François G. Dorais

We prove that the statement “there is a $k$ such that for every $f$ there is a $k$-bounded diagonally non-recursive function relative to 4$f$" does not imply weak König lemma over$\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2$. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every$k$-bounded diagonally non-recursive function computes a$2$-bounded diagonally non-recursive function may fail in the absence of$\mathrm{I}\Sigma^0_2$$.