Translator Disclaimer
2018 On the Uniform Computational Content of the Baire Category Theorem
Vasco Brattka, Matthew Hendtlass, Alexander P. Kreuzer
Notre Dame J. Formal Logic 59(4): 605-636 (2018). DOI: 10.1215/00294527-2018-0016

Abstract

We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete (i.e., “large”) metric space cannot be decomposed into countably many nowhere dense (i.e., small) pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets.

Citation

Download Citation

Vasco Brattka. Matthew Hendtlass. Alexander P. Kreuzer. "On the Uniform Computational Content of the Baire Category Theorem." Notre Dame J. Formal Logic 59 (4) 605 - 636, 2018. https://doi.org/10.1215/00294527-2018-0016

Information

Received: 10 October 2015; Accepted: 21 September 2016; Published: 2018
First available in Project Euclid: 13 October 2018

zbMATH: 06996547
MathSciNet: MR3871904
Digital Object Identifier: 10.1215/00294527-2018-0016

Subjects:
Primary: 03F60

Rights: Copyright © 2018 University of Notre Dame

JOURNAL ARTICLE
32 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.59 • No. 4 • 2018
Back to Top