On 15 Jan., 22:03, Virgil <vir...@ligriv.com> wrote: > In article > <3e9210e4-371e-4fc4-a607-049543041...@bx10g2000vbb.googlegroups.com>, > > > > > > 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.