Date: May 2, 2014 3:35 PM
Author: Virgil
Subject: Re: � 488   Who said this?

In article <>, wrote:
> On Friday, 2 May 2014 18:13:46 UTC+2, Virgil wrote:

> > In article <>,
> > wrote:

> > > On Thursday, 1 May 2014 22:41:27 UTC+2, Dan Christensen wrote:

> > > > On Tuesday, April 29, 2014 2:50:51 AM UTC-4,
> > > > wrote:

> > > > > Couldn't just the seemingly so fruitful hypothesis of the
> > > > > infinite have straightly inserted contradictions into
> > > > > mathematics and have fundamentally distroyed the basic nature
> > > > > of this science which is so proud on its consistency?

> > It didn't!

> It is easy to prove that it did

Then why has WM never provided any actual such proof, but only repeated
claims of its existence?

> Can you quote any "uncountability proof" that constructs an undefinable
> number?

The point is that such INDIVIDUALLY undefineable numbers do not have
individual individual definitions o constructions but are only
defineable as members of infinite sets. Every defined real interval of
positive length, for example, is such an infinite set.

> Or that implies the existence of undefinable numbers?

There are no totally undefineable real numbers but most of reals are
individually undefineable. Which sort of undefineability is is WM asking

> without a real number that has been defined by this proof?
> > If there are countably many members to a set it stays counable.
> Of course.
> > It is the set of all subsets of a countably infinite set that becomes
> > uncountable, which is quite different from what WM s falsely claims
> > above.

> Hessenberg's "proof" constructs one and only one subset. Of course this
> subset is amply defined. I think it has been defined more than 100000 times.
> Therefore we have 100000 space-time quadruples and 100000 names of authors or
> readers and in addition the general definition of this set: The set of all
> natural numbers that are not mapped on sets containing them. Few item in
> mathematics have been better definid.

WM's definition is incomplete and, probabaly deliberately, misleading.

The issue is this: given any non-empty set S and it power set 2^S, can
there be a surjective function f:S -> 2^S.
Any such function defines a subset {s in SL s not in f(S) } of 2^S
which clearly cannot be in the image of f, thus proving the
imposibility of surjecting ANY non-empty set S to istst power set 2^S.

> > But in a set of naturals which is not actually infinite one one cannot
> > have n+1 > n for all n in such a set.

> One can. Otherwise classical mathematics would be self-contradictory

One can have n+1 > n FOR ALL n only for actually infinite sets of n's.
For any finite set of n's it can't work, and potentially infinite sets
are not sets at all, but mere illusions, at least in real mathematics.