Re: wanted: mathematician

On 2008-01-04, David DeLaney <dbd@xxxxxxxxxxxxxxxxxx> wrote:
Aaron Denney <wnoise@xxxxxxx> wrote:
Can you show me the reduction to one of the more standard definitions?

I can try to do cliffs notes...

On 2008-01-04, Gene Ward Smith <gene@xxxxxxxxxxxxx> wrote:
If you multiply together a finite number of finite positive integers the
result is never zero.

Because the way multiplication is defined, the product can't be zero unless
there's a zero as one of the multiplicands. (Furthermore, unless all but one
of them are 1, the product's gonna be larger than any of the factors, but
that's irrelevant here.)

The Axiom of Choice says an arbitrary Cartesian
product of non-empty sets is non-empty.

If I remember how to do this right:

"If you pick one member from each of the non-empty sets, and make a set out
of the sets made by making each of those members into a set, regardless of
which member you picked from each set, the resulting set is non-empty".

Yes. This shows the implication from the Axiom of choice (that you can
do this picking) to Gene's words. I'm looking for the converse.

--
Aaron Denney
-><-
.

Relevant Pages

  • Re: wanted: mathematician
    ... Because the way multiplication is defined, the product can't be zero unless ... "If you pick one member from each of the non-empty sets, ... infinite cardinals, ...
    (rec.arts.sf.written)
  • Re: wanted: mathematician
    ... Cartesian product of non-empty sets is non-empty. ... cardinals, including infinite cardinals, the result is never zero. ... find the idea of multiplying an infinite number of infinities ...
    (rec.arts.sf.written)
  • Re: wanted: mathematician
    ... Cartesian product of non-empty sets is non-empty. ... cardinals, including infinite cardinals, the result is never zero. ...
    (rec.arts.sf.written)
  • Re: Cut a point into two - topological?
    ... Bob Kolker wrote: ... holes can merge into one: ... How do you cut something with zero width? ... so you can't split the point into two non-empty sets. ...
    (sci.math)
  • Re: On writing negative zero - with or without sign
    ... interpretations to the standard as written. ... but I see the phrase "the representation of a positive or zero ... note that Bob was careful to say "one member". ... required to be no prefix for positive values. ...
    (comp.lang.fortran)
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