[FOM] foundations of the pigeonhole principle?

[Robert Tragesser]

Arnon Avron,
     Thank you very much for this.  I did formulate for myself the 
natural generalization about cardinalities...you took it some 
clarifying steps further.  But I don't think that, without further 
argument, this entirely answers my question, albeit my question was 
very badly formulated.
       Very roughly, the question I am trying to ask is this,

		When do fom-ers regard a mathematical principle as having been 
adequately foundationally clarified? [This is in part a request 
relative to a particular case for a clarification of the use of 
'foundations'.]

       I can see why your (i), (ii), (iii) could be regarded as doing 
the trick - it is clarified in relation to general principles about 
cardinality of sets.
       At the same time, when one considers the pigeonhole principle in 
its most informal expressions (mentioning "pigeonholes" or "boxes"), 
cardinality is not the most striking or salient aspect of it.  It is 
rather that Ramsey Theorem clarifies it...which can be regarded as 
another sort of "generalization" of the pigeonhole principle (the 
Ramsey Theorem somehow lets us see "what's going on" with that 
principle), as can indeed combinatorial theory.  Might not, then, 
combinatorial theory be construed as foundationally 
clarifying...combinatorial theory rather than the set theory of 
cardinals(under AC)?
      Does then Reverse Mathematics of Ramsey Theorem add another moment 
or dimension of foundational clarification?
      The principle does then seem to have some traits that make it 
interesting to explore it as a test case or touchstone for getting 
clear about what we want from foundational efforts - it's (what Paul 
Erdös called) purely existential,  it is intuitively clear and 
immediately evident,  its natural logical analysis would have us cast 
it first in second order logic, and then recast it in many-sorted logic 
or set theory (and so two different avenues for two kinds of 
foundational regard).
      It could also be viewed as a principle about natural numbers, and 
so related to mathematical induction (as well as the Inutitionists 
justification of mathematical induction), and so founding the more 
abstract principle (thinking of the notion of finite set deriving from, 
being founded on,  the notion of "finite set" of natural numbers).  It 
could be regarded as a defining characteristic of finite sets.



Robert Tragesser
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