```Date: Jan 24, 2013 2:52 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 198

The following is copied from Mathematics StackExchange andMathOverflow. Small wonder that the sources have been deleted already.How can we distinguish between that infinite Binary Tree that containsonly all finite initial segments of the infinite paths and thatcomplete infinite Binary Tree that in addition also contains allinfinite paths?Let k  denote the L_k th level of the Binary Tree.  The set of allnodes of the Binary Tree defined by the union of all finite initialsegments (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 segmentsof all infinite paths. Does it contain (as subsets) the infinite pathstoo?How could both Binary Trees be distinguished by levels or by nodes?Most mathematicians have no answer and know this. They  agree that animpossible task is asked for. But some of them (the names I will notmention here) offer really exciting ideas.Matheologian 1:  proposed to distinguish between the trees 2^(<w) and2^(=<w). Not all nodes of the tree 2^(=<w) are finite. Nodes at levelw  are not elements of the binary tree 2^(<w) , but they are elementsof the binary tree 2^(=<w) . And yes, I can state with confidence thatnearly all of the experts here support my ideas on this matter.And Matheologian 2 assisted him, addressing me: You?ve demonstratedcopiously over the years in numerous venues that theindistinguishability of 2^(<w)  and  2^(=<w)  is an article of faithfor 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 toplevel, sitting above the levels of 2^(<w) .In case you have not yet figured out what is under discussion, here isa simpler explanation: Try to distinguish the set of all terminatingdecimal fractions and the set of all real numbers of the unit intervalby digits.In case you have grasped the matter, here is another task: Try toexplain why Cantor's diagonal argument is said to apply to actuallyinfinite decimal representations only. Try to understand, why I claimthat everything in Cantor's list happens exclusively inside of finiteinitial segments, such that, in effect, Cantor proves theuncountability of the countable set of terminating decimals.The Binary Tree can be constructed by a sequence such that in everystep a node and with it a finite path is added. If nevertheless allinfinite paths exist in the Binary Tree after all nodes have beenconstructed, then it is obvious that infinite paths can creep inwithout being noticed. If that is proven in the tree, then we can alsoassume that after every line of a Cantor-list has been constructed andchecked to not contain the anti-diagonal, nevertheless all realnumbers and all possible anti-diagonals can creep into the list in thesame way as the infinite paths can creep into the Binary Tree.Regards, WM
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