Date: Apr 9, 2013 12:24 AM
Author: William Elliot
Subject: Re: some amateurish opinions on CH

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?

> 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?

> 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.