Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: george <greeneg@xxxxxxxxxx>
- Date: Thu, 09 Aug 2007 11:43:08 -0700
On Aug 2, 4:59 pm, G. Frege <nomail@invalid> wrote:
Sure, man. Post it! :-)
Well, THAT was a mistake.
Arturo had been asking you for the quote, so I guess this was
needed context, but seriously, this was so silly that it cannot
have helped anything.
On Aug 3, 1:22 pm, Scott <ToaTe...@xxxxxxxxx> wrote:
[1] Potter, M., Set Theory and its Philosophy, 2004.
The context for my original question was an email:
Let S be any non-empty set.
Def 1: The cartesian product, denoted PxQ, is { (x,y) : x in P and y
in Q }.
Lemma 2: M = S x P(S) exists.
Proof: From Proposition 4.7.1 in [1], if P and Q are sets, then PxQ is
a set. Since S and P(S) are sets, SxP(S) is a set. QED.
WHOA, HOSS.
Are you libeling Mr. Potter here?!?
Are Def 1: and Lemma 2: from Potter's book? From Potter's treatment?
If so then you need to just throw that book away, EVEN if it is from
2004.
The problem is, that is a PHILOSOPHY book. This is sci.logic.
In any first-order logical treatment of set theory, you will NOT NEED
to prove that the cross product OF ANY two sets EXISTS. IT MUST
exist. OF COURSE it exists. What you would actually have to prove,
if you wanted to prove existence of cross-product, is existence OF THE
OPERATOR AS A WHOLE, which amounts to proving that THE OPERATOR
IS *functional*, NOT that any particular triple/instance of it
"exists"!
And you CERTAINLY NEVER need to prove
tha anything is a set! In first-order ZFC, EVERYthing is a set!
In that theory, the things that are not sets ARE NOT THINGS, either!
You are wasting time on trivial details that do not even BEAR the
level of
scrutiny you are trying to apply. In first-order logic, we DON'T DO
"existence".
EVERYthing we are talking about is ASSUMED to exist. EXplicitly. The
question
is always whether, among ALL THESE THINGS THAT ARE ALREADY KNOWN
to exist, is the NUMBER of them satisfying (or dissatisfying) some
PROPERTY,
0? Or is it bigger than that?
.
- References:
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: Scott
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: David C . Ullrich
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: Scott
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: Arturo Magidin
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: Scott
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: G . Frege
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: Scott
- Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- Prev by Date: Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- Next by Date: Re: intuitionist logic
- Previous by thread: Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- Next by thread: Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- Index(es):
Relevant Pages
|
