Source: J. Symbolic Logic Volume 75, Issue 1
(2010), 355-371.
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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Mathematical Reviews (MathSciNet):
MR966421
