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

In article <1186512440.514432.38300@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Scott <ToaTerra@xxxxxxxxx> wrote:

Now (hopefully) Prop 4A is a "machine" that, given a subset p, gives
me a function g_p with property P and Prop 5A is a "machine" that,
given a specific subset A, gives me the result that there are no
functions f_A with property P.


In addition to the problems mentioned, I want to once again emphasize
that you are committing a very serious fallacy here, and this is why
this particular line is doomed to failure.

Your "Prop 4A" is a (trivial) result that says that given an element
B of P(S), if S is not empty then you can find a function g_B that
->does<- have B in its range.

Cantor's Argument shows that, given a function f:S->P(S), there is a
set A_f in P(S) that is not in the range of f. Change the function,
you change the set.

But you are trying to shoehorn this into a statement that says that
there is a set A in P(S) which is not in the range of ANY function
f:S->P(S).

That is, you have a statement of the form

Every person has a mother;

and you think you can get it to say, instead

There is a person who is everyone's mother.

You can't. Quantifiers do not easily commute. While (Ex)(Ay)P(x,y)
always implies (Ay)(Ex)P(x,y), in general (Ay)(Ex)P(x,y) does ->NOT<-
imply (Ex)(Ay)P(x,y). In this case, it most definitely does not. That
is why you keep writing nonsense.

You cannot try to use the construction in Cantor's Argument fixing
both function and set. It does not work that way. You fix the
function, but then you have to give way on the subset; the subset will
depend on your function. You CANNOT use the construction applied
simultaneously to a set and a function, which is what you keep trying
to do.

What is most irritating is that if you knew even a little
propositional logic you would see immediately that your attempts are
either unintelligible or trivially false. Yet you persist. You keep
trying to read War & Peace in russian without even learning the
cyrillic alphabet. Why not try to at least learn the basics of the
language first? You keep trying to run marathon when you cannot even
crawl.


--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
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Arturo Magidin
magidin-at-member-ams-org

.