fom
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Re: Matheology § 233
Posted:
Mar 30, 2013 9:55 PM


On 3/30/2013 7:56 PM, Virgil wrote: > In article > <2bc13fff5cbb43dda06a218c68c99849@m9g2000vbc.googlegroups.com>, > WM <mueckenh@rz.fhaugsburg.de> wrote: > >> On 30 Mrz., 22:11, Virgil <vir...@ligriv.com> wrote: >> >>>>> Thus there is always at least one bit of any listed entry disagreeing >>>>> with the antidiagonanl, just as the Cantor proof requires. >>> >>>> In a list containing every rational: Is there always, i.e., up to >>>> every digit, an infinite set of paths identical with the anti >>>> diagonal? Yes or no? >>> >>> The set of paths in any Complete Infinite Binary Tree which agree with >>> any particular path up to its nth node is equinumerous with the set of >>> all paths in the entire tree i.e., is uncountably infinite. >> >> This was the question: In a list containing every rational: Is there >> always, i.e., up to every digit, an infinite set of paths identical >> with the antidiagonal? Yes or no? > > Lists and trees are different. And antidiagonals derive from lists, not > trees. > The entries in list are well ordered. > The entries in a Complete Infinite Binary Tree are densely ordered. > Those order types are incompatible. > So questions, like WM's, which confuse them, are nonsense. > At least outside Wolkenmuekenheim. >
But note that the question also demonstrates WM's complete lack of understanding of the diagonal argument.
He has been told time and time again that it is an argument scheme which only has application under certain assumptions.
He chooses to believe otherwise for the agenda of his fanaticism.
Suppose one is given a countable listing of the rationals (with the appropriate restriction on double representation) according to the infinite listing of an expansion.
Suppose one performs a diagonalization on that listing.
Is the resultant a rational number? No.
What may be concluded? That the rational numbers do not exhaust the capacity of the algorithm to generate representations if that algorithm is to generate a representation for every rational number.
Although the burden of proof lies with WM concerning the nature of the diagonal, it is a simple matter to understand if one uses a Baire space representation instead.
In the Baire space, rationals are in correspondence with eventually constant sequences. Since, by construction, a list of Baire space rationals would exhaust all of the eventually constant sequences, the resultant of a diagonal argument could not have an eventually constant sequence unless the original premise had been false.

