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Located Sets And Reverse Mathematics
 Journal of Symbolic Logic
, 1999
"... Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly l ..."
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Cited by 14 (5 self)
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Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets.
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 9 (2 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 canonical Ramsey theorem and computability theory
"... Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable in ..."
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Cited by 8 (2 self)
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Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and Konig’s Lemma. 1
Separable Banach space theory needs strong set existence axioms
 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 1996
"... We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 1 comprehension, is needed to prove such basic facts as the existence of the weak∗ closure of any normclosed subspace of ℓ1 = c ∗ 0. This is in contrast to earlier w ..."
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Cited by 7 (5 self)
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We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 1 comprehension, is needed to prove such basic facts as the existence of the weak∗ closure of any normclosed subspace of ℓ1 = c ∗ 0. This is in contrast to earlier work [6, 4, 7, 23, 22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π0 2 sentences. En route to our main results, we prove the Krein ˇ Smulian theorem in ACA0, and we give a new, elementary proof of a result of McGehee on weak ∗ sequential closure ordinals.
Vitali’s theorem and WWKL
 Archive for Mathematical Logic
"... Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions ..."
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Cited by 3 (0 self)
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Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0. 1.
POSSIBLE mDIAGRAMS OF MODELS OF ARITHMETIC
"... Abstract. In this paper we investigate the complexity of mdiagrams of models of various completions of firstorder Peano Arithmetic (PA). We obtain characterizations that extend Solovay’s results for open diagrams of models of completions of PA. We first characterize the mdiagrams of models of Tru ..."
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Cited by 1 (1 self)
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Abstract. In this paper we investigate the complexity of mdiagrams of models of various completions of firstorder Peano Arithmetic (PA). We obtain characterizations that extend Solovay’s results for open diagrams of models of completions of PA. We first characterize the mdiagrams of models of True Arithmetic by showing that the degrees of mdiagrams of nonstandard models A of TA are the same for all m ≥ 0. Next, we obtain a more complicated characterization for arbitrary completions of PA. We then provide examples showing that some of the extra complication is needed. Lastly, we characterize sequences of Turing degrees that occur as (deg(T ∩ Σn))n∈ω, where T is a completion of PA. §1. Introduction. We use P (ω) to denote the class of all subsets of ω. Let LPA be the usual language of PA: relations +, ·, S, and <; and constants 0 and 1. We abbreviate True Arithmetic, the theory of the standard model of PA, by the initials TA. We use S n (0) to denote the numeral for n
NECESSARY USE OF Σ 1 1 INDUCTION IN A REVERSAL
"... Abstract. Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base syste ..."
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Abstract. Jullien’s indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA0 plus Σ 1 1 induction, it was shown by Neeman to have strength strictly between weak Σ 1 1 choice and ∆11 comprehension. We prove in this paper that Σ 1 1 induction is needed for the reversal of INDEC, that is for the proof that INDEC implies weak Σ 1 1 choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only Σ 0 1 induction. §1. Introduction. This paper is concerned with a use of strong induction