  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. 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.
