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The Dimensions of Individual Strings and Sequences
 INFORMATION AND COMPUTATION
, 2003
"... 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 ..."
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Cited by 93 (10 self)
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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
Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
 Trans. Amer. Math. Soc
"... 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, o ..."
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Cited by 16 (5 self)
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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.
Partition Theorems and Computability Theory
 Bull. Symbolic Logic
, 2004
"... The computabilitytheoretic 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 the ..."
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Cited by 8 (1 self)
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The computabilitytheoretic 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 colors: 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 2coloring 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 2coloring of pairs of natural numbers.
The Arithmetical Complexity of Dimension and Randomness
, 2003
"... 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]. ..."
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Cited by 6 (3 self)
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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].
Tennenbaum’s Theorem for Models of Arithmetic
, 2006
"... 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 ..."
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Cited by 3 (0 self)
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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.
The surprise examination paradox and the second incompleteness theorem
 Notices of the American Mathematical Society
"... We give a new proof for Gödel’s second incompleteness theorem, based on Kolmogorov complexity, Chaitin’s incompleteness theorem, and an argument that resembles the surprise examination paradox. We then go the other way around and suggest that the second incompleteness theorem gives a possible resolu ..."
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We give a new proof for Gödel’s second incompleteness theorem, based on Kolmogorov complexity, Chaitin’s incompleteness theorem, and an argument that resembles the surprise examination paradox. We then go the other way around and suggest that the second incompleteness theorem gives a possible resolution of the surprise examination paradox. Roughly speaking, we argue that the flaw in the derivation of the paradox is that it contains a hidden assumption that one can prove the consistency of the mathematical theory in which the derivation is done; which is impossible by the second incompleteness theorem. Few theorems in the history of mathematics have inspired mathematicians and philosophers as much as Gödel’s incompleteness theorems. The first incompleteness theorem states that for any rich enough 1 consistent mathematical theory 2, there exists a statement that cannot be proved or disproved within the theory. The second incompleteness theorem states that for any rich enough consistent mathematical theory, the consistency of the theory itself cannot be proved (or disproved) within the theory. The First Incompleteness Theorem Gödel’s original proof for the first incompleteness theorem [Gödel31] is based on the liar paradox. The liar paradox: consider the statement “this statement is false”. The statement can be neither true nor false. Gödel considered the related statement “this statement has no proof”. He showed that this statement can be expressed in any theory that is capable of expressing elementary arithmetic. If the statement has a proof, then it is false; but since in a consistent theory any statement that has a proof must be true, we conclude that if the theory is consistent the statement has no proof. Since
Kolmogorov Complexity and the Second Incompleteness Theorem
 Arch. Math. Logic
, 1997
"... : 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, ..."
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Cited by 2 (0 self)
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: 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 informationtheoretic 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...
ωMODELS OF FINITE SET THEORY
, 2008
"... 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 (ZermeloFraenkel set theory). An ωmodel of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Man ..."
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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 (ZermeloFraenkel set theory). An ωmodel of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the BernaysRieger method of permutations to construct a recursive ωmodel of ZFfin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ωmodels of ZFfin. In particular, we present a perspicuous method for constructing recursive nonstandard ωmodel 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 ωmodel M of ZFfin whose universe contains A and which satisfies the following two conditions: (1) There is parameterfree 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 ωmodels with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ωmodel of ZFfin. Corollary 2. For each infinite cardinal κ there are 2 κ rigid nonisomorphic ωmodels of ZFfin of cardinality κ. Corollary 3. There are continuummany nonisomorphic pointwise definable ωmodels of ZFfin.
Model Theoretic Proofs of the Incompleteness Theorems
"... 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 theor ..."
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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 modeltheoretic 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 modeltheoretic 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