Countable retracing functions and Π<sub>2</sub><sup>0</sup>predicates

Jockusch, Carl G.; McLaughlin, T. G.

doi:10.2140/pjm.1969.30.67

Cited by 34 publications

(35 citation statements)

References 14 publications

(16 reference statements)

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“…We give a proof here for the convenience of the reader. By Jockusch and McLaughlin [28,Theorem 4.13], there is an increasing function f such that X T f and f T g for every function g that dominates f . Now for x 1 < x 2 < · · · < x n , let c(x 1 , .…”

Section: Principles 13mentioning

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: Principles 13mentioning

confidence: 99%

On notions of computability-theoretic reduction between Π21 principles

Hirschfeldt

Jockusch

2016

J. Math. Log.

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“…A string is a function whose domain is a finite initial segment of N. If / is a function and j e N, we define [f] 3 to be the restriction of / to {k: k < j}. Thus if / is total, [/] y is a string for all j.…”

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confidence: 99%

Degrees of members of Π₁⁰classes

Jockusch¹,

Soare²

1972

Pacific J. Math.

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124

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“…(1 ⇐⇒ 2) follows from results of Jockusch and McLaughlin in [7]; see also [16, XII.2.14(d)]. The forward implication (1 =⇒ 3) was proven in Chapter 8 of [15], and its converse is immediate from Theorem 3.2.…”

Section: Every Degree In Autspmentioning

confidence: 81%