An epsilon number is a transfinite number which is a fixed point of an exponential map: ωϵ = ϵ. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑). Namely, the ordinal indexing of epsilon numbers is defined as follows:

and for limit ordinal λ:

Tetration stabilizes at ω:

Every ordinal number α can be uniquely written as

where κ is a natural number, n1, n2, …, nk are positive integers, and β1 > β2 > … > βκ are ordinal numbers (βκ = 0). This decomposition of α is called the Cantor Normal Form of α.Białystok Technical University, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515-519, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990

Agata Darmochwał

Czesław Byliński

Edmund Woronowicz

Grzegorz Bancerek

Tetsuya Tsunetou

Formalized Mathematics

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Epsilon Numbers and Cantor Normal Form

Bancerek, Grzegorz

Repozytorium Uniwersytetu w Białymstoku (University of Bialystok Repository)

FORMALIZED MATHEMATICSVol. 17, No. 4, Pages 249–256, 2009DOI: 10.2478/v10037-009-0032-8Epsilon Numbers and Cantor Normal FormGrzegorz BancerekBiałystok Technical UniversityPolandSummary. An epsilon number is a transfinite number which is a fixedpoint of an exponential map: ωε = ε. The formalization of the concept is donewith use of the tetration of ordinals (Knuth’s arrow notation, ↑↑). Namely, theordinal indexing of epsilon numbers is defined as follows:ε0 = ω ↑↑ω, εα+1 = εα ↑↑ω,and for limit ordinal λ:ελ = limα<λεα =⋃α<λεα.Tetration stabilizes at ω:α ↑↑β = α ↑↑ω for α 6= 0 and β ≥ ω.Every ordinal number α can be uniquely written asn1ωβ1 + n2ωβ2 + · · ·+ nkωβk ,where k is a natural number, n1, n2, . . ., nk are positive integers, and β1 > β2 >. . . > βk are ordinal numbers (βk = 0). This decomposition of α is called theCantor Normal Form of α.MML identifier: ORDINAL5, version: 7.11.04 4.130.1076The notation and terminology used here are introduced in the following papers:[9], [3], [8], [7], [1], [5], [6], [4], [2], and [10].249c© 2009 University of BiałystokISSN 1426–2630(p), 1898-9934(e)Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AM250 grzegorz bancerek1. PreliminariesFor simplicity, we follow the rules: α, β, γ denote ordinal numbers, m, ndenote natural numbers, f denotes a sequence of ordinal numbers, and x denotesa set.One can prove the following proposition(1) If α ⊆ succβ, then α ⊆ β or α = succβ.Let us note that ω is limit ordinal and every empty set is ordinal yielding.One can verify that there exists a transfinite sequence which is non emptyand finite.Let f be a transfinite sequence and let g be a non empty transfinite sequence.One can check that f a g is non empty and g a f is non empty.In the sequel ψ, ψ1, ψ2 denote transfinite sequences.One can prove the following three propositions:(2) If domψ = α + β, then there exist ψ1, ψ2 such that ψ = ψ1 a ψ2 anddomψ1 = α and domψ2 = β.(3) rngψ1 ⊆ rng(ψ1 a ψ2) and rngψ2 ⊆ rng(ψ1 a ψ2).(4) If ψ1aψ2 is ordinal yielding, then ψ1 is ordinal yielding and ψ2 is ordinalyielding.Let f be a transfinite sequence. We say that f is decreasing if and only if:(Def. 1) For all α, β such that α ∈ β and β ∈ dom f holds f(β) ∈ f(α).We say that f is non-decreasing if and only if:(Def. 2) For all α, β such that α ∈ β and β ∈ dom f holds f(α) ⊆ f(β).We say that f is non-increasing if and only if:(Def. 3) For all α, β such that α ∈ β and β ∈ dom f holds f(β) ⊆ f(α).Let us observe that every sequence of ordinal numbers which is increasing isalso non-decreasing and every sequence of ordinal numbers which is decreasingis also non-increasing.We now state the proposition(5) For every transfinite sequence f holds f is infinite iff ω ⊆ dom f.Let us note that every transfinite sequence which is decreasing is also finiteand every sequence of ordinal numbers which is empty is also decreasing andincreasing.Let us consider α. Observe that 〈α〉 is ordinal yielding.Let us consider α. One can check that 〈α〉 is decreasing and increasing.Let us observe that there exists a sequence of ordinal numbers which isdecreasing, increasing, non-decreasing, non-increasing, finite, and non empty.The following propositions are true:(6) For every non-decreasing sequence f of ordinal numbers such that dom fis non empty holds⋃f is the limit of f .Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AMepsilon numbers and Cantor normal form 251(7) If α ∈ β, then n · ωα ∈ ωβ.(8) If 0 ∈ α and for every β such that β ∈ dom f holds f(β) = αβ, then f isnon-decreasing.(9) If α is a limit ordinal number and 0 ∈ β, then αβ is a limit ordinalnumber.(10) If 1 ∈ α and for every β such that β ∈ dom f holds f(β) = αβ, then f isincreasing.(11) If 0 ∈ α and β is a non empty limit ordinal number, then x ∈ αβ iffthere exists γ such that γ ∈ β and x ∈ αγ .(12) If 0 ∈ α and αβ ∈ αγ , then β ∈ γ.2. Tetration (Knuth’s Arrow Notation) of Ordinals1Let α, β be ordinal numbers. The functor α ↑↑β yields an ordinal numberand is defined by the condition (Def. 4).(Def. 4) There exists a sequence ϕ of ordinal numbers such that(i) α ↑↑β = lastϕ,(ii) domϕ = succβ,(iii) ϕ(∅) = 1,(iv) for every ordinal number γ such that succ γ ∈ succβ holds ϕ(succ γ) =αϕ(γ), and(v) for every ordinal number γ such that γ ∈ succβ and γ 6= ∅ and γ is alimit ordinal number holds ϕ(γ) = lim(ϕγ).We now state a number of propositions:(13) α ↑↑ 0 = 1.(14) α ↑↑ succβ = αα ↑↑β.(15) Suppose β 6= ∅ and β is a limit ordinal number. Let ϕ be a sequenceof ordinal numbers. If domϕ = β and for every γ such that γ ∈ β holdsϕ(γ) = α ↑↑ γ, then α ↑↑β = limϕ.(16) α ↑↑ 1 = α.(17) 1 ↑↑α = 1.(18) α ↑↑ 2 = αα.(19) α ↑↑ 3 = ααα .(20) For every natural number n holds 0 ↑↑(2 · n) = 1 and 0 ↑↑(2 · n+ 1) = 0.(21) If α ⊆ β and 0 ∈ γ, then γ ↑↑α ⊆ γ ↑↑β.1Important fact (32)α ↑↑ β = α ↑↑ ω for β ≥ ω and α > 0Brought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AM252 grzegorz bancerek(22) If 0 ∈ α and for every β such that β ∈ dom f holds f(β) = α ↑↑β, thenf is non-decreasing.(23) If 0 ∈ α and 0 ∈ β, then α ⊆ α ↑↑β.(24) If 1 ∈ α and m < n, then α ↑↑m ∈ α ↑↑n.(25) If 1 ∈ α and dom f ⊆ ω and for every β such that β ∈ dom f holdsf(β) = α ↑↑β, then f is increasing.(26) If 1 ∈ α and 1 ∈ β, then α ∈ α ↑↑β.(27) For all natural numbers n, k holds nk = nk.Let n, k be natural numbers. Observe that nk is natural.Let n, k be natural numbers. One can check that n ↑↑ k is natural.Next we state several propositions:(28) For all natural numbers n, k such that n > 1 holds n ↑↑ k > k.(29) For every natural number n such that n > 1 holds n ↑↑ω = ω.(30) If 1 ∈ α, then α ↑↑ω is a limit ordinal number.(31) If 0 ∈ α, then αα ↑↑ω = α ↑↑ω.(32) If 0 ∈ α and ω ⊆ β, then α ↑↑β = α ↑↑ω.3. Critical Numbers2In this article we present several logical schemes. The scheme CriticalNum-ber2 deals with an ordinal number A, a sequence B of ordinal numbers, and aunary functor F yielding an ordinal number, and states that:A ⊆ ⋃B and F(⋃B) = ⋃B and for every β such that A ⊆ β andF(β) = β holds ⋃B ⊆ βprovided the following requirements are met:• For all α, β such that α ∈ β holds F(α) ∈ F(β),• Let given α. Suppose α is a non empty limit ordinal number. Letϕ be a sequence of ordinal numbers. If domϕ = α and for everyβ such that β ∈ α holds ϕ(β) = F(β), then F(α) is the limit ofϕ,• domB = ω and B(0) = A, and• For every α such that α ∈ ω holds B(succα) = F(B(α)).The scheme CriticalNumber3 deals with an ordinal number A and a unaryfunctor F yielding an ordinal number, and states that:There exists α such that A ∈ α and F(α) = αprovided the following requirements are met:• For all α, β such that α ∈ β holds F(α) ∈ F(β), and2F is increasing continuous map of ordinals and α = F(α) is a critical number of FBrought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AMepsilon numbers and Cantor normal form 253• Let given α. Suppose α is a non empty limit ordinal number. Letϕ be a sequence of ordinal numbers. If domϕ = α and for everyβ such that β ∈ α holds ϕ(β) = F(β), then F(α) is the limit ofϕ.4. Epsilon Numbers3Let α be an ordinal number. We say that α is epsilon if and only if:(Def. 5) ωα = α.One can prove the following proposition(33) There exists β such that α ∈ β and β is epsilon.Let us note that there exists an ordinal number which is epsilon.Let α be an ordinal number. The first ε greater than α yielding an epsilonnumber is defined by the conditions (Def. 6).(Def. 6)(i) α ∈ the first ε greater than α, and(ii) for every epsilon number β such that α ∈ β holds the first ε greaterthan α ⊆ β.One can prove the following four propositions:(34) If α ⊆ β, then the first ε greater than α ⊆ the first ε greater than β.(35) Suppose α ∈ β and β ∈ the first ε greater than α. Then the first ε greaterthan β = the first ε greater than α.(36) The first ε greater than 0 = ω ↑↑ω.(37) For every epsilon number e holds ω ∈ e.One can check that every ordinal number which is epsilon is also non emptylimit ordinal.One can prove the following propositions:(38) For every epsilon number e holds ωeω= eeω.(39) For every epsilon number e such that 0 ∈ n holds e ↑↑(n+2) = ωe ↑↑(n+1).(40) For every epsilon number e holds the first ε greater than e = e ↑↑ω.Let α be an ordinal number. The functor εα yields an ordinal number andis defined by the condition (Def. 7).(Def. 7) There exists a sequence ϕ of ordinal numbers such that(i) εα = lastϕ,(ii) domϕ = succα,(iii) ϕ(∅) = ω ↑↑ω,(iv) for every ordinal number β such that succβ ∈ succα holds ϕ(succβ) =ϕ(β) ↑↑ω, and3An ordinal number α is epsilon iff it is a critical number of expnential map: α 7→ ωαBrought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AM254 grzegorz bancerek(v) for every ordinal number γ such that γ ∈ succα and γ 6= ∅ and γ is alimit ordinal number holds ϕ(γ) = lim(ϕγ).The following propositions are true:(41) ε0 = ω ↑↑ω.(42) εsuccα = εα ↑↑ω.(43) Suppose β 6= ∅ and β is a limit ordinal number. Let ϕ be a sequenceof ordinal numbers. If domϕ = β and for every γ such that γ ∈ β holdsϕ(γ) = εγ , then εβ = limϕ.(44) If α ∈ β, then εα ∈ εβ.(45) For every sequence ϕ of ordinal numbers such that for every γ such thatγ ∈ domϕ holds ϕ(γ) = εγ holds ϕ is increasing.(46) Suppose β 6= ∅ and β is a limit ordinal number. Let ϕ be a sequenceof ordinal numbers. If domϕ = β and for every γ such that γ ∈ β holdsϕ(γ) = εγ , then εβ =⋃ϕ.(47) If β is a non empty limit ordinal number, then x ∈ εβ iff there exists γsuch that γ ∈ β and x ∈ εγ .(48)4 α ⊆ εα.(49) Let X be a non empty set. Suppose that for every x such that x ∈ Xholds x is an epsilon number and there exists an epsilon number e suchthat x ∈ e and e ∈ X. Then ⋃X is an epsilon number.(50) Let X be a non empty set. Suppose that(i) for every x such that x ∈ X holds x is an epsilon number, and(ii) for every α such that α ∈ X holds the first ε greater than α ∈ X.Then⋃X is an epsilon number.Let us consider α. Observe that εα is epsilon.The following proposition is true(51) If α is epsilon, then there exists β such that α = εβ.5. Cantor Normal FormLet A be a finite sequence of ordinal numbers. The functor∑A yielding anordinal number is defined by the condition (Def. 8).(Def. 8) There exists a sequence f of ordinal numbers such that∑A = last fand dom f = succ domA and f(0) = 0 and for every natural number nsuch that n ∈ domA holds f(n+ 1) = f(n) +A(n).One can prove the following propositions:(52)∑ ∅ = 0.4Of course not always α ∈ εα because there are critical α’s such that α = εαBrought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AMepsilon numbers and Cantor normal form 255(53)∑〈α〉 = α.(54) For every finite sequenceA of ordinal numbers holds∑Aa〈α〉 =∑A+α.(55) For every finite sequence A of ordinal numbers holds∑〈α〉aA = α+∑A.(56) For every finite sequence A of ordinal numbers holds A(0) ⊆∑A.Let us consider α. We say that α is Cantor component if and only if:(Def. 9) There exist β, n such that 0 ∈ n and α = n · ωβ.Let us note that every ordinal number which is Cantor component is alsonon empty.Let us note that there exists an ordinal number which is Cantor component.Let us consider α, β. The functor β-exponent(α) yields an ordinal numberand is defined by:(Def. 10)(i) ββ-exponent(α) ⊆ α and for every γ such that βγ ⊆ α holds γ ⊆β-exponent(α) if 1 ∈ β and 0 ∈ α,(ii) β-exponent(α) = 0, otherwise.The following propositions are true:(57) α ∈ ωsucc(ω-exponent(α)).(58)5 If 0 ∈ n, then ω-exponent(n · ωα) = α.(59) If 0 ∈ α and γ = ω-exponent(α), then α÷ ωγ ∈ ω.(60) If 0 ∈ α and γ = ω-exponent(α), then 0 ∈ α÷ ωγ .(61) If 0 ∈ α and γ = ω-exponent(α), then α mod ωγ ∈ ωγ .(62) If 0 ∈ α, then there exist n, β such that α = n · ωω-exponent(α) + β and0 ∈ n and β ∈ ωω-exponent(α).(63) If ω-exponent(β) ∈ ω-exponent(α), then β ∈ α.Let A be a sequence of ordinal numbers. We say that A is Cantor normalform if and only if:(Def. 11) For every α such that α ∈ domA holds A(α) is Cantor component andfor all α, β such that α ∈ β and β ∈ domA holds ω-exponent(A(β)) ∈ω-exponent(A(α)).Let us note that every sequence of ordinal numbers which is empty is alsoCantor normal form and every sequence of ordinal numbers which is Cantornormal form is also decreasing and finite.In the sequel C, B are Cantor normal form sequences of ordinal numbers.One can prove the following propositions:(64) If∑ C = 0, then C = ∅.(65) If 0 ∈ n, then 〈n · ωβ〉 is Cantor normal form.Let us note that there exists a sequence of ordinal numbers which is nonempty and Cantor normal form.5α-exponent(β) is the entier of the logarithmBrought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AM256 grzegorz bancerekThe following four propositions are true:(66) Let C, B be sequences of ordinal numbers. Suppose CaB is Cantor normalform. Then C is Cantor normal form and B is Cantor normal form.(67) If C 6= ∅, then there exists a Cantor component ordinal number γ andthere exists B such that C = 〈γ〉 a B.(68) Let C be a non empty Cantor normal form sequence of ordinal numbersand γ be a Cantor component ordinal number. If ω-exponent(C(0)) ∈ω-exponent(γ), then 〈γ〉 a C is Cantor normal form.(69)6 For every ordinal number α there exists a Cantor normal form sequenceC of ordinal numbers such that α =∑ C.References[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathe-matics, 1(1):41–46, 1990.[2] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathema-tics, 1(4):711–714, 1990.[3] Grzegorz Bancerek. Ko¨nig’s theorem. Formalized Mathematics, 1(3):589–593, 1990.[4] Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515–519, 1990.[5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.[6] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281–290, 1990.[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.[8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.[9] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequ-ences. Formalized Mathematics, 9(4):825–829, 2001.[10] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics,1(1):73–83, 1990.Received October 20, 20096α = n1ωβ1 + n2ωβ2 + · · ·+ nkωβkBrought to you by | Biblioteka Uniwersytecka w BialymstokuAuthenticatedDownload Date | 12/3/15 11:01 AM

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Epsilon Numbers and Cantor Normal Form

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