Date: Apr 9, 2013 1:15 AM
Author: William Elliot
Subject: Re: some amateurish opinions on CH

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?