Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 198
Posted:
Jan 26, 2013 6:54 PM


In article <1331a8055b2c426d95f109b12b42da05@u7g2000yqg.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> On 26 Jan., 23:10, Virgil <vir...@ligriv.com> wrote: > > In article > > <054da2be2f0a4290b35610eb0a5e1...@r14g2000yqe.googlegroups.com>, > > > > WM <mueck...@rz.fhaugsburg.de> wrote: > > > On 26 Jan., 01:46, Virgil <vir...@ligriv.com> wrote: > > > > > > > Of interest is this: If the same set of > > > > > nodes has to describe both, the Binary Tree with finite paths and that > > > > > with infinite paths, then it is impossible to discern, alone by nodes, > > > > > whether we work in the former or the latter. > > > > > > There is no such thing as a Complete Infinite Binary Tree with finite > > > > paths. > > > > > So you agree that there is a level omega? > > > > Why should I agree to add another level to the infinitely many finite > > levels that must already exist in order to have a COMPLETE INFINITE > > BINARY TREE at all? > > These levels exist already after constructing all finite initial > segments of all paths, abbreviated by "all finite paths". Or can you > determine a node or level of the complete infinite Binary Tree that > does not exist? The standard definition of a path in the kind of binary trees we are talking about is that it is a MAXIMAL sequence of parentchild nodes.
In other words a path's 'first' node in parentchild order cannot be the child of any node of the tree outside that path and its 'last' node, if any, cannot have any child node outside that path. Since in a Complete Infinite Binary Tree every node has two child nodes, any finite set of nodes must have some last node with no child node in the set, but that node must have two child nodes.
Thus no finite set of nodes can be a maximal sequence of parentchild nodes in a Complete Infinite Binary Tree.
Thus no finite set of nodes can be a PATH in a Complete Infinite Binary Tree.
This proof should be simple enough and straightforward enough for even WM to understand:
In any COMPLETE INFINITE BINARY TREE any sequence of parentchild linked nodes that has a last (most childish) node cannot be a maximal such sequence and thus cannot be a path in any CIBT.
Thus no finite set of nodes in a Complete Infinite Binary Tree can be a path in such a tree.
And it is easily proved, a la Cantor, that there cannot be any surjection from N to the set of all paths in a Complete Infinite Binary Tree, at least outside of Wolkenmuekenheim. 

