A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

Abstract. Π0 1 classes are important to the logical analysis of many parts of mathematics. The Π0 1 classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin’s work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare’s work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of Π0 1 classes) forms an orbit in the lattice of Π01 classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of Π0 1 classes. We remark that the automorphism result is proven via a ∆0 3 automorphism, and demonstrate that this complexity is necessary. 1.

Constructive dimension and constructive strong dimension are effectivizations of the Hausdorff and packing dimensions, respectively. Each infinite binary sequence A is assigned a dimension [0, 1] and a strong dimension Dim(A) [0, 1].

The computability-theoretic and reverse mathematical aspects of various combinatorial principles, such as König’s Lemma and Ramsey’s Theorem, have received a great deal of attention and are active areas of research. We carry on this study of effective combinatorics by analyzing various partition theorems (such as Ramsey’s Theorem) with the aim of understanding the complexity of solutions to computable instances in terms of the Turing degrees and the arithmetical hierarchy. Our main focus is the study of the effective content of two partition theorems allowing infinitely many col-ors: the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our results on the complexity of solutions rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and Konig’s Lemma, and these connections will be emphasized. We also study Ramsey degrees, i.e. those Turing degrees which are able to compute homogeneous sets for every computable 2-coloring of pairs of natural numbers, in an attempt to further understand the effective content of Ramsey’s Theorem for exponent 2. We establish some new results about these degrees, and obtain as a corollary the nonexistence of a “universal ” computable 2-coloring of pairs of natural numbers.

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This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship with the Gödel–Rosser Theorem; and extensions of Tennenbaum’s theorem to diophantine problems in models of arthmetic, especially problems concerning which diophantine equations have roots in some model of a given theory of arithmetic.

: We shall prove the second incompleteness theorem via Kolmogorov complexity. 0. Introduction Kolmogorov complexity is a measure of the quantity of information in nite objects. Roughly speaking, the Kolmogorov complexity of a number n, denoted by K(n), is the size of a program which generates n, and n is called random if n K(n). Kolmogorov showed in 1960's that the set of nonrandom numbers is recursively enumerable but not recursive, and this is a version of Godel's rst incompleteness theorem (cf. Odifreddi [8]). Chaitin also gave information-theoretic formulation of the rst incompleteness theorem in terms of Kolmogorov complexity. Relations between Kolmogorov complexity and the first incompleteness theorem have been discussed in many places (cf. Li and Vitanyi [7]). Our purpose is to show that Kolmogorov complexity also leads to the second incompleteness theorem. While Godel's proof of the rst incompleteness theorem brings the second incompleteness theorem, Kolmogorov's pr...

Abstract. Finite set theory, here denoted ZFfin, is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In particular, we present a perspicuous method for constructing recursive nonstandard ω-model of ZFfin without the use of permutations. We then use this method to establish the following central theorem. Theorem A. For every simple graph (A, F), where F is a set of unordered pairs of A, there is an ω-model M of ZFfin whose universe contains A and which satisfies the following two conditions: (1) There is parameter-free formula ϕ(x, y) such that for all elements a and b of M, M | = ϕ(a, b) iff {a, b} ∈ F; (2) Every element of M is definable in (M, c)c∈A. Theorem A enables us to build a variety of ω-models with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin. Corollary 2. For each infinite cardinal κ there are 2 κ rigid nonisomorphic ω-models of ZFfin of cardinality κ. Corollary 3. There are continuum-many nonisomorphic pointwise definable ω-models of ZFfin.

Introduction Godel's proofs of the incompleteness theorems are frequently discussed in the context of constructive or finitary viewpoint. Godel showed a concrete independent statement of arithmetic and the proof does not contain any transcendental argument. On the other hand, the completeness theorem asserts that a sentence is not derivable from a theory if and only if there exists a model of the theory which does not satisfy the sentence, and it suggests the existence of model-theoretic proofs of the incompleteness theorems. Nevertheless, as Kreisel states, "as far as, e.g., constructive independence proofs are concerned models are, by the nature of the case, useless." ([Kr2], page 381.) However, the history of applications of models for proofs of the incompleteness theorems is not short; Kreisel had already gave a model-theoretic proof of the second incompleteness theorem in 1960's ([Kr2], also). If we leave from such a finitary point of view, we can get some new techniques for pro