On 4/8/2013 11:24 PM, William Elliot wrote: > On Mon, 8 Apr 2013, fom wrote: >>> Remember the engineers' KISS and the beauty of simplicity. >>> What more simple than invoking Occam for V = L and no inaccessible? >>> Face it, that's all the set theory needed for all of math. >> >> Do you believe that? >> >> What about Grothendieck universes arising from >> category theory? > > What good are they? >
Technically, I think they let algebraists work without concern for set-theoretic paradoxes. That would come into play in the representation theory. But, I am not knowledgeable enough to assert that with confidence.
>> http://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theory >> >> MacLane, as a matter of parsimony, assumes one universe >> satisfying Tarski's axiom. Grothendieck assumes one such >> universe for every set. > > That's his problem. > >>> BTW, Quine's NF denies AxC. >> >> I need to look at Quine's work more carefully at this >> point. I doubt I would like it because I do not >> agree with his views on the nature of identity. > > At Quine's time it was assumed AxC was compatible. Decades later, it > turns out to be violated for some large constructed sets. Would you like > the reference for the paper? >
Yes. Thank you.
>> But, it is interesting for other reasons. >> >> The axiom of choice, however, may be construed as >> a necessary axiom for model theory. Models interpret >> the sign of equality as the diagonal of a Cartesian >> product. The statement of axiom of choice in terms >> of Cartesian products is a guarantee that models >> of the sign of equality will exist. >> >> Just something to think about. > > AxC is needed for infinite products of sets to be not empty. > Anyway, I'm a prochoice mathematician. >
Yes. I see what you difference you are making.
Historically, the question of identity is related to Leibniz' principle of identity of indiscernibles. But, Leibniz logic had been intensional. He viewed logical species as more complex than logical genera and his reasoning had been based on the fact that more information is required to describe a species than is required to describe a genus.
I think about identity in those terms. Topologically, that would involve something along the lines of Cantor's intersection theorem for closed sets. So, identity of an individual might require an "infinite description".
In topology, the metric relations and non-metric notions of closeness come together in uniform spaces. And, of course, one can think about the diagonal of a model in relation to the definition of uniformities.
If I am permitted to be ambivalent about the role of model theory, I am in agreement with your prochoice affiliation. Stop worrying about models, and the axiom of determinacy becomes almost preferable.