Pincherle's theorem in reverse mathematics and computability theory

Annals of Pure and Applied Logic 171 (5):102788 (2020)
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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.



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Sam Sanders
Ruhr-Universität Bochum

Citations of this work

Reverse Mathematics of Topology: Dimension, Paracompactness, and Splittings.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (4):537-559.
Representations and the Foundations of Mathematics.Sam Sanders - 2022 - Notre Dame Journal of Formal Logic 63 (1):1-28.

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References found in this work

Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
Open questions in reverse mathematics.Antonio Montalbán - 2011 - Bulletin of Symbolic Logic 17 (3):431-454.
Splittings and Disjunctions in Reverse Mathematics.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (1):51-74.

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