Date: Dec 13, 2012 5:35 PM
Subject: Re: On the infinite binary Tree
WM <firstname.lastname@example.org> wrote:
> On 13 Dez., 20:17, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <f46dba51-46fc-4354-985c-27665000d...@x3g2000yqo.googlegroups.com>,
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 13 Dez., 09:26, Virgil <vir...@ligriv.com> wrote:
> > > > > No. I proved that the number of infinite paths is countable by
> > > > > constructing all nodes of the Binbary Tree by a countable set of
> > > > > infinite paths.
> > > > WM is again, or should I say still, self-deluded in all sorts of ways.
> > > > The only way WM could actually have CONSTRUCTED all nodes of a INFINITE
> > > > binary tree is by completing infinitely many construction steps himself
> > > > which he has often claimed that no one can ever do.
> > > If actual infinity exists (and I assume that for the sake of
> > > contradiction) then the CIBT can be constructed.
> > > > Such trees can exist only in the imagination, as is the case with a
> > > > great many mathematical "constructions".
> > > > But the set of paths of such an imagined tree, to be consistent, must
> > > > have a different path for every different subset of the set of all
> > > > naturals numbers, being the set of levels at which that path branches
> > > > left, and there are uncountably many such subsets of N.
> > > Alas most of them are not definable. Why does no Cantor-list contain
> > > undefinable elements?
> > Undefineable or unreconstructable paths are not needed
> moreover, they cannot be treated in a Cantor list.
If one were to have a list of supposedly all definable reals, the Cantor
process would define an new one. And, in fact, countably many new ones.
So it appears that the set of all definable numbers is not, strictly
speaking, countable, or perhaps it is just not well-defined, thus not a
set at all.
> > to prove
> > uncountability because every list of defineable/constructable paths
> > proves the existence, by explicit definition/construction of it, of a
> > path which has been omitted from that list.
> Explicit construction is onyl possible if every list number is
> defined. That restricts the constructed diagonals to a countable set.
Perhaps that supposed totality of "constructablity" is too ambiguous to
define a set?
> > Thus it is your alleged set of all definable/constructable paths that
> > either does not exist at all or is not countable.
> You say it: The set of all definable real numbers is not countable.
> You just admitted a contradiction of set theory.
Not necessarily. it may be that "definable" is sufficiently ambiguous
that the set of all definable real numbers cannot be made explicit, or
even cannot be claimed to exist.
The problem is, how could you test a number for definability unless you
already have a definition by which to identify it, which begs the
question. We know only indirectly about the necessity of any undefinable
numbers existing, but by their very nature we can never access any of
them without first defining them.
Actually a more accurate description than "undefineable" is
if the set of accessible real numbers, provided it can be shown to be
well defined, is alleged to be countable, as it is alleged, then the set
of inaccessible numbers would have to be uncountable.