| We know that because it is rigorously proven within some axioms. A large, but finite, set falls into the given axiomatic system where it applies, so we know. If you're going to start questioning whether math is real (is the whole universe just a dream type stuff) then sure, do that, but don't try to apply that for math which such fuzzy junk. There is a world of difference between infinite and finite. "large but finite" is not in any way comparable to "infinite". Any attempt to make such a parallel between infinite and large finite is fundamentally flawed to begin with. The pigeonhole principle has been proven and it's by that proof, and the history of math it relies on, and the fact that that math has a higher predictive power than anything else (especially within its own axioms where it's p-much perfect!)... that's why we really know that it works. If you want to question whether it works, then the burden is on you to provide evidence of some form more than some misguided "feeling" that large numbers are relateable to infinite. There already exists an argument in the form of the original proof and it is your burden to make a more compelling one. Also, I'm not sure why you're saying that the principle breaks down with infinitely sized sets. Infinitely sized sets.. Firstly, if both sets are infinite (and your plurality of sets there indicates you mean that), then it doesn't apply at all since m = n. If the sets are of different orders of infinity and you can, in fact, create a bijection between then then it does still hold. However, you state it as "a set and a set with one more element" and then say "infinitely sized", which when combined reads as complete and utter nonsense. Basically, I think you're starting with a flawed premise, leading into a flawed question, and there is no good answer. |