
Re: Matheology § 198
Posted:
Jan 25, 2013 10:04 AM


On 25 Jan., 08:41, William Hughes <wpihug...@gmail.com> wrote: > On Jan 25, 8:32 am, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > On 25 Jan., 01:27, William Hughes <wpihug...@gmail.com> wrote: > > > > On Jan 24, 8:52 am, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > The following is copied from Mathematics StackExchange and > > > > MathOverflow. Small wonder that the sources have been deleted already. > > > > > How can we distinguish between that infinite Binary Tree that contains > > > > only all finite initial segments of the infinite paths and that > > > > complete infinite Binary Tree that in addition also contains all > > > > infinite paths? > > > > > Let k denote the L_k th level of the Binary Tree. The set of all > > > > nodes of the Binary Tree defined by the union of all finite initial > > > > segments (L_1, L_2, ..., L_k) of the sequence of levels U{0 ... oo} > > > > (L_1, L_2, ..., L_k) contains (as subsets) all finite initial segments > > > > of all infinite paths. Does it contain (as subsets) the infinite paths > > > > too? > > > > > How could both Binary Trees be distinguished by levels or by nodes? > > > > They cannot of course. Both have exactly the same levels and the same > > > nodes. > > > > They can of course be distinguished. > > > > In one case you do not include infinite subsets. > > > In the other you do. > > > My question aimed at the posiibility to distinguish the Binary Trees > > by a mathematical criterion, namely that one that is applied in the > > diagonal argument. Of course you have understood that. > > > That does not hinder you to believe in addition in matheological > > concepts that cannot be based on mathematical facts like nodes, > > levels, or digits. > > Nope. The concept is based on nodes, and levels. > > We can use the same set of nodes to make two collections of > sets of nodes. One collection contains all sets of nodes, X, with > the property that there is a node in X with a level greater or > equal to that of any other node in X. > The other collection contains all sets of nodes, Y, with the property > that there is no node in Y with a level greater or equal to that of > any other node Y.
And both sets of nodes are completely exhausted by the same paths with the only difference that one kind is called X and is finite and the other kind is called Y and is infinite. And, of course, there is a bijection between both kinds because they contain same nodes. And, yes, there is a last minor difference, there are far more Y's than X's. But nobody would notice.
Regards, WM

