Some conservation results on week König's lemma

By RCA0, we denote the system of second order arithmetic based on recursive comprehension axioms and Σ1 induction. WKL0 is defined to be RCA0 plus weak König’s lemma: every infinite tree of sequences of 0’s and 1’s has an infinite path. In this paper, we first show that for any countable model M of RCA0, there exists a countable model M ′ of WKL0 whose first order part is the same as that of M , and whose second order part consists of the M -recursive sets and sets not in the second order part of M . By combining this fact with a certain forcing argument over universal trees, we obtain the following result (which has been called Tanaka’s conjecture): if WKL0 proves ∀X∃!Y φ(X,Y ) with φ arithmetical, so does RCA0. We also discuss several improvements of this results.