Date: Jan 24, 2013 2:52 AM Author: mueckenh@rz.fh-augsburg.de Subject: Matheology § 198 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 Cantor-list has been constructed and

checked to not contain the anti-diagonal, nevertheless all real

numbers and all possible anti-diagonals can creep into the list in the

same way as the infinite paths can creep into the Binary Tree.

Regards, WM