# François G. Dorais

### Comparing the strength of diagonally non-recursive functions in the absence of $\Sigma^0_2$ induction

##### By F. G. Dorais, J. L. Hirst and P. Shafer
###### Journal of Symbolic Logic 80 (2015), no. 4, 1211–1235

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$\mathrm{I}\Sigma^0_2$\$.