On the strength of the finite intersection principle

Dzhafarov, Damir D.; Mummert, Carl

doi:10.1007/s11856-012-0150-9

Cited by 11 publications

(15 citation statements)

References 13 publications

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“…In this section, we discuss the reverse mathematical consequences of our work from the previous sections. We discuss the reverse mathematical principles FIP and 2IP, defined by Dzhafarov and Mummert in [4], we define the principle "1-GEN," and we use work from this paper as well as from [2] to prove that FIP and 1-GEN are equivalent over RCA 0 .…”

Section: Reverse Mathematicsmentioning

confidence: 99%

“…All 1-generic sets are hyperimmune. Theorem 1.4 (Dzhafarov and Mummert [4]). Every FIP degree is hyperimmune.…”

Section: Introductionmentioning

confidence: 99%

“…In [3,4], Dzhafarov and Mummert began a systematic study of the proof-theoretical and computability-theoretical strength of various incarnations of the axiom of choice and related principles of finite character. Of concern to the present paper, Dzhafarov and Mummert initiated Downey was supported by the Marsden Fund of New Zealand.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], Dzhafarov and Mummert showed that there was a computable instance of FIP with no computable solution. Definition 1.3 (Dzhafarov and Mummert [4]).…”

Section: Introductionmentioning

confidence: 99%

“…In [4], Dzhafarov and Mummert showed that there was a computable instance of FIP with no computable solution. Definition 1.3 (Dzhafarov and Mummert [4]). A degree a is FIP (or FIP-bounding) if given any computable X as above, a can compute a function f that is FIP with respect to X.…”

Section: Introductionmentioning

confidence: 99%

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Any FIP real computes a 1-generic

Cholak

Downey

Igusa

2017

Trans. Amer. Math. Soc.

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We construct a computable sequence of computable reals X i such that any real that can compute a subsequence that is maximal with respect to the finite intersection property can also compute a Cohen 1-generic. This is extended to establish the same result with 2IP in place of FIP. This is the first example of a classical theorem of mathematics has been found to be equivalent, both proof theoretically and in terms of its effective content, to computing a 1-generic.

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Section: Reverse Mathematicsmentioning

confidence: 99%

“…All 1-generic sets are hyperimmune. Theorem 1.4 (Dzhafarov and Mummert [4]). Every FIP degree is hyperimmune.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [4], Dzhafarov and Mummert showed that there was a computable instance of FIP with no computable solution. Definition 1.3 (Dzhafarov and Mummert [4]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Any FIP real computes a 1-generic

Cholak

Downey

Igusa

2017

Trans. Amer. Math. Soc.

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Inductive inference and reverse mathematics

Hölzl

Jain

Stephan

2016

Annals of Pure and Applied Logic

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Dominating the Erdős–Moser theorem in reverse mathematics

Patey

2017

Annals of Pure and Applied Logic

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The Erdős-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT 2 2 ) by providing an alternate proof of RT 2 2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (ω-model) of simultaneously EM, weak König's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdös-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner. LUDOVIC PATEYJockusch, Lempp and Slaman [13] proved that COH contains a model with no diagonally non-computable function, thus COH does not imply SRT 2 2 over RCA 0 . Cooper [6] proved that every degree above 0 ′ is the jump of a minimal degree. Therefore there exists a p-cohesive set of minimal degree. The Erdős-Moser theoremThe Erdős-Moser theorem is a principle coming from graph theory. It provides together with the ascending descending principle (ADS) an alternative proof of Ramsey's theorem for pairs (RT 2 2 ). Indeed, every coloring f : [ω] 2 → 2 can be seen as a tournament R such that R(x, y) holds if x < y and f (x, y) = 1, or x > y and f (y, x) = 0. Every infinite transitive subtournament induces a linear order whose infinite ascending or descending sequences are homogeneous for f .EM is the statement "Every infinite tournament T has an infinite transitive subtournament." SEM is the restriction of EM to stable tournaments.Bovykin and Weiermann [1] introduced the Erdős-Moser theorem in reverse mathematics and proved that EM together with the chain-antichain principle (CAC) is equivalent to RT 2 2 over RCA 0 . This was refined into an equivalence between EM + ADS and RT 2 2 by Montalbán (see [1]), and the equivalence still holds between the stable versions of the statements. Lerman, Solomon and Towsner [19] proved that EM is strictly weaker than RT 2 2 by constructing an ωmodel of EM which is not a model of the stable ascending descending sequence (SADS). SADS is the restriction of ADS to linear orders of order type ω + ω * [14]. The author noticed in [22] that their construction can be adapted to obtain a separation of EM from the stable thin set theorem for pairs (STS (2)). Wang strengthened this separation by constructing in [31] a standard model of many theorems, including EM, COH and weak König's lemma (WKL) which is neither a model of STS(2) nor a model of SADS. The author later refined in [24,26] the forcing technique of Lerman, Solomon and Towsner and showed that it is strong enough to obtain the same separations as Wang.On the lower bounds side, Lerman, Solomon and Towsner [19] showed that EM implies the omitting partial types principle (OPT) over RCA 0 + BΣ 0 2 and Kreuzer proved in [18] that SEM implies BΣ 0 2 over RCA 0 . The stat...

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On the strength of the finite intersection principle

Cited by 11 publications

References 13 publications

Any FIP real computes a 1-generic

Any FIP real computes a 1-generic

Inductive inference and reverse mathematics

Dominating the Erdős–Moser theorem in reverse mathematics

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