[FOM] foundations of the pigeonhole principle
robert at thesavvydog.com
Thu Nov 20 17:46:36 EST 2003
Speculating that the considerations in the neighborhood of Simpson
SUBSYTEMS OF SECOND ORDER ARITHMETIC p.72, II.3 Primitive Recursion,
especially Lemma II.3.7) provide the means to a foundations of the
pigeonhole principle, I am finding some questions that are hard for me
There are such a number of ways of formalizing the pigeonhole
principle. I'm inclined to view it as a fundamental principle.
Or is it best to view it as a special case of a more general principle?
Does it suffice to formulate it for natural numbers (sets of natural
Maybe it is conceptually/foundationally best to think of it as
characterizing finite sets (not necessarily finite sets of natural
numbers)? In that case, wouldn't formulating the pigeonhole principle
for sets of natural numbers make it too easy, because it's too easy to
say what a finite set of natural numbers is? I suppose we'd have to be
convinced that the very idea of a finite set is founded on the idea of
a finite set of natural numbers (as I am, but should I be)?
Allowing that it's foundationally sound/adequate to say that it
suffices to formulate the pigeonhole principle over natural numbers, is
it right to say that it is equivalent to (the proposition in Simpson
Lemma II.3.7... but not restricting phi to Sigma1,2): "Either there
exists a finite set X such that (n)(n el X iff phi(n)), or there exists
a 1-1 function f from N to N such that every n such that phi(n) is the
image under f of an m of N). Then the foundations of the pigeonhole
principle are relativized to conditions of phi ?
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