Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?

Scott wrote:
On Aug 8, 3:32 pm, magi...@xxxxxxxxxxxxxxxxx (Arturo Magidin) wrote:

[...]
You aren't picking "something challenging". You are trying to disprove
a basic theorem in set theory. Why?

Because there is something fundamental about the proof of Cantor's
Theorem that I do not understand and would like to. In this thread, I
have two proofs Prop4 and Prop5 (which is flawed as you pointed out).
Why, if Cantor's Theorem proves there are no functions with A_f in its
range does Prop4 have a function with A_f in its range?

There's nothing important I'm going to be able to add to what others
have said (most notably Arturo Magidin). However, it's possible
that if the situation is put in concrete terms you will understand
what you've been missing.

Suppose you have a bag of marbles and a tray of cups. (The bag
of marbles is our set S and the tray of cups is P(S).)

Cantor's Theorem says that, no matter how you put the marbles
in the cups, there will be at least one empty cup left over.

Your attempt to refute that is to say, "But that's not right!
Point to any cup and I can put a marble in it."

But that's not what CT says. It does not say "There is a particular
cup that is always empty." It says "Somewhere there is an
empty cup, no matter how you distribute the marbles."

The proof of Cantor's Theorem takes a description of how
the marbles are placed and uses it to describe one cup
that has to be empty (on pain of self-contradiction).
The description of the empty cup can be different for
each different distribution of marbles. That doesn't
matter, because the point is only that some cup is
always empty.

What makes Cantor's result deep is that it is so general.
Nowhere in the proof is any specific fact about the set S used,
not even if it's finite or infinite. Thus we arrive at the
conclusion that there are different "sizes" of infinity.

I am enlisting your intuitions about finite groups of objects
in order to make a point about possibly infinite groups of
objects. I admit, there is the potential for cheating there.
The problem with your approach, though, shows up well
before any properties of finite or infinite sets can be
used. It's the difference between "some cup that's
always empty" and "always some empty cup".

Jim Burns
.

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