Iterative Forcing and Hyperimmunity in Reverse Mathematics

Patey, Ludovic

doi:10.1007/978-3-319-20028-6_30

Cited by 24 publications

(44 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, Seetapun's theorem (separating

{sans-serifRT}_{2}^{2}

from

{ACA}_{0}

), Wang's separation of the free set and thin set theorems from

{ACA}_{0}

, and various recent separations of Patey can be streamlined by using models of

sans-serifRWKL

in place of models of

sans-serifWKL

. Many computability‐theoretic properties are preserved by both

{sans-serifRT}_{2}^{2}

and

sans-serifWKL

, such as cone avoidance , hyperimmunity and fairness . Explicit use of models of

sans-serifRWKL

is helpful when proving that

{sans-serifRT}_{2}^{2}

preserves a property which is not preserved by

sans-serifWKL

, such as constant‐bound‐enumeration avoidance .…”

Section: Introductionmentioning

confidence: 99%

On the logical strengths of partial solutions to mathematical problems

Bienvenu

Patey

Shafer

2017

Trans. London Math. Soc.

Self Cite

101

View full text Add to dashboard Cite

We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to K\"onig's lemma, such as restrictions of K\"onig's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak K\"onig's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak K\"onig's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak K\"onig's lemma and algorithmic randomness by showing that Ramsey-type weak weak K\"onig's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak K\"onig's lemma. This answers a question of Flood.Comment: 43 page

show abstract

“…For example, Seetapun's theorem (separating

{sans-serifRT}_{2}^{2}

from

{ACA}_{0}

), Wang's separation of the free set and thin set theorems from

{ACA}_{0}

, and various recent separations of Patey can be streamlined by using models of

sans-serifRWKL

in place of models of

sans-serifWKL

. Many computability‐theoretic properties are preserved by both

{sans-serifRT}_{2}^{2}

and

sans-serifWKL

, such as cone avoidance , hyperimmunity and fairness . Explicit use of models of

sans-serifRWKL

is helpful when proving that

{sans-serifRT}_{2}^{2}

preserves a property which is not preserved by

sans-serifWKL

, such as constant‐bound‐enumeration avoidance .…”

Section: Introductionmentioning

confidence: 99%

On the logical strengths of partial solutions to mathematical problems

Bienvenu

Patey

Shafer

2017

Trans. London Math. Soc.

Self Cite

101

View full text Add to dashboard Cite

show abstract

“…Let m be the maximum value among {g(y) : y < x}, {g u (x) : u < v} and µ(x). 19). As seen in Lemma 2.9, µ-transitivity ensures that µ-largeness of the intervals over a set D is fully specified by the µ-largeness of its adjacent intervals.…”

Section: Definition 21 Fix a Countable Collection Of Variablesmentioning

confidence: 99%

“…The situation becomes different when preserving 2 hyperimmunities. Indeed, the author [19,Lemma 25 and Lemma 27] proved that RT 1 2 does not admit strong preservation of 2 hyperimmunities, while Dzhafarov and Jockusch [8] proved that RT 1 2 can strongly avoid multiple cones simultaneously. Question 5.4.…”

Section: Definition 53 (Preservation Of Hyperimmunities)mentioning

confidence: 99%

Ramsey-Like Theorems and Moduli of Computation

Patey¹

2020

J. symb. log.

Self Cite

View full text Add to dashboard Cite

Ramsey's theorem asserts that every k-coloring of [ω] n admits an infinite monochromatic set. Whenever n ≥ 3, there exists a computable k-coloring of [ω] n whose solutions compute the halting set. On the other hand, for every computable k-coloring of [ω] 2 and every non-computable set C, there is an infinite monochromatic set H such that C ∕ ≤ T H. The latter property is known as cone avoidance. In this article, we design a natural class of Ramsey-like theorems encompassing many statements studied in reverse mathematics. We prove that this class admits a maximal statement satisfying cone avoidance and use it as a criterion to re-obtain many existing proofs of cone avoidance. This maximal statement asserts the existence, for every k-coloring of [ω] n , of an infinite subdomain H ⊆ ω over which the coloring depends only on the sparsity of its elements. This confirms the intuition that Ramsey-like theorems compute Turing degrees only through the sparsity of its solutions.

show abstract

“…Patey [35,Theorem 28] proved that for every set A and every hyperimmune function g, there is an infinite subset H of A or A such that g is H-hyperimmune. A natural question is whether this result can be effectivized in the case of A and f being ∆ 0 2 .…”

Section: Lemma 44 (Simpsonmentioning

confidence: 99%

Some results concerning the SRT 2 2 vs. COH problem

Cholak

Dzhafarov

Hirschfeldt

et al. 2020

COM

Self Cite

View full text Add to dashboard Cite

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

show abstract

Iterative Forcing and Hyperimmunity in Reverse Mathematics

Cited by 24 publications

References 12 publications

On the logical strengths of partial solutions to mathematical problems

On the logical strengths of partial solutions to mathematical problems

Ramsey-Like Theorems and Moduli of Computation

Some results concerning the SRT 2 2 vs. COH problem

Contact Info

Product

Resources

About