Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Thu, 9 Aug 2007 02:51:26 +0000 (UTC)
In article <1186615575.807542.279810@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Scott <ToaTerra@xxxxxxxxx> wrote:
For every function f:S -> P(S), let A_f = { x : in S : x notin f(x) }.
Clearly it can be shown that A_f notin range(f). My question has
nothing to do with putting f: x |-> p into a subset A_f, because I
fully understand A_f content changes based upon f. What I am trying to
understand is how it can be claimed that there are no functions with
A_f in the range. Example:
S = {1}, P(S) = { 0, {1} }. S x P(S) = { (1,0), (1,{1}) }.
Let f1 = { (1,0) } and f2 = { (1,{1} ) }.
Let A_f1 = { x in S : x notin f1(x) } so A_f1 = { 1 }
It follows from CT that there are no functions with A_f1 in its range.
No. Again: Cantor's Theorem asserts ONLY that A_f1 is not in the range
->of f1<-. It says nothing about whether A_f1 is or is not in the
range of other functions.
Cantor's Theorem says: given a function f:S->P(S), there exists a set A_f,
WHICH DEPENDS ON f, such that A_f is not in the range of the original
function f.
That is ALL it says.
But there is, f2.
Duh.
There are two functions from S to P(S), your f1 and your f2. The ONLY
thing that Cantor's Theorem says when applied to S is:
The set A_f1 is not in the range of f1 (true);
The set A_f2 is not in the range of f2 (true).
It does NOT say that "there are no functions S->P(S) with A_f1 in
its[sic] range". It does NOT say that "there are no functioins[six]
with A_f2 in its[sic] range." It says, simply: A_f1 is not in the
range of f1, and A_f2 is not in the range of f2.
Let A_f2 = { x in S : x notin f2(x) } so A_f2 = 0
It follows from CT that there are no functioins with A_f2 in its
range.
No, no such thing follows.
But there is, f1.
Duh. Because the assertion you make is NOT what Cantor's Theorem says.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
.
- Follow-Ups:
- 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: 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?
- Prev by Date: Re: Does logic reasoning need quantifier?
- Next by Date: sci.logic scheduled for sporge flood
- 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
|
