In article <firstname.lastname@example.org>, WM <email@example.com> 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.
> 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.
> 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, but then its set of paths cannot be listed or counted. > > 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.
> 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?
> 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.
Outside of WMytheology that makes for infinitely many natural numbers. --