In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 25 Jan., 09:26, Virgil <vir...@ligriv.com> wrote: > > > > How do you express actual infinity by means of nodes or levels? > > > Try it. > > > Write a sequence like xxxxxxxxxx... and do never stop. > > > > It is the knowledge that it never stops that makes it NOT FINITE.
To have extra knowledge beyond what WM has is a necessary qualification for every true mathematician, and a lot of non-matjematicians, for that matter. > > Yes you need some external "knowledge" or better say trust in > matheology.
If "matheology" is what goes on outside Wolkenmuekenheim, we should all learn it. > > > It is also the knowledge that no finite initial segment of natural > > numbers (fison) surjects onto it that makes it infinite. > > That is correct in every case. The reason could be that N does not > exist as something completed.
It exists as a completed idea, which is a much existence as anything purely mathematical has. > > > > > Tell me when you have expressed an infinite sequence. > > > > Tell me when you can surject a fison onto it. > > I am fairly sure that I can surject a FISON on that sequence that you > may have produced today, even if you have started very early.
WM is often fairly sure of his ability, but his claims far exceed his grasp of mathematics. > > > > > > > > > > We can use the same set of nodes to make two collections of > > > > sets of nodes. One collection contains all sets of nodes, X, with > > > > the property that there is a node in X with a level greater or > > > > equal to that of any other node in X. > > > > The other collection contains all sets of nodes, Y, with the property > > > > that there is no node in Y with a level greater or equal to that of > > > > any other node Y. > > > > > No. You cannot express the latter by nodes and levels. If you don't > > > believe me, try it. > > > > We have already done so to our own satisfaction, by showing that no > > fison surjects onto any a path ( as a sequnce of nodes) in a real > > Complete Infinite Binary Tree > > So the complete infinite paths must extend beyond every finit path in > the tree?
Just like |N exceeds every FISON.
|N is essentially a Complete Infinite Unary Tree.
And if the tree contains only all finite paths, then we have > something like the bed of Procrustes. What shall we do? Cut the tails > that are too long? I would be glad to do so, but I have never seen > any. Perhaps it is only an apparition of matheology if you see such > tails?
Just as outside of Wolkenmuekenheim there can be a union of all FISONS which is an infinite path in a Complete Infinite Unary Tree, unions of nested sequences of finite sets of rooted node chains form paths . --