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Comparing the strength of diagonally nonrecursive functions in the absence of $$\Sigma_2^0$$ induction. (English) Zbl 1373.03016
It is well-known that provably in the theory $$\mathrm{RCA}_0$$, the statement “for every set $$X$$ there is a $$2$$-bounded diagonally nonrecursive function relative to $$X$$” is equivalent to weak König’s lemma. Moreover, it is known that for any $$k \geq 2$$, a DNR function that is $$k$$-bounded computes one that is $$2$$-bounded. Stephen Simpson had asked whether the latter result is also provable in $$\mathrm{RCA}_0$$. The authors of the present paper show that the usual proof goes through in $$\mathsf{RCA}_0$$ extended by the $$\Sigma^0_2$$ induction axiom. On the other hand, in the absence of $$\Sigma^0_2$$ induction, “all standard $$k$$ are alike, but each nonstandard $$k$$ is nonstandard in its own unique way”.
More precisely, the paper contains two main results. Firstly, the statement “there exists $$k$$ such that for every $$X$$ there is a $$k$$-bounded DNR function relative to $$X$$” does not imply $$\mathrm{WKL}$$ over $$\mathrm{RCA}_0$$, even in the presence of the $$\Sigma^0_2$$ collection axiom. Secondly, the statement “for every $$X$$ there is a bounded DNR function relative to $$X$$” is even weaker, and cannot be used to obtain a bound independent of $$X$$ within $$\mathrm{RCA}_0$$ + $$\Sigma^0_2$$ collection.
To prove these theorems, the authors refine a known construction of an (unbounded) DNR function $$g$$ that does not compute any $$2$$-bounded DNR function, in a technically nontrivial way. Taking advantage of the failure of $$\Sigma^0_2$$ induction, they are able to implement the construction in a $$\Sigma^0_2$$ -definable proper cut, thus making the DNR function $$g$$ bounded. The main results are obtained by iterating this basic construction.
The paper also contains a section on connections between bounded DNR functions and graph colourings.

##### MSC:
 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments 03D30 Other degrees and reducibilities in computability and recursion theory
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