Pincherle's theorem in Reverse Mathematics and computability theory

TL;DR

This paper explores the logical and computational distinctions between Pincherle's theorem and compactness in uncountable mathematics, revealing fundamental differences in their reverse mathematical and computability properties.

Contribution

It demonstrates that Pincherle's theorem differs from compactness in reverse mathematics and computability, providing new insights into local-to-global principles.

Findings

Pincherle's theorem has different logical strength than compactness.

Computational properties of Pincherle's theorem differ from those of compactness.

Countable choice and Lindelöf lemma are crucial in analyzing these principles.

Abstract

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 Lindel\"of lemma.