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)