Induction, Bounding, Weak Combinatorial
                    Principles, and the Homogeneous Model
                    Theorem

Hirschfeldt, Denis R.; Lange, Karen; Shore, Richard A.

doi:10.1090/memo/1187

Cited by 3 publications

(1 citation statement)

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“…If RCA 0 + Q P then P ω Q, but the example of RT n+1 k and RT n k , where n 3, shows that it is not necessarily the case that P c Q. Conversely, even having P su Q is not enough to ensure that RCA 0 + Q P , although it is somewhat more difficult to find a counterexample. To do so, we consider the principle Π 0 1 GA, introduced by Hirschfeldt, Lange, and Shore [23], as a "miniaturization" of the principle Π 0 1 G studied by Hirschfeldt, Shore, and Slaman [25]. Definition 1.6.…”

Section: Introductionmentioning

confidence: 99%

On notions of computability-theoretic reduction between Π21 principles

Hirschfeldt

Jockusch

2016

J. Math. Log.

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Abstract. Several notions of computability theoretic reducibility between Π 1 2 principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey's Theorem and related principles under these notions. Among other results, we show that for each n 3, there is an instance of RT n 2 all of whose solutions have PA degree over ∅ (n−2) , and use this to show that König's Lemma lies strictly between RT 2 2 and RT 3 2 under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [ta] on comparing versions of Ramsey's Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey theoretic properties, we show that for each m 2, there is an (m + 1)-coloring c of N such that every m-coloring of N has an infinite homogeneous set that does not compute any infinite homogeneous set for c, and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa [ta]. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to ω-models of RCA 0 , and related notions that allow us to count how many applications of a principle P are needed to reduce another principle to P . Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman [2001].

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Section: Introductionmentioning

confidence: 99%