Re: Why? [was Re: Cantor`s powerset theorem is false?]


georgie wrote:
It seems rather odd that the "set of all sets exists" is false.

Anyone's subjective assessment of oddity is going to depend on
what he has previously been exposed to as context. In the modern
context, sets are often taken to be well-founded, i.e., not to
contain themselves. In that context the non-existence of a set
of all sets does not seem odd at all. If you revise that context
(qv. Aczel), if you acknowledge that well-foundedness is not all
THAT important, that some sets SHOULD be thought of as
containing themselves, then you can consistently have a universal
set. You could just google "Set theories with a universal set" and
see what you get.

I believe you say that because a theory that
allows such a thing might lead to a paradox.

Well, in the presence of cetain other valued axioms that
we'd rather not yield, it doesn't just MIGHT lead to a paradox;
it PROVABLY leads to a paradox.

But Cantor's
diagonal proof that leads to "larger than infinitely
sized" sets is not looked at as paradoxical.

Paradoxicality is NOT a matter of how ANYthing LOOKS!
It IS a matter of whether you can or cannot PROVE a contradiction!

It sure is strange what some people think is meaningful.

It is strange that you think that that sentence is meaningful, in
THIS context. The WHOLE point about THIS (logic-oriented)
context is that NO ONE NEED GIVE A *** about what
"meaning" anything might have. All you HAVE to care about
is what consequences follow from what axioms.

.