Reverse mathematics and ordinal exponentiation

Hirst, Jeffry L.

doi:10.1016/0168-0072(94)90076-0

Cited by 48 publications

(44 citation statements)

References 3 publications

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“…Girard [12, Section 5.4] (cf. also the computability-theoretic proof by J. Hirst [13]). We can deduce the following: Proof.…”

Section: By Construction We Havementioning

confidence: 93%

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Predicative Collapsing Principles

Freund¹

2019

J. symb. log.

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We show that arithmetical transfinite recursion is equivalent to a suitable formalization of the following: For every ordinal α there exists an ordinal β such that 1 + β · (β + α) (ordinal arithmetic) admits an almost order preserving collapse into β. Arithmetical comprehension is equivalent to a statement of the same form, with β · α at the place of β · (β + α). We will also characterize the principles that any set is contained in a countable coded ω-model of arithmetical transfinite recursion resp. arithmetical comprehension.

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“…Girard [12, Section 5.4] (cf. also the computability-theoretic proof by J. Hirst [13]). We can deduce the following: Proof.…”

Section: By Construction We Havementioning

confidence: 93%

“…Precise definitions and proofs can be found in the references that are given in the right column. 13] the ω-jump of every set exists X → ε X [14,3] arithmetical transfinite recursion…”

Section: Introductionmentioning

confidence: 99%

Predicative Collapsing Principles

Freund¹

2019

J. symb. log.

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“…The techniques of reverse mathematics have been applied to a number of theorems that could be classified as arithmetic on countable well-orderings (e.g. [l], [2], [3], and Chapter 5 of [6]). The goal of this paper is to extend this work by analyzing statements concerning multiplication of countable well-orderings within second order arithmetic.…”

Section: Introductionmentioning

confidence: 99%

“…That is, for any well-orderings a , P , and y, RCAo proves that aP is well-ordered, a ( P y ) E ( a @ ) y , and a(P + y) aP + a y . The right distributive rule (P + y ) a < Pa + y a , known as SHERMAN'S Inequality, is equivalent to ATRo (see [3]). In fact, ATRo is necessary for a reasonably complete development of ordinal multiplication, as demonstrated by the results of the next two sections.…”

Section: Introductionmentioning

confidence: 99%

Reverse Mathematics and Ordinal Multiplication

Hirst

1998

Mathematical Logic Qtrly

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This paper uses the framework of reverse mathematics to analyze the proof theoretic content of several statements concerning multiplication of countable well-orderings. In particular, a division algorithm for ordinal arithmetic is shown t o be equivalent t o the subsystem ATRo.Mathematics Subject Classification: 03F35, 03E10.The techniques of reverse mathematics have been applied to a number of theorems that could be classified as arithmetic on countable well-orderings (e.g. [l], [2], [3], and Chapter 5 of [6]). The goal of this paper is to extend this work by analyzing statements concerning multiplication of countable well-orderings within second order arithmetic. The subsystems RCAo and ATRo are used in this paper. The weak base system RCAo includes basic arithmetic axioms, an induction scheme restricted to Ey formulas, and a comprehension axiom which asserts the existence of recursive sets. The subsystem ATRo appends a comprehension axiom for sets definable by transfinite iterations of arithmetical comprehension. More detailed descriptions of these subsystems can be found in [2], [5], and [6].Via suitable coding, the language of second order arithmetic can express statements about countable well-orderings and functions between them. We will use lowercase greek letters to denote countable well-orderings. As a notational convenience, we will use 0 to denote the least element of any well-ordering, and 1 to denote the next smallest element. Consequently, the theorems presented here bear a strong resemblance to standard developments of ordinal arithmetic. However, here a represents a particular countable well-ordering, not an equivalence class of orderings under isomorphism. If there is an order preserving bijection between a and an initial segment of p, we say that a is strongly less than or equal to p and write a S S p. We write a z S /3 if a S s /3 and , f 3 s s a . Similarly, a 5 p denotes weak comparabzlity, that is, the existence of an order preserving injection of a into p. If a statement holds with both forms of comparability, we will drop the subscripts. For example, the following theorem holds with 5 replaced by 2 or 5 s . See [2] for a proof. ')

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“…(See [6] or [7] for a proof.) As shown below, ACA 0 is also necessary and sufficient to prove the equivalence of two natural definitions of strict inequality for ordinals.…”

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confidence: 99%

Ordinal inequalities, transfinite induction, and reverse mathematics

Hirst¹

1999

J. symb. log.

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If α and β are ordinals, α ≤ β, and β ≰ α then α+ 1 < β. The first result of this paper shows that the restriction of this statement to countable well orderings is provably equivalent to ACA0, a subsystem of second order arithmetic introduced by Friedman. The proof of the equivalence is reminiscent of Dekker's construction of a hypersimple set. An application of the theorem yields the equivalence of the set comprehension scheme ACA0 and an arithmetical transfinite induction scheme.

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Reverse mathematics and ordinal exponentiation

Cited by 48 publications

References 3 publications

Predicative Collapsing Principles

Predicative Collapsing Principles

Reverse Mathematics and Ordinal Multiplication

Ordinal inequalities, transfinite induction, and reverse mathematics

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