Abstract

Using the techniques of reverse mathematics, we characterize subsets X (0; 1) in terms of the strength of HB(X), the Heine-Borel Theorem for the subset. We introduce W(X), formalizing the notion that the Heine-Borel Theorem for X is weak, and S(X), formalizing the notion that the theorem is strong. Using these, we can prove the following three results: RCA0' W(X)! HB(X), RCA0' S(X)! (HB(X)! WKL0), and ATR0' X exists! (W(X)_ S(X)). 2010 Mathematics Subject Classification 03B30, 03F35 (primary); 03F60, 26E40 (secondary)

Highlights

  • One of the earliest results involving WKL0 is Friedman’s theorem showing the equivalence of WKL0 and the Heine-Borel Theorem for [0, 1]. (See Friedman’s abstracts [2].) In response to a question of Friedman, Hirst [4] shows that the Heine-Borel Theorem for closed subsets of Q ∩ [0, 1] is equivalent to WKL0

  • S(X) indicates that the Heine-Borel Theorem for X is stronger; WKL0 can be deduced from HB(X)

  • The deduction of WKL0 from a version of the Heine-Borel Theorem is generally carried out via some construction paralleling that of the Cantor middle third set

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Summary

Introduction

One of the earliest results involving WKL0 is Friedman’s theorem showing the equivalence of WKL0 and the Heine-Borel Theorem for [0, 1]. (See Friedman’s abstracts [2].) In response to a question of Friedman, Hirst [4] shows that the Heine-Borel Theorem for closed subsets of Q ∩ [0, 1] is equivalent to WKL0. Closed subsets can be encoded as the complements of open sets. S(X): X is a subset of [0, 1] and there is a countable dense in itself set Y which is contained in every closed superset of X . Proof Suppose S(X) holds and the Heine-Borel theorem holds for every closed subset of X .

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