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

Virgil said:
> 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!
Learn to read. I said the rationals constitute a sort of enumeration of the
reals, but as a representation, it is reall a 2D array of naturals, and should
be considered to be N^2 in size. This is smaller than the reals. My point was
that they are certainly NOT an equivalent set to the naturals. That's
balderdash.
>
> 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.
No he hasn't, really. Leanr to read.
>
> 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.
Actually I did give a method for providing a precise measure for all sets that
are generated by functions on the naturals. If you missed it, sux 2 b u.
>
> Are there sets whose 'sizes' cannot be compared?
Undoubtedly. Which is larger, the set of grains of sand in the Sahar, or the
set of stars in the galaxy? How would you compare these sets? Perhps you would
turn to a method like set density......
>
> 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.
So what? I succeeded. Are you jealous? that's what it sound like.
>
> If TO succeeds, it will be a marvel.
>
> But he has a long way to go.
>
As long as I waste my time here, perhaps.

--
Smiles,

Tony
.

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