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$Π_1^0$ sets and models of $\text{WKL}_0$. (English)
Simpson, Stephen G. (ed.), Reverse mathematics 2001. Wellesley, MA: A K Peters; Urbana, IL: Association for Symbolic Logic (ASL) (ISBN 1-56881-263-9/hbk; 1-56881-264-7/pbk). Lecture Notes in Logic 21, 352-378 (2005).
The author proves that any two nonempty $Π^0_1$ Medvedev complete subsets of $2^ω$ are computably homeomorphic. This result is used to derive several theorems about models of WKL$_0$. For example, he proves that there is a countable $ω$-model of WKL$_0$ in which every definable element is computable. He also constructs a $Π^0_1$ formula, $ϕ(X)$, with one free set-variable such that WKL$_0$ proves “there are countably many $X$ such that $ϕ(X)$”, but WKL$_0$ does not prove “there is a computable $X$ such that $ϕ(X)$”, refuting a conjecture of Kazuyuki Tanaka.
Reviewer: Jeffry L. Hirst (Boone)