# From Vitali to Heine-Borel in reverse mathematics

The Vitali and Heine-Borel covering theorems are house-hold names of analysis, and rightly well-studied in reverse mathematics. As shown in Simpson's excellent monograph , for countable coverings of the unit interval, the Heine-Borel theorem is equivalent to WKL (weak Koenig's lemma), while the Vitali covering theorem is equivalent to WWKL (weak weak Koenig's lemma). The theorem numbers in  are IV.1.2 and X.1.13.

My question is then as follows:

Is there a natural statement X such that [WWKL +X ] $$\leftrightarrow$$ WKL, say over RCA$$_0$$?

Here, $$X$$ should be weaker than WKL, obviously. Results in related frameworks (computability theory, Weihrauch reducibility, constructive math, ...) are also welcome.

PS: I am asking this question because in the case of uncountable coverings, such an X does exist.

 Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, 2009.

## 1 Answer

In the Weihrauch reducibility framework, my hunch is that the answer is no. Of course, "natural statement" does not lend itself to easily disprove existence, so I can't rule out changing my mind in the future.

First, we are looking at principles below $$\mathrm{WKL}$$ which are incomparable with $$\mathrm{WWKL}$$. To my knowledge, the only such principles which were studied in the literature are variants of convex choice $$\mathrm{XC}_{[0,1]^n}$$. All these variants are themselves below the principle $$\mathrm{XC}_{[0,1]}^\diamond$$, which lets us invoke choice for subsets of the unit interval finitely many times (with subsequent queries depending on previous answers). Note that $$\mathrm{XC}_{[0,1]}$$ is equivalent to the intermediate value theorem.

However, $$(\mathrm{XC}_{[0,1]} \sqcup \mathrm{WWKL})^\diamond <_{\mathrm{W}} \mathrm{WKL}$$ (the left hand side is "make finitely many calls to both $$\mathrm{XC}_{[0,1]}$$ and to $$\mathrm{WWKL}$$). The reason for this is that on a computable input, $$\mathrm{XC}_{[0,1]}$$ can always return something computable, and $$\mathrm{WWKL}$$ can always return some ML-random. Thus, the left-hand side can always return something which is computable from a ML-random, but this doesn't hold for $$\mathrm{WKL}$$, as $$\mathrm{PA}$$-degrees are not computable from ML-randoms.

Another piece of evidence is that there is multivalued function $$g$$ at all such that $$\mathrm{WKL} \leq_{\mathrm{W}} \mathrm{WWKL} \circ g$$. On the right, we have that every input has a positive measure of potential solutions, and $$\mathrm{WKL}$$ has instances where the Turing-upper cone of solutions is measure $$0$$.