email@example.com wrote: > Aatu Koskensilta wrote: >> Obviously it makes no sense to agree or disagree with the group axioms. >> It does make perfect sense to agree or disagree, or remain agnostic >> about, that there exists a measurable cardinal, or that a particular >> primitive recursive ordering of naturals has no primitive recursive >> infinite descending sequences. > > Well, this is a bit out of my depth, but my understanding is that, say, > regarding the Continuum Hypothesis, one can have no opinion, one can > assert it, or one can assert its negation.
Why? Do you think similarly one can have no opinion regarding the consistency of ZFC and that one can assert it or one can assert its negation?
> In each case, there are a different set of theorems which can be proven > under these assumptions. And a different universe of sets which can be > said to exist or not exist.
The problem with this idea is that set theoretical principles have consequences that concern things mathematicians think of as uniquely determined, such as the naturals, reals and so forth. While in set theoretical contexts talk about "different universes" might make some marginal sense, surely it makes no sense to speak of there being "different natural number systems".
> How is this conceptually different for you than considering on the one > hand the usual axioms of group theory, and on the other assuming in > addition commutativity, so that all groups are Abelian?
The difference is that in group theory one studies various structures with various properties. The axioms in that cases are used just to pick the sort of structure one is interested in. Set theoretical axioms, axioms about well-orderedness of primitive recursive orderings, etc. aren't used to pick any structures, but rather to provide information about one structure.
-- Aatu Koskensilta (firstname.lastname@example.org)
"Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus