Chris J. Conidis

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In 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e. for every complete atomic decidable theory T , there is a prime model M computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for ∆2 sets. Some of these(More)
Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [Mil], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a(More)
We answer a recent question of Bienvenu, Muchnik, Shen, and Vereshchagin. In particular, we prove an effective version of the standard fact from analysis which says that, for any ε > 0 and any Lebesgue-measurable subset of Cantor space, X ⊆ 2, there is an open set Uε ⊆ 2, Uε ⊇ X, such that μ(Uε) ≤ μ(X) + ε, where μ(Z) denotes the Lebesgue measure of Z ⊆ 2.(More)
We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X . In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA0 and so RCA0 +2-RAN implies the(More)
We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0; x1; : : : ; xN 2 G, N 2 N, then one can always …nd a uniformly computably enumerable (i.e. uniformly 1) ascendant sequence of order type ! + 1 of subgroups in G beginning with hx0; x1; : : : ; xN iG, the subgroup(More)
This article expands upon the recent work by Downey, Lempp, and Mileti [3], who classified the complexity of the nilradical and Jacobson radical of commutative rings in terms of the arithmetical hierarchy. Let R be a computable (not necessarily commutative) ring with identity. Then it follows from the definitions that the prime radical of R is Π1, and the(More)
We prove that if S is an ω-model of weak weak König’s lemma and A ∈ S, A ⊆ ω, is incomputable, then there exists B ∈ S, B ⊆ ω, such that A and B are Turing incomparable. This extends a recent result of Kučera and Slaman who proved that if S0 is a Scott set (i.e. an ω-model of weak König’s lemma) and A ∈ S0, A ⊆ ω, is incomputable, then there exists B ∈ S0,(More)
We construct a Π1-class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing dimension for(More)
We construct a Π1-class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing dimension for(More)
We show that the fact that the first player (“white”) wins every instance of Galvin’s “racing pawns” game (for countable trees) is equivalent to arithmetic transfinite recursion. Along the way we analyse the satisfaction relation for infinitary formulas, of “internal” hyperarithmetic comprehension, and of the law of excluded middle for such formulas.