In 1975 H. Friedman introduced two statements of hyperarithmetic analysis, SL 0 (sequential limit system) and ABW 0 (arithmetic Bolzano-Weierstrass), which are motivated by standard and well-known theorems from analysis such as the Bolzano-Weierstrass theorem for F σ and G δ sets of reals. In this article we characterize the reverse mathematical strength of ABW 0 by comparing it to most known theories of hyperarithmetic analysis. In particular we show that, over RCA 0 + IΣ 1 1 , SL 0 is equivalent to Σ 1 1 − AC 0 , and that ABW 0 is implied by Σ 1 1 − AC 0 , and implies weak − Σ 1 1 − AC 0. We then use Steel's method of forcing with tagged trees to show that ABW 0 is incomparable with INDEC (i.e. Jullien's Theorem) and Δ 1 1 − CA 0. This makes ABW 0 the first theory of hyperarithmetic analysis that is known to be incomparable with other (known) theories of hyperarithmetic analysis. We also examine the reverse mathematical strength of the Bolzano-Weierstrass theorem in the context of open, closed, F σ , G δ , and other types of sets.

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [14], 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 real of effective packing dimension one (on the other hand, it is known via [10. 3. 7] that every real of strictly positive effective Hausdorff dimension computes reals whose effective packing dimensions are arbitrarily close to, but not necessarily equal to, one).

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 computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

Abstract. 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 RCA 0 and so RCA 0 + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA 0 for arithmetic sentences. Thus, from the CsimaMileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over1 -sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P − + I Σ 1 and no stronger than P − + B Σ 2 . §1. Introduction. One of the benefits to having a precise formulation of the concept "random infinite binary sequence" is that one can ask and precisely answer questions about what types of objects can be computed from random input and about what sorts of theorems can be proven from the existence of a random sequence.In [3], Csima and Mileti gave an intriguing example of the first type. Their example concerns the Rainbow Ramsey Theorem, which states that if C is a coloring of size-k subsets of N such that there is a uniform finite bound on the number of sets assigned to any particular color, then there is an infinite set X such that C is injective on the size-k subsets of X , i.e. X is a C -rainbow. As is described below, they give a proof of the Rainbow Ramsey Theorem for pairs (k = 2) by showing that if R is a sufficiently random sequence relative to C , then R can be used to compute a C -rainbow. In a sentence, Csima and Mileti show how a random source can be used to produce a solution to a infinitary-combinatorial problem. Further, since they also show that there is a recursive such C with no recursive rainbow, any general method to produce rainbows for colorings must be driven by some such non-recursive data.In this article, we analyze the second question with respect to theorems about finite sets. For this question, we shift the setting from the examination of the computation