The Journal of Symbolic Logic
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 f" does not imply weak K\"onig's lemma over RCA0+BΣ02. 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 IΣ02.
Dorais, Francois G.; Hirst, Jeffry L.; and Shafer, Paul, "Comparing the strength of diagonally non-recursive functions in the absence of Σ02 induction" (2015). Open Dartmouth: Faculty Open Access Articles. 2362.