Date: Nov 4, 2010 1:40 AM
Author: Bill Taylor
Subject: Re: Mathematics as a language

On Nov 3, 8:35 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Even if we think of the category of sets as the result of some sort of
> iterative construction, there are many possible resulting structures
> (just as there are many possible fields).


Indeed so!

> Now, some of us have the idea
> that the "real" structure of sets is one of these constructions,
> perhaps, and the aim of the axioms is to characterize that real
> structure somehow,


Or at least to describe it as well as possible.
Characterisation (in the technical sense) doesn't seem possible.

There are AT LEAST two views on this, though...

1) the maximalist view, that every set that can conceivably
be brought within the purview of "the iterative hierarchy"
is really part of "the true set theoretic universe";

2) the minimalist view, that only sets that are required to exist
by the "constructive" (?) axioms of ZF, are really part of
"the true set theoretic universe".

(1) would allow AC and as many inaccessibles as you can get;
it is silent on CH.

(2) probably means V=L, and thus solves CH at least,
but seems to be massively unpopular anong set theorists.

Probably many working mathies have a view somewhere in between.

> but I confess I just don't have that intuition at all.

And there're probably a lot like that as well!

Basically, our intuitions just aren't up to confidently
dealing with general set theory. (OC some claim theirs ARE.)

> > We may also observe that infintary set theoretic claims have
> > arithmetic consequences -- "Theory T is inconsistent", <etc>


> I have no difficulties with these (interesting) observations.

Me neither. And they give some urgency to the questions above!

-- Baffled Bill