Partial orders and immunity in reverse mathematics

Patey, Ludovic

doi:10.3233/com-170071

Cited by 11 publications

(10 citation statements)

References 26 publications

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“…They also showed that none of these principles imply WKL 0 over RCA 0 . The fact that CAC is properly stronger than ADS was first proved by Lerman, Solomon, and Towsner [LST13], and then given a simpler proof by Patey [Pat16]. These results support the idea that RT 2 2 , in contrast to the big five, is not robust (Montalbán [Mon11] informally defined a theory to be robust "if it is equivalent to small perturbations of itself").…”

Section: Reverse Mathematicsmentioning

confidence: 70%

The Reverse Mathematics of wqos and bqos

Marcone

2020

Trends in Logic

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In this paper we survey wqo and bqo theory from the reverse mathematics perspective. We consider both elementary results (such as the equivalence of different definitions of the concepts, and basic closure properties) and more advanced theorems. The classification from the reverse mathematics viewpoint of both kinds of results provides interesting challenges, and we cover also recent advances on some long standing open problems.

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Section: Reverse Mathematicsmentioning

confidence: 70%

The Reverse Mathematics of wqos and bqos

Marcone

2020

Trends in Logic

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“…No arrows reverse. Proofs of these implications and separations may be found in [6,8,16,17,21,22,26,29,30].…”

Section: Reverse Mathematics Backgroundmentioning

confidence: 99%

(Extra)ordinary equivalences with the ascending/descending sequence principle

Fiori-Carones¹,

Marcone²,

Shafer³

et al. 2021

Preprint

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We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival-Sands theorem for infinite partial orders of arbitrary finite width is equivalent to IΣ 0 2 + ADS over RCA0. For each fixed k ≥ 3, the Rival-Sands theorem for infinite partial orders of width ≤ k is equivalent to ADS over RCA0. The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to SADS over RCA0. Here RCA0 denotes the recursive comprehension axiomatic system, IΣ 0 2 denotes the Σ 0 2 induction scheme, ADS denotes the ascending/descending sequence principle, and SADS denotes the stable ascending/descending sequence principle. To our knowledge, these versions of the Rival-Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by IΣ 0 2 + ADS, by ADS, and by SADS. Furthermore, we give a new purely combinatorial result by extending the Rival-Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over RCA0.

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“…Patey [Pat15] proved that EM admits avoidance of constant-bound traces for ω closed sets. On the other hand, Rice [Ric] constructed a computable instance of EM such that every solution computes a diagonally non-computable function, while Patey [Pat16a] constructed a ∆ 0 2 immune set A such that every diagonally non-computable function computes an infinite subset of A. This shows that EM does not admit preservation of 1 immunity.…”

Section: The Hierarchy Of Preservationsmentioning

confidence: 99%

Relationships between computability-theoretic properties of problems

Downey¹,

Greenberg²,

Harrison-Trainor³

et al. 2019

Preprint

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A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computabilitytheoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set A and a computable instance of a problem P, to find a solution relative to which A is still non-computable.In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of ω cones, of 1 hyperimmunity or of 1 non-Σ 0 1 definition. We also prove that the hierarchies of preservation of hyperimmunity and non-Σ 0 1 definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting.Definition 1.1. A Turing ideal is a collection of reals S ⊆ 2 ω which is closed under the effective join and downward-closed under the Turing reduction. In other wordsMany statements studied in reverse mathematics can be formulated as mathematical problems, with instances and solutions. For example, weak König's lemma (WKL) asserts that every infinite, finitely branching subtree of 2 <ω has an infinite path. Here, an instance is such a tree T , and a solution to T is an infinite path 1 The authors are thankful to Mariya Soskova for interesting comments and discussions about cototal degrees.through it. An ω-structure M with second-order part S is a model of a problem P (written M ⊧ P) if every instance in S has a solution in it. In this case we also say that P holds in S. In order to separate a problem P from another problem Q in reverse mathematics, one usually constructs a Turing ideal S in which P holds, but not Q. However, when closing the Turing ideal with solution to instances of P, one must be careful not to make it a model of Q. This motivates the use of preservation properties. Definition 1.2. Fix a collection of sets W ⊆ 2 ω downward-closed under Turing reduction. A problem P admits preservation of W if for every set Z ∈ W and every Z-computable instance X of P, there is a solutionThe following basic lemma is at the core of separations in reverse mathematics.Lemma 1.3. Suppose a problem P admits preservation of some collection W, but another problem Q does not. Then there is a Turing ideal S ⊆ W in which P holds, but not Q.Proof. Since Q does not admit preservation of W, there is some Z ∈ W, and a Q-instance X Q ⩽ T Z such that for every solution Y to X Q , Z ⊕ Y ∈ W. We will build a Turing ideal S ⊆ W containing Z and in which P holds. In particular, Q cannot hold in any such Turing ideal. We build a countable sequence of sets Z 0 , Z 1 , . . . such that for every n ∈ ω, ⊕ s

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Partial orders and immunity in reverse mathematics

Cited by 11 publications

References 26 publications

The Reverse Mathematics of wqos and bqos

The Reverse Mathematics of wqos and bqos

(Extra)ordinary equivalences with the ascending/descending sequence principle

Relationships between computability-theoretic properties of problems

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