Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]

In article <MPG.1ced6107b1babfda989c32@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> > If A being a proper subset of B does not mean that A is "smaller than"
> > B, in your system, as you remarks about the rationals and reals would
> > imply, then why are not the naturals, the odd naturals and the odd
> > primes all the same size?
> A proper subset IS smaller than the superset, as I have stated. It is you who
> believe this rule evaporates at infinity. When did I ever make such a
> statement?

When TO claimed that the sets of rational and reals are of the same
size!

Since, for example, sqrt(2) is real but not rational, the rationals are
a proper subset of the reals, so there must be, by TO's own definition,
fewer of them, but TO has elsewhere said otherwise.

Also, TO has not explained how one compares sets when neither is a
subset of the other. TO has offered no rule covering such cases.

Are there sets whose 'sizes' cannot be compared?

Many people, of considerable talents in mathematics, have tried, and
failed, to construct any 'size' definition which incorporates proper
subsets as always being smaller that the superset.

If TO succeeds, it will be a marvel.

But he has a long way to go.
.

Relevant Pages

  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >>> thinking on the rationals, that one can get arbitrarily close to any ... >>> that it therefore constitutes an enumeration of the reals. ... > A proper subset IS smaller than the superset, ...
    (sci.math)
  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >> thinking on the rationals, that one can get arbitrarily close to any real, ... >> that it therefore constitutes an enumeration of the reals. ... > imply, then why are not the naturals, the odd naturals and the odd ... A proper subset IS smaller than the superset, ...
    (sci.math)
  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >> What you call inconsistencies are clashes between what you want to hold ... > thinking on the rationals, that one can get arbitrarily close to any real, ... > that it therefore constitutes an enumeration of the reals. ... If A being a proper subset of B does not mean that A is "smaller than" ...
    (sci.math)
  • Re: Computable functions/reasls: followup.
    ... The computable-function definition above still applies, ... Russian-style constructivism is BISH + MP, ... which were reals that you couldn't tell whether or not were rational. ... special about the rationals; they could be replaced by the integers, ...
    (sci.logic)
  • Re: Cantor Confusion
    ... The "number" pi is definitely a merely fictitious element of continuum. ... naturals, integers, rationals, irrationals, or reals. ... intergers and naturals are genuine. ... genuine numbers to the reals is tempting but not justified. ...
    (sci.math)