Re: wanted: mathematician

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".

This gets relevant because the _amount_ of different Cartesian products you
can get this way is the multiplicative product of the amount of members in
each of the non-empty sets. (A set with 4 things in it and a set with 5 can
give you 20 different result sets that contain a pair of things, one from
each of the sets, for example.)

[I may be misremembering things and the 'Cartesian product' is the collection
of all the paired-off result sets... <wiki> okay, yes, I am. So it's actually

"If you pick one member from each of the non-empty sets, and make a set out
of them, and take the collection of _all such possible sets_ for each different
choice of picked members, the resulting set is non-empty".

and "the cardinality of the Cartesian product of the sets is the multiplicative
product of the cardinalities of the sets themselves".]

This means if you multiply any
number, including an infinite number, of nonzero cardinals, including
infinite cardinals, the result is never zero.

.... because the Cartesian product you get from them as sets, rather than as
numbers, is non-empty.

I find the idea of
multiplying an infinite number of infinities together and getting zero as a
result objectionably counter-intuitive.

Thanks.

Does that help any?

Dave
--
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Relevant Pages

  • 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: wanted: mathematician
    ... Cartesian product of non-empty sets is non-empty. ... cardinals, including infinite cardinals, the result is never zero. ... standard definition. ...
    (rec.arts.sf.written)
  • Re: wanted: mathematician
    ... I can try to do cliffs notes... ... 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, ...
    (rec.arts.sf.written)
  • 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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