Re: Video Words

Roland Hutchinson filted:

Jens Brix Christiansen wrote:

Glenn Knickerbocker skrev:
On Fri, 29 May 2009 17:41:33 -0700, Evan Kirshenbaum wrote:
notion of "the set of all sets that don't contain themselves". The
solution that the barber isn't a man is like saying that that isn't a
set.

Well, yeah. One explanation for the layman that I read long ago talked
about the notion of algebras defining sets in such a way as to exclude
sets from being members of sets, or from being members of themselves, or
from not being members of themselves. (I never dug any further to know
whether or how they might be useful.)

Looks like a description of the Zermelo-Fraenkel axiomatic approach to
set theory, slightly distorted over the years. In this system, sets are
defined in such a way that sets can be members of sets, but not of
themselves. For mathematicians, this is a very useful notion. It is one
of the most important foundations of all mathematics.

The key idea in the ZF approach that sidesteps the paradoxes is that you can
consists of the members of another (larger) set that meet some specified
criterion -- but you always have to start by specifying the larger set.

You're not given "the set of all sets" as a starting point, so you can't use
it construct "the set of all sets that do not contain themselves".

But the cardinality of "the set of all sets" and of "the set of all sets that do
not contain themselves" is the same: countably infinite (aleph-null)...neither
is therefore "larger"....r


--
A pessimist sees the glass as half empty.
An optometrist asks whether you see the glass
more full like this?...or like this?
.

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