In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 5 Dez., 20:06, Virgil <Vir...@home.esc> wrote: > > > > That is potential infinity. > > > For every node there is a path surpassing it > > > For every path there is a node surpassing it. > > > > While I can guess what you mean by saying "For every node there is a > > path surpassing it", I have no idea what you might mean by "For every > > path there is a node surpassing it". > > There is a symmetry: > For every path of the form 0.000...000111... there is a node 0 at the > outmost left side of the tree not covered by that path. > And for every node 0 at the outmost left side of the tree, there is a > path of the form 0.000...000111... covering it.
Then why did you not say so instead of being deliberately ambiguous? And while you re, for once correct, your statements are trivial and of no relevance to the falsities you are trying to establish. > > > > > > There is no "all nodes" and no "all paths". > > > > There is in ZF and in most set theories. > > > > If there is no "all nodes" your claim of "For every node" is impossible. > > If there is no "all paths" your claim of "For every path" is impossible. > > > > You can't have one without the other. > > Correct.
The why do you keep claiming one without the other?
> You can't have an infinite number of natural numbers without > having also an infinite natural number (which is a self- > contradiction).
Both a non sequitur,and a false statement!!! ZF has an infinite number of naturals but does not have any infinite naturals, at least according to any definition of "natural number" allowed in ZF.
What goes on in WM's Wolkenmuekenheim is, of course, irrelevant to mathematics.
> > > > > > > > > Actual infinity infinity or set theory, what is the same, claim that > > > there is a path containing all nodes > > > > Only in unary trees. In binary trees no path contains all nodes. > > > > > (namely the path 0.000...) > > > > Which does not contain the node 0.1 > > Which does contain all nodes of the outmost left side of the tree.
That is an entirely different matter which, as you left it out of your original statement, makes you original statement false, as I previously said it was. > > > > > and > > > that the union of all paths > > > 0.111... > > > 0.0111... > > > 0.00111... > > > ... > > > contains the path 0.000... > > > > Not outside of Wolkenmuekenheim > > > > > but that there is no single path of the list containing all nodes of > > > 0.000... > > > > That path, 0.000..., certainly contains all nodes of that path!!!!! > > But there are not all nodes of that form.
If one is not embedded in Wolkenmuekenheim, there are. > > > > > > > That is simply impossible. The union of paths of the list contains > > > exactly as many nodes of 0.000... as one of those paths to be unioned. > > > > WRONG!!! Each such path contains only finitely many nodes of 0.000..., > > but the union of all of them contains more than any finite number of > > those nodes. > > That is nonsense.
What WM claims makes no sense within his Wolkenmuekenheim makes good sense everywhere else
> Each path starts at the root and stretches until a > finite number is reached.
In a standard infinite binary tree, none of the uncountably many infinite paths has such a last node. So whatever WM is talking about, it is not a standard infinite binary tree.
> Even an infinity of finite paths does not > contain a path which contains an infinity of nodes.
I do not know where WM is getting his paths, but he is being fed a very non-standard and inferior quality bunch of them, and, no doubt, being grossly overcharged into the bargain.