On Apr 9, 8:34 am, fom <fomJ...@nyms.net> wrote: > On 4/9/2013 12:15 AM, William Elliot wrote: > > > > > > > > > > > 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. > > Sadly, set theory is philosophy. It should > not be, but it is. When Kunen or Jech are > deferring to first-order logic with identity, > they are deferring to this > > http://plato.stanford.edu/entries/identity-relative/#1 > > Perhaps it would be more correct to simply say > that deciding on "any favorite set theory" that > sometimes appears in texts is a hard decision. > It throws philosophy in your face even if you > did not really mean to pursue it. > > > > > > > > > > > > >> 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? > > The axiom of determinacy is inconsistent with the > axiom of choice. Under the axiom of determinacy, > every set of reals is Lebesgue measurable. > > It probably trades one set of needless entities for > another.
The universe of ZF + AD appears to be, in all respects, 'smaller' than ZF+AC . So, it does eliminate a set of entities,but it doesn't seem to generate another . Anyway, may be a little off topic, I found this paper interesting ,even though I don't agree with everything it says : http://arxiv.org/abs/0905.1675