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