Re: bijection of R: R <--> Rx.....xR

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.

.

Relevant Pages

  • Re: Cantors diagonal proof wrong?
    ... >> So you claim that all infinite sets have the same cardinality? ... >What I don't understand is what the notion of cardinality really is. ... on about, it's not mathematics, it's a confession of faith (and ... or on whether "axioms" in one system or another ...
    (sci.math)
  • Re: An uncountable countable set
    ... You agreed that it is not a cardinality ordering. ... 'infinite') in one ordering, which is not a cardinality ordering, is ... problem is that you spout about non-standard analysis while you haven't ... and mathematics that that book uses as that book is written ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... >>very accurate description of the size of an infinite set, ... >>integers, and no more integers than odds, based on cardinality measure? ... > used by mathematics in universal terms.They're talking about tangents. ...
    (sci.math)
  • Re: Galileos Paradox
    ... Note, I said 'infinite', not 'infinity'. ... about the term "cardinality", but I will respond to that later. ... The words 'prime' and 'even' have meanings outside of mathematics. ... correspondence) and less squares than naturals. ...
    (sci.math)
  • Re: Cantor Confusion
    ... impossible to construct a bijection between W and N. ... Look at just about any set theory text book for such ... But we need it to determine this cardinality. ... Consider a binary tree which has infinite ...
    (sci.math)