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Splittings and disjunctions in reverse mathematics. (English) Zbl 1462.03009

The author examines equivalences in higher order reverse mathematics for conjunctions and disjunctions of formulas. Such results have long been of interest in traditional reverse mathematics. For example, P. A. Cholak et al. [J. Symb. Log. 66, No. 1, 1–55 (2001; Zbl 0977.03033)] proved the equivalence of Ramsey’s theorem for pairs with the conjunction of the stable Ramsey theorem and the principle COH. H. Friedman et al. [Ann. Pure Appl. Logic 62, No. 1, 51–64 (1993; Zbl 0781.03048)] proved the equivalence of the disjunction of \(\mathsf{WKL}_0\) and \(\Sigma^0_2\) induction with the assertion that finite iterations of continuous functions are continuous. Both types of results can lead to novel proof strategies. The first group of results here list splittings (conjunctions) equivalent to a modulus of uniform continuity MUC principle corresponding to the existence of the intuitionistic fan functional. The base system for these results is \(\mathsf{RCA}^\omega_0 +\mathsf{QF}\text{-}\mathsf{AC}^{2,0}\). Additional splittings of the functional principles \(\exists^2\) and \(\exists^3\) are related to \(\neg\mathsf{MUC}\). A second group of results proven over \(\mathsf{RCA}^\omega_0\) includes many disjunctions equivalent to \(\mathsf{WKL}_0 \lor\mathsf{I}\Sigma^0_2\). Additional disjunctions equivalent to \(\mathsf{WWKL}\) follow. After a section describing the foundational motivations of the work, a table gives a partial overview of the results. Note that, for example, Theorem 3.8 provides a scheme for generating additional results too numerous for inclusion in the table.

MSC:

03B30 Foundations of classical theories (including reverse mathematics)
03D65 Higher-type and set recursion theory
03F35 Second- and higher-order arithmetic and fragments
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