Date: Jan 15, 2013 4:18 PM
Subject: Re: WMatheology § 191
On 15 Jan., 22:03, Virgil <vir...@ligriv.com> wrote:
> In article
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 15 Jan., 19:54, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <e3bfe180-1cbe-415a-a2c9-0f1dd676f...@w3g2000yqj.googlegroups.com>,
> > > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 15 Jan., 08:23, Virgil <vir...@ligriv.com> wrote:
> > > > > > But here is the list: All finite initial segments of all decimal
> > > > > > expansions are included.
> > > > > That is not a list.
> > > > The set is countable. There exists a bijection with |N. So list-
> > > > fetishists should be able to set up a list of that set.
> > > Your set is not a list until that bijection, or at least a surjection,
> > > from |N to your set has been explicitly established, at which point an
> > > antidiagonal which is not listed can be shown to exist.
> > The set is countable with no doubt.
> Until it is proved so by being listed, there can be legitimate doubt.
Doubt that the terminating rationals are countable?
Doubt that the definable tails are countable?
Doubt that aleph_0 * aleph_0 = aleph_0?
Not even in matheology.
> > An anti-diagonal cannot differ from every number of the set because
> > the set contains all numbers.
> Only as finite strings so that any infinite string will differ from
> every finite string.
Not at a digit at a finite place.
> > Compare the Binary Tree where no anti-
> > diagonal can be found (in the finite realm).
> But the complete infinite binary tree itself does not exist in any
> finite realm,
The complete infinite Binary Tree exists within the infinite set of
all finite levels. There is no further place where it could exist. A
life after life belongs to theology. A definition afterall finite
definability belongs to matheology.
> > And there is no infinite realm.
> Maybe not in WMYTHEOLOGY, but there are more things in heaven and earth,
> WM, than are dreamt of your philosophy.
Perhaps in heaven, but mathematics does not belong to heaven.
> > So if there are infinitely many paths
> > in the Binary Tree, then they must cross at least one finite level
> > together.
> Paths of finite trees don't "cross" any level together, so why should
> any other tree differ?
All paths cross every level, but not at distinct nodes. If however a
number n of paths of the Binary Tree is claimed, then ther must be a
level with n nodes.
> > But that is not the case. Hence they can only become
> > infinitely many beyond every finite level. But that is the realm of
> > matheology. In mathematics there does nothing follow beyond every
> > finite level.
> In a sequence of levels, either there is a last level or no last level.
> If there is a last level then there are only finitely many levels.
> If there is no last level then there are infinitely many levels.
> In the set of naturals numbers, beyond each natural there is another
> natural, so there more than any finite number of naturals.
Nevertheless every natural n is finite and the index n can be the last
one of a finite initial segment.