Journal of Symbolic Logic

On Tao's “finitary” infinite pigeonhole principle

Jaime Gaspar and Ulrich Kohlenbach

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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.

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

First available in Project Euclid: 25 January 2010

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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.

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