Natural well-orderings

Crossley, John N.; Kister, Jane Bridge

doi:10.1007/bf02017491

Cited by 14 publications

(1 citation statement)

References 71 publications

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“…For a history of the Bachmann method of describing constructive ordinals, and how it gave way to the more modern approach, we refer to Crossley and Bridge Kister [11] and the preface of Buchholz, Feferman, Pohlers and Sieg [7].…”

Section: Introductionmentioning

confidence: 99%

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

Buchholtz¹,

Jäger²,

Strahm³

2016

Concepts of Proof in Mathematics, Philosophy, and Computer Science

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The purpose of this article is to present a range of theories with proof-theoretic ordinal ψ(Γ Ω+1 ). This ordinal parallels the ordinal of predicative analysis, Γ 0 , and our theories are parallel to classical theories of strength Γ 0 such as ID <ω , FP 0 , ATR 0 , Σ 1 1 -DC 0 + (SUB), and Σ 1 1 -AC 0 + (SUB). We also relate these theories to the unfolding of ID 1 which was already presented in the PhD thesis of the first author as a system of strength ψ(Γ Ω+1 ).

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Section: Introductionmentioning

confidence: 99%

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

Buchholtz¹,

Jäger²,

Strahm³

2016

Concepts of Proof in Mathematics, Philosophy, and Computer Science

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An order‐theoretic characterization of the Schütte‐Veblen‐Hierarchy

Weiermann

1993

Mathematical Logic Qtrly

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For f : On -+ On let supp(f) := {( : f(C) # 0}, and let S := {f : On + On : supp(f) finite}. For f,g E S define f 5 g : e (3 : O n + On)[ h one-to-one A (V() f(() 5 g ( h ( ( ) ) ] . A function $ : S -+ On is called monotonic increasing, if f(() 5 $(f) and if f 5 g implies $(f) 5 $(g). For a mapping $ : S -+ On let Clw(0) be the least set T of ordinals which contains 0 as an element and which is closed under the following rule: If f E S, range(f) c T and supp(f) C T, then $(f) E T. Let cp be the enumeration function of the class {( : (37)" = d]}. Let @ be the induced Sch~tte-KIanimertenrrfunction (see [Ill) which generates the Schiitte-Veblen-hierarchy of ordinals. Then CP is monotonic increasing. We show: If $ : S -+ On is monotonic increasing, then otyp(Cl+(O)) 5 min{( : ( = @(It)}, where lc(a) = 1 if Q = ( and lc(a) = 0 if a # (. For r an ordinal let S r := {f E S : supp(f) 2 r}. A function $ : S , 4 On is called r-monotonic increasing if f(() 5 $(f) and if f 5 g implies $(f) 5 $(g). For a function $ : Sr + O n let Cl+(O) 1 r be the least set T of ordinals, which contains 0 as an element, such that i f f E S, and range(f) c T, then $(f) E T. We show: Zf r 2 2 and if $ : S , -+ O n is r-monotonic increasing, then otyp(Cl+(O) I r) 5@ ( l r ) . We also prove a generalization of this theorem in terms of well-partial orderings. MSC: 03F15, 03E10, 06A06.

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Calculating Maximal Order Types for Finite Rooted Unstructured Labeled Trees

Schmidt

Meeren

Weiermann

2020

The Legacy of Kurt Schütte

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Diana Schmidt, in her Habilitationsschrift in 1979, completely classified the maximal order types of the natural tree embeddability relations for finite rooted structured labeled trees. Her results since have found interesting applications in proof theory and reverse mathematics. The question concerning the maximal order types of unstructured trees has been left open for years, and a conclusive answer will be given in this article. Moreover, we provide an answer to a question of Harvey Friedman regarding Γ 0 and binary two-labeled unstructured trees.

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Natural well-orderings

Cited by 14 publications

References 71 publications

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

An order‐theoretic characterization of the Schütte‐Veblen‐Hierarchy

Calculating Maximal Order Types for Finite Rooted Unstructured Labeled Trees

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Natural well-orderings

Cited by 14 publications

References 71 publications

Theories of Proof-Theoretic Strength Ψ (ΓΩ +1)

Theories of Proof-Theoretic Strength Ψ (ΓΩ +1)

An order‐theoretic characterization of the Schütte‐Veblen‐Hierarchy

Calculating Maximal Order Types for Finite Rooted Unstructured Labeled Trees

Contact Info

Product

Resources

About

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)

Theories of Proof-Theoretic Strength Ψ (Γ_Ω +1)