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Topic: The uncountability infinite binary tree.
Replies: 118   Last Post: Dec 17, 2012 8:49 PM

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Rupert

Posts: 3,806
Registered: 12/6/04
Re: The uncountability infinite binary tree.
Posted: Dec 16, 2012 3:39 PM
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On Dec 16, 7:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:
> Al-Jofar writes:
>
> 1: "This is a simple Corollary of Cantor's diagonal argument actually.
> Proof: Let G be any countable subtree of the infinite binary tree that
> is 0 rooted and such that all paths ending by 0-0-0-... or by
> 1-1-1-...are among the paths of G. "
>
> 'Let G be a subtree of the IBBT including all expansions representing
> fractions by powers of two (there are two of those for each of those
> values).'
>
> 2: "Let f be a bijective function from the domain N of all
> naturals(except 0) to the set of all infinite paths of G. "
>
> For the countable subtree only containing fractions by powers of two
> and their dual representations, that's countable (and not obviously
> all the paths).
>
> 3: "Construct the diagonal path d_f in the following manner: The root
> (i.e. the 1st) node of d_f is 0 labeled. Now for n=1,2,3,.. ; The n
> +1_th node of d_f is labeled by a label that is opposite to the label
> of the n+1_th node of the path of G that f sends n to. "
>
> Let G be all the paths.  In lexicographic order, here as described as
> BT _oo (binary tree's breadth-first traversal at infinity), augmented
> with .0111... being first to reflect the modification of "the" anti-
> diagonal, the generated anti-diagonal path is .0111..., that is the
> least element of G in the ordering.
>


No, it's not.

> With an un-modified construction of the antidiagonal, the generated
> anti-diagonal path is .111... and at the end of any course-of-passage
> through the paths in their natural order, i.e., never before the end.
> That is, there's a simpler corollary that is the same as the binary
> anti-diagonal argument.
>
> 4:  "Now clearly the diagonal d_f (actually the anti-diagonal but it
> shall be called the diagonal for short) is a path and clearly it is
> labeled in a way that is different from labeling of all paths of G. So
> d_f is missing from G. "
>
> In the course of passage with the natural order of the paths, d_f =
> f(0) or here f(1) and is not missing from G.
>


Nonsense.

> 5:  "So any countable subtree of the infinite binary tree, that is 0
> rooted and that has all paths ending with 0-0-0-.. or with 1-1-1..
> among its paths; would be missing a path of the infinite binary tree.
> "
>
> That doesn't follow for the natural ordering of the paths by their
> content as expansions.  Then though while the paths are totally
> ordered, well-ordering them would be as well-ordering the reals.
>


That's irrelevant.

> Then I'll happily accept that the anti-diagonal argument is much the
> same for the list of expansions or the tree of expansions, then that
> EF's antidiagonal is .111... and at the end of the list and BT's
> antidiagonal is .111... and rightmost of the tree.
>
> Thanks,
>
> Ross Finlayson




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12/14/12
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Zaljohar@gmail.com
12/14/12
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Virgil
12/15/12
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12/15/12
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Rupert
12/15/12
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12/15/12
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12/15/12
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12/15/12
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12/15/12
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Virgil
12/16/12
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12/16/12
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12/16/12
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12/16/12
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12/16/12
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12/16/12
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Virgil
12/16/12
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ross.finlayson@gmail.com
12/16/12
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Virgil
12/16/12
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12/16/12
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Zaljohar@gmail.com
12/16/12
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Virgil
12/17/12
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12/17/12
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Virgil
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Virgil
12/15/12
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Tanu R.
12/15/12
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Virgil
12/16/12
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mueckenh@rz.fh-augsburg.de
12/16/12
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Virgil
12/17/12
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12/17/12
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Virgil
12/15/12
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Virgil
12/16/12
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12/16/12
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Virgil
12/17/12
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12/17/12
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Virgil
12/15/12
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Rupert
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Virgil
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namducnguyen
12/15/12
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Tanu R.
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12/16/12
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Rupert
12/16/12
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12/16/12
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Rupert
12/16/12
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Rupert
12/16/12
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ross.finlayson@gmail.com
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Rupert
12/16/12
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Rupert
12/16/12
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Rupert
12/16/12
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Rupert
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Rupert
12/16/12
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Rupert
12/17/12
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12/17/12
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Virgil
12/17/12
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12/17/12
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Virgil
12/16/12
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Rupert
12/17/12
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Virgil
12/17/12
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Virgil
12/16/12
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Virgil
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Virgil
12/16/12
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m. m. m.
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Virgil
12/16/12
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Rupert
12/16/12
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Rupert
12/17/12
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12/17/12
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12/17/12
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fom
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Virgil
12/16/12
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m. m. m.
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Virgil
12/16/12
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m. m. m.
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Virgil
12/16/12
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ross.finlayson@gmail.com
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trj
12/16/12
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Zaljohar@gmail.com
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mueckenh@rz.fh-augsburg.de
12/16/12
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Virgil
12/16/12
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Virgil
12/15/12
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Virgil
12/15/12
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Tanu R.
12/15/12
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Virgil
12/15/12
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Virgil
12/15/12
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Virgil

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