Your claim is that "no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path."
A: actually infinite paths exist, B: the infinite tree contains a path p that can be distinguished from every path of P.
A ==> B
So if A is true then B is true and your claim is false.
You want to show
~A [Follows from ( A ==> B, ~B) ==> ~A ]
by proving ~B
(Note that assuming ~A is circular)
> > > > 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"? >
"all nodes of t are in the tree" There is no node that is in t but is not in the tree
"all nodes of t are in P" There is no node that is in t but is not in an element of P
WM: the list of paths P is the same as the tree
Nope. You have agreed that the tree contains a subset of nodes that is not contained in one element of the list of paths P.