Measure, Π 1 0 -classes and complete extensions of PA

Kučera, Antonín

doi:10.1007/bfb0076224

Cited by 173 publications

(213 citation statements)

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“…We then have that S f is 0 -computable and the set of branches of S f has positive measure. Relativizing a result of Kučera [11], there exists a branch g of S f such that g ≤ T A. Let h be a g-computable branch of T f .…”

Section: Theorem 310 Suppose That F : [ω]mentioning

confidence: 89%

The strength of the rainbow Ramsey Theorem

Csima

Mileti

2009

J. symb. log.

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Abstract. The Rainbow Ramsey Theorem is essentially an "anti-Ramsey" theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey's Theorem, even in the weak system RCA 0 of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey's Theorem for pairs over RCA 0 . The separation involves techniques from the theory of randomness by showing that every 2-random bounds an ω-model of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin's theorem that the hyperimmune degrees have measure one.

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Section: Theorem 310 Suppose That F : [ω]mentioning

confidence: 89%

The strength of the rainbow Ramsey Theorem

Csima

Mileti

2009

J. symb. log.

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“…The original, unrelativized concept is due to Martin-Löf [17] and has been studied by Kučera [15] and many others. Kučera 1985). We can construct a universal Martin-Löf test.…”

Section: Randomnessmentioning

confidence: 99%

Almost everywhere domination and superhighness

Simpson

2007

Mathematical Logic Qtrly

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Let ω denote the set of natural numbers. For functions f, g : ω → ω, we say that f is dominated by g if f (n) < g(n) for all but finitely many n ∈ ω. We consider the standard "fair coin" probability measure on the space 2 ω of infinite sequences of 0's and 1's. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω , each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′′ is truth-table computable from B ′ , the Turing jump of B.

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“…Restating the theorem Kučera [78] we see the following: X is 1-random iff X is a Poincaré point for the shift operator with respect to the collection of effectively closed subsets of 2 ω .…”

Section: Randomness Is the Same As Differentiabilitymentioning

confidence: 99%

“…Theorem 10 (Kućera [78], Gács [61]). For every set X, there is a 1-random Y such that X ≤ wtt Y , where ≤ wtt is Turing reducibility with use bounded by a computable function.…”

Section: Computability and Randomnessmentioning

confidence: 99%