🚀 go-pugleaf

In article <1992Sep4.172833.20527@guinness.idbsu.edu>, holmes@opal.idbsu.edu (Randall Holmes) writes:

> >Thus you might as well reject the proof rather than the concept
> >of set of all sets. However if you define sets with ZF it seems
> >that you must accept the proof. The separation "axiom" which is
> >in facta theorem of ZF,  implies that the above dubious set be
> >a proper set. (Am I wrong ?)
> 
> The usual terminology is "proper class".

I think he just means "... implies that the above set be an honest-to-goodness
set, where we're assuming the universe to be a set". So no, he's not wrong.

> >
> >The question is now, whether or not alternative axiomatizations
> >have been proposed that are consistent with the existence of the
> >set of all sets ?
> 
> 	There are a number of such axiomatizations.  The nicest is
> Jensen's version NFU of Quine's set theory NF ("New Foundations").  NF
> itself is not recommended, because it is not known to be consistent,
> and it also has the curious feature of disproving the Axiom of Choice.
> Jensen's system is known to be consistent, and can be extended with
> the Axiom of Choice and as many axioms of strong infinity as one might
> desire (loosely speaking).

Hmm. One could alternatively argue that NFU is not recommended because it is
known to have a model within ZFC, so it's not likely to tell us anything we
didn't know already...
Seriously, ZF isn't known to be consistent either, just believed to be so
because we've been playing with it for ages and not found any holes. (As I'm
quite sure you're perfectly well aware!) Why shouldn't we play with NF for
a while and see if it falls over? If it doesn't -- well, perhaps we'll start
to trust NF the way we presently trust ZF, and perhaps it will give some new
insights.
I don't mean to malign NFU or anything; I just take exception to the idea that
a set theory that's not provably weaker than ZF can't be worth studying!