Kees
Posts:
136
Registered:
8/24/05
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Re: bijection of R: R <--> Rx.....xR
Posted:
Sep 7, 2005 4:51 PM
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> Timothy Golden http://www.BandTechnology.com wrote:
> > I think I see what you mean.
> > But I will counter that such a mapping would have
> to be systematic and
> > because these are real numbers that system should
> be functional in
> > nature.
>
> At its basic set-theoretic level, a function is just
> a set of
> ordered pairs. If there exists a bijection between 2
> infinite
> sets, then there exist infinitely many, most without
> any
> describable structure at all. As has been explained,
> you're
> not likely to find a "nice" bijection between R and
> R^2 that
> takes into account their mathematical structures in
> ways that
> you might like, because these structures are
> sufficiently
> different. But these sets do have the same
> cardinality.
>
> > Intuitively I have a hard time seeing the
> cardinality of RxR matching
> > R.
> > To split hairs the digit crunch method leaves the
> length of the
> > resultant twice as long as its two sources. That is
> not a symmetrical
> > relationship.
> > I really care about information more than
> cardinality and so this whole
> > bijection argument may be irrelevant.
>
> This is an important point. The concept of
> cardinality of
> sets ignores most of the useful distinctions of
> mathematics. Two
> sets may have the same cardinality and yet differ in
> many
> significant mathematical properties. So for some
> applications
> cardinality is a useful tool, but for others you need
> more detail.
> The category Set has much less structure than most of
> the useful
> categories of mathematics.
>
I agreed with what you said up to the point about the category of sets. Indeed many categories of structures have lots of properties, but the category of sets enjoyes a lot of these properties. It has all limits and colimits and it is a topos. While a set is 'merely' just a set and other mathematical structures which are based on sets appear to be a set with 'much more', the category of 'just sets' is very reach.
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