Journal of Symbolic Logic

On Tao's “finitary” infinite pigeonhole principle

Jaime Gaspar and Ulrich Kohlenbach

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Abstract

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP₁. That turned out to not be the proper formulation and so we proposed an alternative version FIPP₂. Tao himself formulated yet another version FIPP₃ in a revised version of his essay.

We give a counterexample to FIPP₁ and discuss for both of the versions FIPP₂ and FIPP₃ the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP₂ and IPP ↔ FIPP₃ in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the “big five” subsystems of second order arithmetic.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 1 (2010), 355-371.

Dates
First available in Project Euclid: 25 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1264433926

Digital Object Identifier
doi:10.2178/jsl/1264433926

Mathematical Reviews number (MathSciNet)
MR2605899

Zentralblatt MATH identifier
1188.03045

Citation

Gaspar, Jaime; Kohlenbach, Ulrich. On Tao's “finitary” infinite pigeonhole principle. J. Symbolic Logic 75 (2010), no. 1, 355--371. doi:10.2178/jsl/1264433926. https://projecteuclid.org/euclid.jsl/1264433926


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