Re: infinity ...


William Hughes wrote:
> albstorz@xxxxxx wrote:
> > David R Tribble wrote:
> >
> > >
> > > Consider the set of reals in the interval [0,1], that is, the set
> > > S = {x in R : 0 <= x <= 1}. The elements of this set cannot be
> > > enumerated by the naturals (which is why it is called an "uncountably
> > > infinite" set). But all sets have a size, so this set must have a
> > > size that is not a natural number. It is meaningless (and just
> > > plain false) to say this set "has no size" or "is not a set".
> >
> >
> > I'm not shure if the reals build a set in spite of you and Cantor and
> > others are shure.
> > A set is defined by consisting of discrete, distinguishable, individual
> > elements. Now tell me: what separates a point on a line from the very
> > next point on the line to be discrete? What separates sqrt(2) from the
> > very next real number to be discrete?
> > If you look only on individual points, you may have a set. But if you
> > look on all of them?
> >
> > So, your above argumentation has no relevance to me. Proof the reals to
> > be a set, then let's talk again.
>
> What is your difficulty with the standard definitions?
>
> Certainly the Integers { ... -3,-2,-1,0,1,2,3 ... }
> form a set

Yes. If we accept infinite sets.


>
> Take the set of all pairs of integers (i,j) where
> the second integer is not 0.
>
> Take some equivalence classes of the above and we
> have the rationals. (Note the rationals are not
> discrete).
>
> Take paris of sets of rationals (A,B) where all
> the rationals in A are less than those in B and
> where A union B is all the rationals. Now
> we have the reals.

You think about Dedekind cuts?

>
> At which step do we fail to have a set?
>
> -William Hughes


All this don't proof anything.

You know that the constructable real numbers are denumerable infinite.
If all real numbers are nondenumerable, there must be nondenumerable
infinite many real numbers between every pair of constructable reals.

Proof that this numbers are entities.

Regards
AS

.

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