> > 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.
> > > 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. You can't have an infinite number of natural numbers without having also an infinite natural number (which is a self- contradiction). > > > > > 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. > > > 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. > > > > 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. Each path starts at the root and stretches until a finite number is reached. Even an infinity of finite paths does not contain a path which contains an infinity of nodes. The union of paths however is a path.