Some conservation results on weak König's lemma
The main conservation theorem of this paper is that if \( \text{WKL}_0\) proves \(\forall X \exists ! Y \varphi (x,y)\) with \(\varphi\) arithmetical, then so does \( \text{RCA}_0\). This result extends some of the previous work of \textit{T. Yamazaki} [Math. Log. Q. 46, No. 1, 105-110 (2000; Zbl 0942.03061)] and \textit{A. M. Fernandes} [Arch. Math. Log. 41, No. 1, 55-63 (2002)]. The proof technique utilizes forcing and the fact that for any countable model \(M\) of \(\text{RCA}_0\), there is a countable model \(M^\prime\) of \(\text{WKL}_0\) such that \(M\) and \(M^\prime\) have the same integers and the intersection of their set universes consists exactly of the \(M\)-recursive subsets of the integers of \(M\). For a more extensive treatment of advances in models of \(\text{WKL}_0\), see \textit{S. G. Simpson}'s ``\(\Pi^0_1\) sets and models of \(\text{WKL}_0\)'', which will appear in the forthcoming volume of collected articles on reverse mathematics which he is editing.
scientific article; zbMATH DE number 1838867
scientific article