In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 17 Jun., 19:01, William Hughes <wpihug...@hotmail.com> wrote: > > > A: actually infinite paths exist, > > B: the infinite tree contains a path p that can be > > distinguished from every path of P. > > > > You agree > > > > A ==> B > > > > So if A is true your claim is false. > > That is nonsense. > If (A) pi has a rational presentation, then (B) it can be written as p/ > q. > B is false. And A cannot be true without B.
That depends on P. If A. and (P is any countable set of paths), then B If A and (P is the uncountable set of all paths), then ~B > > > > You want to show > > > > ~A [Follows from ( A ==> B, ~B) ==> ~A ] > > > > by proving ~B > > > > (Note that assuming ~A is circular) > > > > <snip> > > > > > > Which is the first statement you disagree with > > > > > > all nodes of t are in the tree > > > > t is in the tree > > > > all nodes of t are in P > > > > t is not an element of P > > > > <snip evasion> > > > > Please answer the question > > What do you understand by "all nodes"?
In what context? All nodes of the tree, or all nodes of a path, or all nodes of a set of paths or something else? > > > > Note: The list of paths of P is not > > the tree in slightly different > > notation. > > Note the list of paths P is the same as the tree in slightly different > notation.
Not at all. No node is a member of any set of paths, but every node is a member of the tree.
> > >If all the nodes of a path h are in the > > tree then h is in the tree, however, if all the > > nodes of h are in P, h may or may not be an element > > of P. > > That is a self-contradiction
Not to those who understand the difference between being a member of a set and being a subset of a set.
(With exception of mathemagicians' tricks, of course.)
The only "trick" is noting that a subset of set need no be a member of that set and a member of a set need not be a subset of it.
In dealing with sets and their members and their subsets, the distinction between subsets and members is important, even if WM cannot keep it straight in his head.
Assuming that each node in a tree contains a pointer to any parent and pointers to any children, then a tree is merely a suitable pointer-connected set of nodes. A path is also a suitably pointer-connected set of nodes, though infinite paths are not binary trees and infinite binary trees are not paths.
Furthermore, no set of paths IS an infinite binary tree, though its union may be.
It is possible to have a set of paths whose union is a maximal infinite binary tree without its being the set of all paths of a maximal infinite binary tree. And any countable set of paths is like this.