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
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.