence in this chapter

Equation form expr-4b68ab3847feda7d

XX

Read as: capital X

Means: capital X

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

2 occurrences in this chapter

Equation form expr-50357c7599c18542

{n2:n∈ω}\Setabs{\nicefrac{n}{2}}{n \in \omega}

Read as: the set of the fraction n over two such that n belongs to omega

Means: the set of the fraction n over two such that n belongs to omega

1 occurrence in this chapter

Equation form expr-53fec2e06a93fdda

δ+1<β\delta\ordplus 1 < \beta

Read as: delta ordinal plus one is less than beta

Means: delta ordinal plus one is less than beta

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

1 occurrence in this chapter

Equation form expr-546c92f2db1415b8

γ<α\gamma < \alpha

Read as: gamma is less than alpha

Means: gamma is less than alpha

1 occurrence in this chapter

Equation form expr-559aead08264d579

AA

Read as: capital A

Means: capital A

5 occurrences in this chapter

Equation form expr-560f0f89dfe2e1a5

g(γ)=f(γ,0)g(\gamma) = f(\gamma, 0)

Read as: g of gamma equals f of gamma and zero

Means: g of gamma equals f of gamma and zero

1 occurrence in this chapter

Equation form expr-573b250f62878407

(α+β)+0=α+β=α+(β+0)(\alpha \ordplus \beta) \ordplus 0 = \alpha \ordplus \beta = \alpha \ordplus (\beta \ordplus 0)

Read as: open scope, alpha ordinal plus beta, close scope ordinal plus zero equals alpha ordinal plus beta equals alpha ordinal plus open scope, beta ordinal plus zero, close scope

Means: open scope, alpha ordinal plus beta, close scope ordinal plus zero equals alpha ordinal plus beta equals alpha ordinal plus open scope, beta ordinal plus zero, close scope

1 occurrence in this chapter

Equation form expr-57b3be130bf4416d

α+0=αα+(β+1)=(α+β)+1α+β=lsubδ<β(α+δ)if β is a limit ordinal\alpha\ordplus 0 &= \alpha\\ \alpha \ordplus (\beta\ordplus 1) &= (\alpha \ordplus \beta) \ordplus 1\\ \alpha \ordplus \beta &= \supstrict_{\delta < \beta}(\alpha \ordplus \delta) && \text{if $\beta $ is a limit ordinal}

Read as: Source-ordered display. alpha ordinal plus zero equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus one. Then, alpha ordinal plus beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal plus delta, close scope, if beta is a limit ordinal. End display

Means: Source-ordered display. alpha ordinal plus zero equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus one. Then, alpha ordinal plus beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal plus delta, close scope, if beta is a limit ordinal. End display

1 occurrence in this chapter

Equation form expr-58bb0103cc434b09

rank(A∪B)=max(α,β)\setrank{A \cup B} = \max(\alpha, \beta)

Read as: the rank of capital A union capital B equals the maximum of alpha, then beta

Means: the rank of capital A union capital B equals the maximum of alpha, then beta

1 occurrence in this chapter

Equation form expr-5e5957f0d058f1a9

finfun(α,γ)\text{finfun}(\alpha, \gamma)

Read as: finfun, open scope, alpha, then gamma, close scope

Means: finfun, open scope, alpha, then gamma, close scope

1 occurrence in this chapter

Equation form expr-5f67a793a81728e3

α,β\alpha, \beta

Read as: alpha, then beta

Means: alpha, then beta

3 occurrences in this chapter

Equation form expr-5feceb66ffc86f38

00

Read as: zero

Means: zero

2 occurrences in this chapter

Equation form expr-609d091fd0e36ec3

α(0)=1α(β+1)=α(β)·αα(β)=⋃δ<βα(δ)when β is a limit ordinal\ordexpo{\alpha}{0} &= 1\\ \ordexpo{\alpha}{\beta\ordplus 1} &=\ordexpo{\alpha}{\beta} \ordtimes \alpha\\ \ordexpo{\alpha}{\beta} &= \bigcup_{\delta < \beta}\ordexpo{\alpha}{\delta}& & \text{when $\beta$ is a limit ordinal}

Read as: Source-ordered display. ordinal exponentiation of alpha to the power zero equals one. Then, ordinal exponentiation of alpha to the power beta ordinal plus one equals ordinal exponentiation of alpha to the power beta ordinal times alpha. Then, ordinal exponentiation of alpha to the power beta equals the union over delta is less than beta of ordinal exponentiation of alpha to the power delta when beta is a limit ordinal. End display

Means: Source-ordered display. ordinal exponentiation of alpha to the power zero equals one. Then, ordinal exponentiation of alpha to the power beta ordinal plus one equals ordinal exponentiation of alpha to the power beta ordinal times alpha. Then, ordinal exponentiation of alpha to the power beta equals the union over delta is less than beta of ordinal exponentiation of alpha to the power delta when beta is a limit ordinal. End display

1 occurrence in this chapter

Equation form expr-6118280e7afa1b89

(α+β)+γ=lsubδ<γ((α+β)+δ)=lsubδ<γ(α+(β+δ))=α+lsubδ<γ(β+δ)=α+(β+γ)(\alpha \ordplus \beta) \ordplus \gamma & = \supstrict_{\delta < \gamma}((\alpha \ordplus \beta) \ordplus \delta) \\ &= \supstrict_{\delta < \gamma}(\alpha \ordplus (\beta \ordplus \delta))\\ &= \alpha \ordplus \supstrict_{\delta < \gamma}(\beta \ordplus \delta)\\ & = \alpha \ordplus (\beta \ordplus \gamma)

Read as: Source-ordered display. open scope, alpha ordinal plus beta, close scope ordinal plus gamma equals the strict supremum over delta is less than gamma of open scope, alpha ordinal plus beta, close scope ordinal plus delta. Then, equals the strict supremum over delta is less than gamma of alpha ordinal plus open scope, beta ordinal plus delta, close scope. Then, equals alpha ordinal plus the strict supremum over delta is less than gamma of beta ordinal plus delta. Then, equals alpha ordinal plus open scope, beta ordinal plus gamma, close scope. End display

Means: Source-ordered display. open scope, alpha ordinal plus beta, close scope ordinal plus gamma equals the strict supremum over delta is less than gamma of open scope, alpha ordinal plus beta, close scope ordinal plus delta. Then, equals the strict supremum over delta is less than gamma of alpha ordinal plus open scope, beta ordinal plus delta, close scope. Then, equals alpha ordinal plus the strict supremum over delta is less than gamma of beta ordinal plus delta. Then, equals alpha ordinal plus open scope, beta ordinal plus gamma, close scope. End display

1 occurrence in this chapter

Equation form expr-654beaadb374454b

f(α)=⟨0,1⟩f(\alpha) = \tuple{0, 1}

Read as: f of alpha equals the ordered pair zero, then one

Means: f of alpha equals the ordered pair zero, then one

1 occurrence in this chapter

Equation form expr-65da559a940a9b79

rank(x)≤rank(A)\setrank{x} \leq \setrank{A}

Read as: the rank of x is less than or equal to the rank of capital A

Means: the rank of x is less than or equal to the rank of capital A

1 occurrence in this chapter

Equation form expr-6738a1de2d4131f3

rank(⟨A,B⟩)=max(α,β)+2\setrank{\tuple{A,B}} = \max(\alpha, \beta) \ordplus 2

Read as: the rank of the ordered pair capital A, then capital B equals the maximum of open scope, alpha, then beta, close scope ordinal plus two

Means: the rank of the ordered pair capital A, then capital B equals the maximum of open scope, alpha, then beta, close scope ordinal plus two

1 occurrence in this chapter

Equation form expr-67610b0632683369

γ\gamma

Read as: gamma

Means: gamma

3 occurrences in this chapter

Equation form expr-68a50860c6f708f4

α·0=0α·(β+1)=(α·β)+αα·β=lsubδ<β(α·δ)when β is a limit ordinal.\alpha \ordtimes 0 &= 0\\ \alpha \ordtimes (\beta \ordplus 1) &= (\alpha \ordtimes \beta) \ordplus \alpha\\ \alpha \ordtimes \beta &= \supstrict_{\delta < \beta}(\alpha \ordtimes \delta) && \text{when $\beta$ is a limit ordinal}.

Read as: Source-ordered display. alpha ordinal times zero equals zero. Then, alpha ordinal times open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus alpha. Then, alpha ordinal times beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal times delta, close scope, when beta is a limit ordinal. End display

Means: Source-ordered display. alpha ordinal times zero equals zero. Then, alpha ordinal times open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus alpha. Then, alpha ordinal times beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal times delta, close scope, when beta is a limit ordinal. End display

1 occurrence in this chapter

Equation form expr-69e508a667cb655a

α=γ+1\alpha = \gamma\ordplus 1

Read as: alpha equals gamma ordinal plus one

Means: alpha equals gamma ordinal plus one

2 occurrences in this chapter

Equation form expr-6b86b273ff34fce1

11

Read as: one

Means: one

1 occurrence in this chapter

Equation form expr-6c7cc04ee472acf2

α+β=α+γ\alpha \ordplus \beta = \alpha\ordplus \gamma

Read as: alpha ordinal plus beta equals alpha ordinal plus gamma

Means: alpha ordinal plus beta equals alpha ordinal plus gamma

1 occurrence in this chapter

Equation form expr-6cee96e44036247b

α=β+γ\alpha = \beta \ordplus \gamma

Read as: alpha equals beta ordinal plus gamma

Means: alpha equals beta ordinal plus gamma

1 occurrence in this chapter

Equation form expr-6cf3fedf7b95831f

f:(α⊔1)→(1⊔α)f \colon (\alpha \disjointsum 1) \to (1 \disjointsum \alpha)

Read as: f colon open scope, alpha disjoint sum one, close scope maps to open scope, one disjoint sum alpha, close scope

Means: f colon open scope, alpha disjoint sum one, close scope maps to open scope, one disjoint sum alpha, close scope

1 occurrence in this chapter

Equation form expr-77329e8cfcdf8717

f:(α⊔1)→αf \colon (\alpha \disjointsum 1) \to \alpha

Read as: f colon open scope, alpha disjoint sum one, close scope maps to alpha

Means: f colon open scope, alpha disjoint sum one, close scope maps to alpha

1 occurrence in this chapter

Equation form expr-7a6fbc20930ad7c0

A∈℘(A)A \in \Pow{A}

Read as: capital A belongs to the power set of capital A

Means: capital A belongs to the power set of capital A

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-7f42c976925d2255

2(3)=8<9=3(2)\ordexpo{2}{3} = 8 < 9 = \ordexpo{3}{2}

Read as: ordinal exponentiation of two to the power three equals eight is less than nine

Means: ordinal exponentiation of two to the power three equals eight is less than nine

1 occurrence in this chapter

Equation form expr-80ea4d0b8d80bb7b

2·ω=ω<ω·22 \ordtimes \omega = \omega < \omega \ordtimes 2

Read as: two ordinal times omega equals omega is less than omega ordinal times two

Means: two ordinal times omega equals omega is less than omega ordinal times two

1 occurrence in this chapter

Equation form expr-840142d87a0eb6f4

rank(℘(A))≤α+1\setrank{\Pow{A}} \leq \alpha \ordplus 1

Read as: the rank of the power set of capital A is less than or equal to alpha ordinal plus one

Means: the rank of the power set of capital A is less than or equal to alpha ordinal plus one

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-8960b9ad00940429

∢\rlexless

Read as: is less than in reverse lexicographic order

Means: is less than in reverse lexicographic order

2 occurrences in this chapter

Equation form expr-8bddf64d9d4172ee

α·β=α·γ\alpha \ordtimes \beta = \alpha\ordtimes\gamma

Read as: alpha ordinal times beta equals alpha ordinal times gamma

Means: alpha ordinal times beta equals alpha ordinal times gamma

1 occurrence 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

2 occurrences in this chapter

Equation form expr-90e19c4b02f79730

ω·2\omega \ordtimes 2

Read as: omega ordinal times two

Means: omega ordinal times two

1 occurrence in this chapter

Equation form expr-91c7df5e88e3759f

rank(A)=α\setrank{A} = \alpha

Read as: the rank of capital A equals alpha

Means: the rank of capital A equals alpha

1 occurrence in this chapter

Equation form expr-91e88acb54883d43

α+1\alpha\ordplus 1

Read as: alpha ordinal plus one

Means: alpha ordinal plus one

1 occurrence in this chapter

Equation form expr-93b91265eff125c7

g∘f\comp{f}{g}

Read as: g composed with f

Means: g composed with f

1 occurrence in this chapter

Equation form expr-9660eb5ccbd75438

α+(β+γ)=(α+β)+γ\alpha \ordplus (\beta \ordplus \gamma) = (\alpha \ordplus \beta) \ordplus \gamma

Read as: alpha ordinal plus open scope, beta ordinal plus gamma, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus gamma

Means: alpha ordinal plus open scope, beta ordinal plus gamma, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus gamma

1 occurrence in this chapter

Equation form expr-97aa0725397ca691

α·(β+γ)=(α·β)+(α·γ)\alpha \ordtimes (\beta \ordplus \gamma) = (\alpha \ordtimes \beta )\ordplus (\alpha\ordtimes \gamma)

Read as: alpha ordinal times open scope, beta ordinal plus gamma, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus open scope, alpha ordinal times gamma, close scope

Means: alpha ordinal times open scope, beta ordinal plus gamma, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus open scope, alpha ordinal times gamma, close scope

1 occurrence in this chapter

Equation form expr-9a70a3b2427db72f

rank(A×B)≤max(α,β)+2\setrank{A \times B} \leq \max(\alpha, \beta) \ordplus 2

Read as: the rank of capital A times capital B is less than or equal to the maximum of open scope, alpha, then beta, close scope ordinal plus two

Means: the rank of capital A times capital B is less than or equal to the maximum of open scope, alpha, then beta, close scope ordinal plus two

1 occurrence in this chapter

Equation form expr-9cd666c82e17249b

rank({A,B})=max(α,β)+1\setrank{\{A, B\}} = \max(\alpha, \beta) \ordplus 1

Read as: the rank of the set containing capital A, then capital B equals the maximum of open scope, alpha, then beta, close scope ordinal plus one

Means: the rank of the set containing capital A, then capital B equals the maximum of open scope, alpha, then beta, close scope ordinal plus one

1 occurrence in this chapter

Equation form expr-a040d23c32730664

f(γ0)<g(γ0)f(\gamma_0) < g(\gamma_0)

Read as: f of gamma subscript zero is less than g of gamma subscript zero

Means: f of gamma subscript zero is less than g of gamma subscript zero

1 occurrence in this chapter

Equation form expr-a0a7ed650db6bc1e

1+α=1+(β+γ)=(1+β)+γ=lsubδ<β(1+δ)+γ=β+γ=α.1 \ordplus \alpha = 1 \ordplus (\beta \ordplus \gamma) = (1 \ordplus \beta) \ordplus \gamma = \supstrict_{\delta < \beta} (1 \ordplus \delta) \ordplus \gamma = \beta \ordplus \gamma = \alpha.

Read as: one ordinal plus alpha equals one ordinal plus open scope, beta ordinal plus gamma, close scope equals open scope, one ordinal plus beta, close scope ordinal plus gamma equals the strict supremum over delta is less than beta of open scope, one ordinal plus delta, close scope ordinal plus gamma equals beta ordinal plus gamma equals alpha

Means: one ordinal plus alpha equals one ordinal plus open scope, beta ordinal plus gamma, close scope equals open scope, one ordinal plus beta, close scope ordinal plus gamma equals the strict supremum over delta is less than beta of open scope, one ordinal plus delta, close scope ordinal plus gamma equals beta ordinal plus gamma equals alpha

1 occurrence in this chapter

Equation form expr-a0be0b095b7f0867

ω+1\omega \ordplus 1

Read as: omega ordinal plus one

Means: omega ordinal plus one

1 occurrence in this chapter

Equation form expr-a38c055ee6304ace

⟨α⊔β,∢⟩\tuple{\alpha \disjointsum \beta, \rlexless}

Read as: the ordered pair alpha disjoint sum beta, then is less than in reverse lexicographic order

Means: the ordered pair alpha disjoint sum beta, then is less than in reverse lexicographic order

1 occurrence in this chapter

Equation form expr-a51f348d58159132

(α+β)+δ=α+(β+δ)(\alpha \ordplus \beta) \ordplus \delta = \alpha \ordplus (\beta \ordplus \delta)

Read as: open scope, alpha ordinal plus beta, close scope ordinal plus delta equals alpha ordinal plus open scope, beta ordinal plus delta, close scope

Means: open scope, alpha ordinal plus beta, close scope ordinal plus delta equals alpha ordinal plus open scope, beta ordinal plus delta, close scope

2 occurrences in this chapter

Equation form expr-a560b4dc73de5ac7

α+γ≤β+γ\alpha \ordplus \gamma \leq \beta \ordplus \gamma

Read as: alpha ordinal plus gamma is less than or equal to beta ordinal plus gamma

Means: alpha ordinal plus gamma is less than or equal to beta ordinal plus gamma

1 occurrence in this chapter

Equation form expr-a5e17f3790240a7c

α+β=lsubδ<β(α+δ)\alpha \ordplus \beta = \supstrict_{\delta< \beta}(\alpha\ordplus \delta)

Read as: alpha ordinal plus beta equals the strict supremum over delta is less than beta of alpha ordinal plus delta

Means: alpha ordinal plus beta equals the strict supremum over delta is less than beta of alpha ordinal plus delta

1 occurrence in this chapter

Equation form expr-ab582afcf822a356

1+ω1 \ordplus \omega

Read as: one ordinal plus omega

Means: one ordinal plus omega

1 occurrence in this chapter

Equation form expr-ae3660560a54702c

⇒\Rightarrow

Read as: implies

Means: implies

5 occurrences in this chapter

Equation form expr-aeaf3d791d30f0c3

1+α=α1 \ordplus \alpha = \alpha

Read as: one ordinal plus alpha equals alpha

Means: one ordinal plus alpha equals alpha

2 occurrences in this chapter

Equation form expr-b540b5e3442aa867

f⊏gf \sqsubset g

Read as: f is a proper initial segment of g

Means: f is a proper initial segment of g

1 occurrence in this chapter

Equation form expr-b635eccbb7107f13

α+0=ord((α×{0})∪(0×{1}),∢)=ord((α×{0})∪{0},∢)=αα+(β+1)=ord((α×{0})∪(β+×{1}),∢)=ord((α×{0})∪(β×{1}),∢)+1=(α+β)+1\alpha \ordplus 0 & = \ordtype{(\alpha \times \{0\}) \cup (0 \times \{1\}), \rlexless} \\ &= \ordtype{(\alpha \times \{0\}) \cup \{0\}, \rlexless}\\ &= \alpha\\ \alpha \ordplus (\beta \ordplus 1) &= \ordtype{(\alpha\times \{0\}) \cup (\ordsucc{\beta}\times \{1\}), \rlexless} \\ &= \ordtype{(\alpha\times \{0\}) \cup (\beta \times \{1\}), \rlexless} \ordplus 1\\ &= (\alpha \ordplus \beta) \ordplus 1

Read as: Source-ordered display. alpha ordinal plus zero equals the order type of open scope, alpha times the set containing zero, close scope union open scope, zero times the set containing one, close scope is less than in reverse lexicographic order. Then, equals the order type of open scope, alpha times the set containing zero, close scope union the set containing zero is less than in reverse lexicographic order. Then, equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals the order type of open scope, alpha times the set containing zero, close scope union open scope, the ordinal successor of beta times the set containing one, close scope is less than in reverse lexicographic order. Then, equals the order type of open scope, alpha times the set containing zero, close scope union open scope, beta times the set containing one, close scope is less than in reverse lexicographic order ordinal plus one. Then, equals open scope, alpha ordinal plus beta, close scope ordinal plus one. End display

Means: Source-ordered display. alpha ordinal plus zero equals the order type of open scope, alpha times the set containing zero, close scope union open scope, zero times the set containing one, close scope is less than in reverse lexicographic order. Then, equals the order type of open scope, alpha times the set containing zero, close scope union the set containing zero is less than in reverse lexicographic order. Then, equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals the order type of open scope, alpha times the set containing zero, close scope union open scope, the ordinal successor of beta times the set containing one, close scope is less than in reverse lexicographic order. Then, equals the order type of open scope, alpha times the set containing zero, close scope union open scope, beta times the set containing one, close scope is less than in reverse lexicographic order ordinal plus one. Then, equals open scope, alpha ordinal plus beta, close scope ordinal plus one. End display

1 occurrence in this chapter

Equation form expr-b66990ae723bd473

α·γ≤β·γ\alpha \ordtimes \gamma \leq \beta \ordtimes \gamma

Read as: alpha ordinal times gamma is less than or equal to beta ordinal times gamma

Means: alpha ordinal times gamma is less than or equal to beta ordinal times gamma

1 occurrence in this chapter

Equation form expr-ba816dcdb55cd716

g:(1⊔α)→αg \colon (1 \disjointsum \alpha) \to \alpha

Read as: g colon open scope, one disjoint sum alpha, close scope maps to alpha

Means: g colon open scope, one disjoint sum alpha, close scope maps to alpha

1 occurrence in this chapter

Equation form expr-bc27413f3fedb185

(α+β)+(δ+1)=((α+β)+δ)+1=(α+(β+δ))+1=α+((β+δ)+1)=α+(β+(δ+1))(\alpha \ordplus \beta) \ordplus (\delta \ordplus 1) & = ((\alpha \ordplus \beta) \ordplus \delta)\ordplus 1\\ & = (\alpha \ordplus (\beta \ordplus \delta)) \ordplus 1\\ & = \alpha \ordplus ((\beta \ordplus \delta)\ordplus 1)\\ & = \alpha \ordplus (\beta \ordplus (\delta\ordplus 1))

Read as: Source-ordered display. open scope, alpha ordinal plus beta, close scope ordinal plus open scope, delta ordinal plus one, close scope equals open scope, open scope, alpha ordinal plus beta, close scope ordinal plus delta, close scope ordinal plus one. Then, equals open scope, alpha ordinal plus open scope, beta ordinal plus delta, close scope, close scope ordinal plus one. Then, equals alpha ordinal plus open scope, open scope, beta ordinal plus delta, close scope ordinal plus one, close scope. Then, equals alpha ordinal plus open scope, beta ordinal plus open scope, delta ordinal plus one, close scope, close scope. End display

Means: Source-ordered display. open scope, alpha ordinal plus beta, close scope ordinal plus open scope, delta ordinal plus one, close scope equals open scope, open scope, alpha ordinal plus beta, close scope ordinal plus delta, close scope ordinal plus one. Then, equals open scope, alpha ordinal plus open scope, beta ordinal plus delta, close scope, close scope ordinal plus one. Then, equals alpha ordinal plus open scope, open scope, beta ordinal plus delta, close scope ordinal plus one, close scope. Then, equals alpha ordinal plus open scope, beta ordinal plus open scope, delta ordinal plus one, close scope, close scope. End display

1 occurrence in this chapter

Equation form expr-c1cf61a4fa5e8b4a

α1,α2,β1,β2\alpha_1, \alpha_2, \beta_1, \beta_2

Read as: alpha subscript one, then alpha subscript two, then beta subscript one, then beta subscript two

Means: alpha subscript one, then alpha subscript two, then beta subscript one, then beta subscript two

1 occurrence in this chapter

Equation form expr-c2048e02de4999a2

f:α→βf \colon \alpha \to \beta

Read as: f colon alpha maps to beta

Means: f colon alpha maps to beta

1 occurrence in this chapter

Equation form expr-c21d47aca340fe21

⟨α1,α2⟩∢⟨β1,β2⟩ iff either α2∈β2or both α2=β2 and α1∈β1\tuple{\alpha_1, \alpha_2} \rlexless \tuple{\beta_1, \beta_2}\text{ iff }& \text{either $\alpha_2 \in \beta_2$}\\ & \text{or both $\alpha_2 = \beta_2$ and $\alpha_1 \in \beta_1$}

Read as: Source-ordered display. the ordered pair alpha subscript one, then alpha subscript two is less than in reverse lexicographic order the ordered pair beta subscript one, then beta subscript two if and only if either alpha subscript two belongs to beta subscript two. Then, or both alpha subscript two equals beta subscript two and alpha subscript one belongs to beta subscript one. End display

Means: Source-ordered display. the ordered pair alpha subscript one, then alpha subscript two is less than in reverse lexicographic order the ordered pair beta subscript one, then beta subscript two if and only if either alpha subscript two belongs to beta subscript two. Then, or both alpha subscript two equals beta subscript two and alpha subscript one belongs to beta subscript one. End display

1 occurrence in this chapter

Equation form expr-c2490f52044ecc34

β+1\beta \ordplus 1

Read as: beta ordinal plus one

Means: beta ordinal plus one

1 occurrence in this chapter

Equation form expr-c8c50b9fd52112ab

ord(finfun(α,β),⊏)\ordtype{\text{finfun}(\alpha, \beta), \sqsubset}

Read as: the order type of finfun, open scope, alpha, then beta, close scope is a proper initial segment of

Means: the order type of finfun, open scope, alpha, then beta, close scope is a proper initial segment of

1 occurrence in this chapter

Equation form expr-c9f6af631d1996e0

rank(⋃A)=α\setrank{\bigcup A} = \alpha

Read as: the rank of the union of capital A equals alpha

Means: the rank of the union of capital A equals alpha

2 occurrences in this chapter

Equation form expr-ca3c103245cb2587

A×B⊆℘(℘(A∪B))A \times B \subseteq \Pow{\Pow{A \cup B}}

Read as: capital A times capital B is a subset of the power set of the power set of capital A union capital B

Means: capital A times capital B is a subset of the power set of the power set of capital A union capital B

1 occurrence in this chapter

Equation form expr-ca6aa6b30c713e06

rank(c)=γ\setrank{c} = \gamma

Read as: the rank of c equals gamma

Means: the rank of c equals gamma

1 occurrence in this chapter

Equation form expr-cce8da7b0bad451c

γ=δ+1\gamma = \delta\ordplus 1

Read as: gamma equals delta ordinal plus one

Means: gamma equals delta ordinal plus one

1 occurrence in this chapter

Equation form expr-d0892578eb0ea6e0

rank(B)=β\setrank{B} = \beta

Read as: the rank of capital B equals beta

Means: the rank of capital B equals beta

1 occurrence in this chapter

Equation form expr-d0a0b033be0537f2

rank(A×B)=max(rank(A),rank(B))\setrank{A \times B}= \max(\setrank{A}, \setrank{B})

Read as: the rank of capital A times capital B equals the maximum of the rank of capital A the empty expression, then the rank of capital B

Means: the rank of capital A times capital B equals the maximum of the rank of capital A the empty expression, then the rank of capital B

1 occurrence in this chapter

Equation form expr-d3c2c72cf2d558f1

ω≤α\omega \leq \alpha

Read as: omega is less than or equal to alpha

Means: omega is less than or equal to alpha

1 occurrence in this chapter

Equation form expr-d59ebe183fd26fa8

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-d5a73e89f5709742

β<γ\beta < \gamma

Read as: beta is less than gamma

Means: beta is less than gamma

2 occurrences in this chapter

Equation form expr-d758bfb1a1a86d63

β≠∅\beta \neq \emptyset

Read as: beta is not equal to the empty set

Means: beta is not equal to the empty set

1 occurrence in this chapter

Equation form expr-d82dda87fbb57f20

2(ω)=⋃δ<ω2(δ)=ω\ordexpo{2}{\omega} = \bigcup_{\delta < \omega}\ordexpo{2}{\delta} = \omega

Read as: ordinal exponentiation of two to the power omega equals the union over delta is less than omega of ordinal exponentiation of two to the power delta

Means: ordinal exponentiation of two to the power omega equals the union over delta is less than omega of ordinal exponentiation of two to the power delta

1 occurrence in this chapter

Equation form expr-da814de5df013f7e

1+ω=lsubn<ω(1+n)=ω∈ω∪{ω}=ω+=ω+11 \ordplus \omega = \supstrict_{n < \omega} (1 \ordplus n) = \omega \in \omega \cup \{\omega\} = \ordsucc{\omega} = \omega \ordplus 1

Read as: one ordinal plus omega equals the strict supremum over n is less than omega of one ordinal plus n equals omega belongs to omega union the set containing omega equals the ordinal successor of omega equals omega ordinal plus one

Means: one ordinal plus omega equals the strict supremum over n is less than omega of one ordinal plus n equals omega belongs to omega union the set containing omega equals the ordinal successor of omega equals omega ordinal plus one

1 occurrence in this chapter

Equation form expr-deb8538b3eaf8bdd

γ=0\gamma = 0

Read as: gamma equals zero

Means: gamma equals zero

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-e1ac8c86a3676366

β+\ordsucc{\beta}

Read as: the ordinal successor of beta

Means: the ordinal successor of beta

1 occurrence in this chapter

Equation form expr-e1bb16088f6263a3

α∉ω\alpha\notin \omega

Read as: alpha does not belong to omega

Means: alpha does not belong to omega

1 occurrence in this chapter

Equation form expr-e230ccc102f6823b

2·ω2 \ordtimes \omega

Read as: two ordinal times omega

Means: two ordinal times omega

1 occurrence in this chapter

Equation form expr-e7c927775de66fce

δ<β\delta < \beta

Read as: delta is less than beta

Means: delta is less than beta

2 occurrences in this chapter

Equation form expr-ea4fcf93f33f81fb

finfun(α,β)\text{finfun}(\alpha,\beta)

Read as: finfun, open scope, alpha, then beta, close scope

Means: finfun, open scope, alpha, then beta, close scope

2 occurrences in this chapter

Equation form expr-eb683663e82c11f8

f(γ)=⟨γ,0⟩f(\gamma) = \tuple{\gamma, 0}

Read as: f of gamma equals the ordered pair gamma, then zero

Means: f of gamma equals the ordered pair gamma, then zero

1 occurrence in this chapter

Equation form expr-ecf9f989cbf707cd

α+β\alpha \ordplus \beta

Read as: alpha ordinal plus beta

Means: alpha ordinal plus beta

3 occurrences in this chapter

Equation form expr-edcf12781d91b782

γ=α+δ\gamma = \alpha \ordplus \delta

Read as: gamma equals alpha ordinal plus delta

Means: gamma equals alpha ordinal plus delta

1 occurrence in this chapter

Equation form expr-f090b0c115f6f729

α≈α+1\alpha \approx \alpha\ordplus 1

Read as: alpha is approximately equal to alpha ordinal plus one

Means: alpha is approximately equal to alpha ordinal plus one

1 occurrence in this chapter

Equation form expr-f3f3804480e8551a

β\beta

Read as: beta

Means: beta

17 occurrences in this chapter

Equation form expr-f4ddfb7be3f1bc22

rank(℘(A))=α+1\setrank{\Pow{A}} = \alpha\ordplus 1

Read as: the rank of the power set of capital A equals alpha ordinal plus one

Means: the rank of the power set of capital A equals alpha ordinal plus one

1 occurrence in this chapter

Equation form expr-f7c889ee296bd2ed

β=γ\beta = \gamma

Read as: beta equals gamma

Means: beta equals gamma

2 occurrences 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-f89412665cfd3b55

X⊆α⊔βX \subseteq \alpha \disjointsum \beta

Read as: capital X is a subset of alpha disjoint sum beta

Means: capital X is a subset of alpha disjoint sum beta

1 occurrence in this chapter

Equation form expr-fab30dbcc4c008a4

α≠0\alpha \neq 0

Read as: alpha is not equal to zero

Means: alpha is not equal to zero

2 occurrences in this chapter

Equation form expr-fb1b60b3fbb25c13

f≠gf \neq g

Read as: f is not equal to g

Means: f is not equal to g

1 occurrence in this chapter

Equation form expr-fc31480b031d127a

rank(⋃A)=γ\setrank{\bigcup A} = \gamma

Read as: the rank of the union of capital A equals gamma

Means: the rank of the union of capital A equals gamma

2 occurrences in this chapter

Definition one in this chapter

This source definition contains, in source order: capital A; then capital B; then capital A disjoint sum capital B equals open scope, capital A times the set containing zero, close scope union open scope, capital B times the set containing one, close scope. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition two in this chapter

This source definition contains, in source order: alpha subscript one, then alpha subscript two, then beta subscript one, then beta subscript two; then Source-ordered display. the ordered pair alpha subscript one, then alpha subscript two is less than in reverse lexicographic order the ordered pair beta subscript one, then beta subscript two if and only if either alpha subscript two belongs to beta subscript two. Then, or both alpha subscript two equals beta subscript two and alpha subscript one belongs to beta subscript one. 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. the ordered pair alpha subscript one, then alpha subscript two is less than in reverse lexicographic order the ordered pair beta subscript one, then beta subscript two if and only if either alpha subscript two belongs to beta subscript two. Then, or both alpha subscript two equals beta subscript two and alpha subscript one belongs to beta subscript one. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition three in this chapter

This source definition contains, in source order: alpha; then beta; then alpha ordinal plus beta equals the order type of alpha disjoint sum beta is less than in reverse lexicographic order. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma one in this chapter

This source lemma contains, in source order: the ordered pair alpha disjoint sum beta, then is less than in reverse lexicographic order; then alpha; then beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition one in this chapter

This source proposition contains, in source order: alpha ordinal plus one equals the ordinal successor of alpha; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma two in this chapter

This source lemma contains, in source order: alpha, then beta; then Source-ordered display. alpha ordinal plus zero equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus one. Then, alpha ordinal plus beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal plus delta, close scope, if beta is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math two in this chapter

This source display math contains, in source order: Source-ordered display. alpha ordinal plus zero equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus one. Then, alpha ordinal plus beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal plus delta, close scope, if beta is a limit ordinal. 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: Source-ordered display. alpha ordinal plus zero equals the order type of open scope, alpha times the set containing zero, close scope union open scope, zero times the set containing one, close scope is less than in reverse lexicographic order. Then, equals the order type of open scope, alpha times the set containing zero, close scope union the set containing zero is less than in reverse lexicographic order. Then, equals alpha. Then, alpha ordinal plus open scope, beta ordinal plus one, close scope equals the order type of open scope, alpha times the set containing zero, close scope union open scope, the ordinal successor of beta times the set containing one, close scope is less than in reverse lexicographic order. Then, equals the order type of open scope, alpha times the set containing zero, close scope union open scope, beta times the set containing one, close scope is less than in reverse lexicographic order ordinal plus one. Then, equals open scope, alpha ordinal plus beta, close scope ordinal plus one. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma three in this chapter

This source lemma contains, in source order: alpha, then beta, then gamma; then beta is less than gamma; then alpha ordinal plus beta is less than alpha ordinal plus gamma; then alpha ordinal plus beta equals alpha ordinal plus gamma; then beta equals gamma; then alpha ordinal plus open scope, beta ordinal plus gamma, close scope equals open scope, alpha ordinal plus beta, close scope ordinal plus gamma; then alpha is less than or equal to beta; then alpha ordinal plus gamma is less than or equal to beta ordinal plus gamma. 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: Source-ordered display. open scope, alpha ordinal plus beta, close scope ordinal plus open scope, delta ordinal plus one, close scope equals open scope, open scope, alpha ordinal plus beta, close scope ordinal plus delta, close scope ordinal plus one. Then, equals open scope, alpha ordinal plus open scope, beta ordinal plus delta, close scope, close scope ordinal plus one. Then, equals alpha ordinal plus open scope, open scope, beta ordinal plus delta, close scope ordinal plus one, close scope. Then, equals alpha ordinal plus open scope, beta ordinal plus open scope, delta ordinal plus one, close scope, close scope. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math five in this chapter

This source display math contains, in source order: Source-ordered display. open scope, alpha ordinal plus beta, close scope ordinal plus gamma equals the strict supremum over delta is less than gamma of open scope, alpha ordinal plus beta, close scope ordinal plus delta. Then, equals the strict supremum over delta is less than gamma of alpha ordinal plus open scope, beta ordinal plus delta, close scope. Then, equals alpha ordinal plus the strict supremum over delta is less than gamma of beta ordinal plus delta. Then, equals alpha ordinal plus open scope, beta ordinal plus gamma, close scope. End display. 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 two in this chapter

This source proposition contains, in source order: one ordinal plus omega equals omega is less than omega ordinal plus one. 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 rank of capital A equals alpha; then the rank of capital B equals beta; then the rank of the power set of capital A equals alpha ordinal plus one; then the rank of the set containing capital A, then capital B equals the maximum of open scope, alpha, then beta, close scope ordinal plus one; then the rank of capital A union capital B equals the maximum of alpha, then beta; then the rank of the ordered pair capital A, then capital B equals the maximum of open scope, alpha, then beta, close scope ordinal plus two; then the rank of capital A times capital B is less than or equal to the maximum of open scope, alpha, then beta, close scope ordinal plus two; then the rank of the union of capital A equals alpha; then alpha; then the rank of the union of capital A equals gamma; then alpha equals gamma ordinal plus one. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise two in this chapter

This source exercise contains, in source order: capital A; then capital B; then the rank of capital A times capital B equals the maximum of the rank of capital A the empty expression, then the rank of capital B; then capital A; then capital B; then the rank of capital A times capital B the maximum of open scope, the rank of capital A the empty expression, then the rank of capital B, close scope ordinal plus two. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Lemma five in this chapter

This source lemma contains, in source order: alpha; then alpha does not belong to omega; then alpha; then omega is less than or equal to alpha; then one ordinal plus alpha equals alpha; then alpha is approximately equal to alpha ordinal plus one; then alpha; then alpha ordinal plus one; then alpha. The complete surrounding source prose remains in the continuous listener stream.

Source

Definition four in this chapter

This source definition contains, in source order: alpha, then beta; then alpha ordinal times beta equals the order type of alpha times beta is less than in reverse lexicographic order. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma six in this chapter

This source lemma contains, in source order: the ordered pair alpha times beta, then is less than in reverse lexicographic order; then alpha; then beta. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma seven in this chapter

This source lemma contains, in source order: alpha, then beta; then Source-ordered display. alpha ordinal times zero equals zero. Then, alpha ordinal times open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus alpha. Then, alpha ordinal times beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal times delta, close scope, when beta is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math six in this chapter

This source display math contains, in source order: Source-ordered display. alpha ordinal times zero equals zero. Then, alpha ordinal times open scope, beta ordinal plus one, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus alpha. Then, alpha ordinal times beta equals the strict supremum over delta is less than beta of open scope, alpha ordinal times delta, close scope, when beta is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Lemma eight in this chapter

This source lemma contains, in source order: alpha, then beta, then gamma; then alpha is not equal to zero; then beta is less than gamma; then alpha ordinal times beta is less than alpha ordinal times gamma; then alpha is not equal to zero; then alpha ordinal times beta equals alpha ordinal times gamma; then beta equals gamma; then alpha ordinal times open scope, beta ordinal times gamma, close scope equals open scope, alpha ordinal times beta, close scope ordinal times gamma; then alpha is less than or equal to beta; then alpha ordinal times gamma is less than or equal to beta ordinal times gamma; then alpha ordinal times open scope, beta ordinal plus gamma, close scope equals open scope, alpha ordinal times beta, close scope ordinal plus open scope, alpha ordinal times gamma, close scope. The complete surrounding source prose remains in the continuous listener stream.

Source

Proposition three in this chapter

This source proposition contains, in source order: two ordinal times omega equals omega is less than omega ordinal times two. 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

Definition five in this chapter

This source definition contains, in source order: Source-ordered display. ordinal exponentiation of alpha to the power zero equals one. Then, ordinal exponentiation of alpha to the power beta ordinal plus one equals ordinal exponentiation of alpha to the power beta ordinal times alpha. Then, ordinal exponentiation of alpha to the power beta equals the union over delta is less than beta of ordinal exponentiation of alpha to the power delta when beta is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Display math seven in this chapter

This source display math contains, in source order: Source-ordered display. ordinal exponentiation of alpha to the power zero equals one. Then, ordinal exponentiation of alpha to the power beta ordinal plus one equals ordinal exponentiation of alpha to the power beta ordinal times alpha. Then, ordinal exponentiation of alpha to the power beta equals the union over delta is less than beta of ordinal exponentiation of alpha to the power delta when beta is a limit ordinal. End display. The complete surrounding source prose remains in the continuous listener stream.

Source

Exercise four in this chapter

This source exercise contains, in source order: ordinal exponentiation of alpha to the power beta equals the order type of finfun, open scope, alpha, then beta, close scope is a proper initial segment of. The complete surrounding source prose remains in the continuous listener stream. The exercise is preserved as stated and no solution is supplied.

Source

Cross-reference reference-001414

chapter “Ordinals”

Source occurrence

Cross-reference reference-001415

chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001416

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

Source occurrence

Cross-reference reference-001417

proposition five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001418

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001419

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001420

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

Source occurrence

Cross-reference reference-001421

lemma two in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001422

theorem five in chapter “Ordinals”

Source occurrence

Cross-reference reference-001423

item 3 of lemma three in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001424

lemma two in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001425

lemma two in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001426

lemma three in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001427

section “Hilbert's Hotel” in chapter “Infinite Sets”

Source occurrence

Cross-reference reference-001428

definition seven in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001429

proposition five in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001430

item 1 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001431

item 2 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001432

proposition five in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001433

item 3 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001434

proposition five in chapter “Stages and Ranks”

Source occurrence

Cross-reference reference-001435

item 4 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001436

item 2 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001437

item 5 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001438

item 4 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001439

item 6 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001440

item 5 of lemma four in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001441

item 1 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001442

item 2 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001443

item 2 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001444

item 3 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001445

item 3 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001446

item 4 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001447

item 4 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001448

item 5 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001449

item 5 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001450

item 1 of lemma five in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001451

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

Source occurrence

Cross-reference reference-001452

lemma one in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001453

proposition two in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001454

lemma six in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001455

lemma seven in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001456

lemma eight in chapter “Ordinal Arithmetic”

Source occurrence

Cross-reference reference-001457

(Michael Potter, 2004, p. 199)

Source occurrence

Cross-reference reference-001458

definition five in chapter “Ordinal Arithmetic”

Source occurrence