Re: Post Axiom Syndrome

On 11 Jul 2005 12:39:00 -0700, Ross A. Finlayson <raf@xxxxxxxxxxxxxxx> said:
> ...
> There are a variety of modern theories with universal sets. They are
> not ZF, they generally share the axioms of ZF besides regularity.

If they share the axioms of ZF other than regularity (a.k.a.
foundation), then they don't have universal sets -- the proof that there
is no such set in ZF does not involve regularity. Notably, Aczel's
non-well-founded set theory AFA -- which fits exactly your description
of a theory that "shares the axioms of ZF besides regularity -- has no
universal set. Rather than simply dropping his name, you might actually
try *studying* Aczel's work. It's quite beautiful.

> ZF, founded in the 1930's or so,

Try 1908.

> has been generally useful for formalists,

No more so, or less so, than any other axiomatic theory.

> and has largely been traduced by NBG/GBN, NFU, Aczel,

It's not even coherent to talk about one theory "traducing" another. As
for Aczel traducing ZF, well, it's utterly ludicrous.

> and from my perspective NAST, the null axiom set theory, an axiom-free
> theory.

And a sadly vague and uninformed perspective it is. As has been noted
before, your "null axiom set theory" is just first-order logic. It's
not a theory of anything, at least, as opposed to anything else. (I
suspect you think that's a virtue...)

> There are a variety of set theorists who see an imperative for a
> universal set in a set theory, ...

Whether or not a theory with a universal set is desirable for one reason
or another is neither here nor there. Though most well-known set
theories don't allow one, there are a few that do, and that might even
be useful in some contexts, or at least formally interesting. The
original problem here (among other things) was your ridiculous claim
that ZF is inconsistent simply because it lacks one.

> ...ranging from Cantor,

In fact, Cantor had a relatively sophisticated grasp of the distinction
between sets and proper classes, and was entirely clear on the fact that
the sets themselves did not constitute a further set, but rather an
"absolutely infinite multiplicity," by which it is quite clear he meant
what we now call a proper class. Read, e.g., Hallett's book Cantorian
Set Theory and Limitation of Size if you actually want to know the
history and the math.

> Skolemize, your model is countable. In the generic extension N^G
> contains no elements not in N.

Good grief, more word salad. You don't have the slightest idea what
Skolemization is or what a generic extension is, and I think you know
it. Really, it's much more satisfying (and much less embarrassing)
actually to know what you're talking about. Give it a try.

Well, that's more than enough from me to you. Back to work...

.

Relevant Pages

  • Re: Post Axiom Syndrome
    ... they generally share the axioms of ZF besides regularity. ... >> universal set in a set theory, ...
    (sci.logic)
  • Re: Question about universal set
    ... ZF is a particular set of axioms, which does include regularity. ... the existence of a universal set is incompatible with both ...
    (sci.math)
  • Re: a question for the anti-Cantorians
    ... In set theory, people generally use a set of axioms, axiomatic set ... One of those axioms is called regularity or sometimes foundation. ... is that quantification over sets implies a universal set. ...
    (sci.math)
  • Re: Skolems Paradox
    ... sci.math_20050214.rtf:I suggest discarding all the non-logical axioms ... where in general theorems of ZFC minus regularity are theorems. ... Zermelo-Fraenkel Set Theory, is inconsistent, because of regularity. ... "p-adic integers") may well be infinite in precision and extent. ...
    (sci.logic)
  • Re: Post Axiom Syndrome
    ... Chris Menzel wrote: ... > If they share the axioms of ZF other than regularity (a.k.a. ... AS WE KNOW IT quantifies over A DOMAIN OF quantification! ...
    (sci.logic)