On Mon, 8 Apr 2013, fom wrote: > 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.
It's likely as useful ast the category theory topology, pointless topology, ie pointless.
> > > > 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.
Ernst P. Specker, "The Axiom of Choice in Quine's New Foundations for Mathematical Logic," pp 972-975, Vol. 39, 1653, Proc. N.A.S.
I'd be interested in your comments.
> > 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. > Philosphy isn't math.
> 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". > Just the DNA and the google governement file on the person which has superceeded the old fashion time, date and location of birth.
> 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.
What's that? The determination to needlessly multiply entities?