> "Jesse F. Hughes" <email@example.com> writes: > >> For me (and I'd wager for many others), it makes no more sense to ask >> whether infinite sets exist than to ask whether the field axioms are >> really true. > > The field axioms are not about anything. Set theoretic statements in > standard set theory are about sets, as envisaged in the iterative > conception.
Yes, I thought about that difference, but it still doesn't seem to make much difference to me.
Even if we think of the category of sets as the result of some sort of iterative construction, there are many possible resulting structures (just as there are many possible fields). Now, some of us have the idea that the "real" structure of sets is one of these constructions, perhaps, and the aim of the axioms is to characterize that real structure somehow, but I confess I just don't have that intuition at all.
> We may also observe that infintary set theoretic claims have > arithmetic consequences -- "Theory T is inconsistent", "Theory T is > consistent", "Algorithm A terminates on all inputs", ... -- so in so > far as we regard arithmetical claims, computational claims, as > meaningful in themselves, as true or false as a matter of mathematical > fact, we must ask, perhaps not if infinite sets exist, but at least > whether the arithmetical consequences we draw reasoning with them are > true or not. (There's loads of insightful rambling and illuminating > rumination on this issue in Torkel's dissertation.)
I have no difficulties with these (interesting) observations.
-- Jesse F. Hughes "Basically there are two angry groups. I am a harsh force of one. Against me is a society of mathematicians. So far it's been a draw." -- JSH gives another display of keen insight.