We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to (...) prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma. (shrink)

Along the line of Hirst‐Mummert and Dorais, we analyze the relationship between the classical provability of uniform versions Uni(S) of Π2‐statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π2‐statement S of some syntactical form, if its uniform version derives the uniform variant of over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semi‐intuitionistic systems (...) including bar induction in all finite types but also nonconstructive principles such as Kőnig's lemma and uniform weak Kőnig's lemma. Our result is applicable to many mathematical principles whose sequential versions imply. (shrink)

Kierstead showed that every computable marriage problem has a computable matching under the assumption of computable expanding Hall condition and computable local finiteness for boys and girls. The strength of the marriage theorem reaches or if computable expanding Hall condition or computable local finiteness for girls is weakened. In contrast, the provability of the marriage theorem is maintained in even if local finiteness for boys is completely removed. Using these conditions, we classify the strength of variants of marriage theorems in (...) the context of reverse mathematics. Furthermore, we introduce another condition that also makes the marriage theorem provable in, and investigate the sequential and Weihrauch strength of marriage theorems under that condition. (shrink)

We characterize Weihrauch reducibility in $ \operatorname {\mathrm {E-PA^{\omega }}} + \operatorname {\mathrm {QF-AC^{0,0}}}$ and all systems containing it by the provability in a linear variant of the same calculus using modifications of Gödel’s Dialectica interpretation that incorporate ideas from linear logic, nonstandard arithmetic, higher-order computability, and phase semantics.

We show that \\) cannot be proved with one typical application of \\) in an intuitionistic extension of \ to higher types, but that this does not remain true when the law of the excluded middle is added. The argument uses Kohlenbach’s axiomatization of higher order reverse mathematics, results related to modified reducibility, and a formalization of Weihrauch reducibility.