In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 30 Mrz., 19:03, Virgil <vir...@ligriv.com> wrote: > > In article > > <050b4a95-d2b0-433b-98b6-d63c34635...@m9g2000vbc.googlegroups.com>, > > > > > > > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 29 Mrz., 19:40, Virgil <vir...@ligriv.com> wrote: > > > > In article > > > > <ce3c22f2-9116-4621-b3b4-e722fe51a...@a14g2000vbm.googlegroups.com>, > > > > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 26 Mrz., 22:47, Virgil <vir...@ligriv.com> wrote: > > > > > > > > But a tree that contains paths for all binary rationals will > > > > > > contain a > > > > > > path for all limits of a sequences of binary rationals. > > > > > > > Does a sequence always contain its limit? > > > > > > Depends on the sequence, of course. but a sequence of paths in a > > > > Complete Infinite Binary Tree in which the nth path must share at least > > > > n nodes with each of its successors will always converge, though not > > > > neccessarily to a binary rational. > > > > > A sequence of numbers may converge, but not necessarily to a limit > > > that is a term of the sequence. > > > > Precisely my point! > > Precisely not your point.
It is still my point, even though clearly WM does not understand it.
> The infinite path is not in the infinite > sequence of finite paths which are used to construct the complete > tree.
Each node of the infinite limit path is in all but finitely many of the infinite sequence of infinite but binary-rational paths of which is a limit.
otherwise that path would not be a limit or the tree would not be a CIBT. > > > > > A sequence of paths may converge, but not necessarily to a limit that > > > is a term of the sequence. > > > > So WM acknowledges that A sequence of binary rational paths can converge > > to a path that is not a binary rational > > and that is not in the tree of all binary rationals.
A tree having all binary rationals AS INFINITE PATHS is the only sort that can be a CIBT. So whatever sort of trees WM is talking about they cannot be Complete Infinite Binary Trees. > > > > > > > > > In mathematics more precision is required. > > > > Certainly more than WM is capable of producing,. > > You intermingle the paths of the tree and the limits which are neither > paths nor belong to the tree.
In CIBTs all paths are infinite, so WM must be talking about other types of trees > > > > > > > > In a COMPLETE INFINITE BINARY TREE, all paths are actually infinite > > > > -- > > > > > This is again a simple statement of countermathematical belief > > > > It is matter of simple definition. > > No.
WM does not get to decide what can be or cannot be a definition. > > > > The definition of a COMPLETE Infinite Binary Tree requires that no path > > in such a tree can terminate. > > An infinite sequence does not terminate. Nevertheless its limit is in > general not in the sequence. Correct or not?
If they are sequences of paths in a Complete Infinite Binary Tree which
There are a lot of infinite sequences that are "eventually constant" in a CIBT if one identifies each path with an infinite sequence of 0's and 1's in the binary representations of numbers in [0,1], and those sequences which are eventually all 0's, and only those, are "eventually constant".
But most of the sequences in a CIBT, even if convergent, are NOT eventually constant, --