
Matheology § 198
Posted:
Jan 24, 2013 2:52 AM


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?
Most mathematicians have no answer and know this. They agree that an impossible task is asked for. But some of them (the names I will not mention here) offer really exciting ideas.
Matheologian 1: proposed to distinguish between the trees 2^(<w) and 2^(=<w). Not all nodes of the tree 2^(=<w) are finite. Nodes at level w are not elements of the binary tree 2^(<w) , but they are elements of the binary tree 2^(=<w) . And yes, I can state with confidence that nearly all of the experts here support my ideas on this matter.
And Matheologian 2 assisted him, addressing me: You?ve demonstrated copiously over the years in numerous venues that the indistinguishability of 2^(<w) and 2^(=<w) is an article of faith for you, and that you are either unwilling or unable to learn better. One tree is 2^(<w) ; the other is 2^(=<w) , which has 2^w as its top level, sitting above the levels of 2^(<w) .
In case you have not yet figured out what is under discussion, here is a simpler explanation: Try to distinguish the set of all terminating decimal fractions and the set of all real numbers of the unit interval by digits.
In case you have grasped the matter, here is another task: Try to explain why Cantor's diagonal argument is said to apply to actually infinite decimal representations only. Try to understand, why I claim that everything in Cantor's list happens exclusively inside of finite initial segments, such that, in effect, Cantor proves the uncountability of the countable set of terminating decimals.
The Binary Tree can be constructed by a sequence such that in every step a node and with it a finite path is added. If nevertheless all infinite paths exist in the Binary Tree after all nodes have been constructed, then it is obvious that infinite paths can creep in without being noticed. If that is proven in the tree, then we can also assume that after every line of a Cantorlist has been constructed and checked to not contain the antidiagonal, nevertheless all real numbers and all possible antidiagonals can creep into the list in the same way as the infinite paths can creep into the Binary Tree.
Regards, WM

