[SPORK, FUNNY] Donald Rumsfeld... the poet?

Russell Turpin deafbox at hotmail.com
Tue Apr 15 13:37:01 PDT 2003


>At first thought, that the "finitely  describable
>reals" should be countable seems correct.  Only finitely many statements of 
>a given length possible,
>so list them all in order of length, and the list should provide a 
>counting, QED.

While first thoughts are often flawed, in this case,
it is quite correct.

>But why is e one? Or pi?  For proper definition they both require a 
>limiting process. And once limits are allowed into the class of processes 
>of finite describability, the camel's nose is under the tent.  Because 
>every real number is the limit of some sequence of rational numbers.  And 
>the Cantor diagonal construction shows easily that any listing of those is 
>not complete.

This objection misses the fact that some limit
sequences are finitely describable, but the bulk of
them are not. Both e and pi have finitely describable
limit sequences. One can write a computer program
that will spit out ever finer estimate of pi, getting
as much precision as desired, as long as one has the
patience and computation resources. That program is a
finite description of an infinite sequence. These
finite descriptions are countable, whether they are
phrased as Turing machines, Python programs, or
essays in English. The MIT graduate is quite correct
that the set of infinite sequences is uncountable.
This restates the fact that the bulk of real numbers
are shy. (Or if you prefer, dark.)



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