Re: Cardinality of Set of Computable Numbers?

From: George Greene (greeneg_at_greeneg-cs.cs.unc.edu)
Date: 12/30/03 

Date: 30 Dec 2003 13:13:19 -0500

"Russell Easterly" <logiclab@comcast.net> writes:
 : Let a computable number be defined as the unbounded, but finite output tape
 : of a TM.

This is just idiotic, Russ. You CANNOT MEAN "computable number"
by this. What you have defined here is a FINITE STRING.
This is NOT ANY kind of number, not without some VASTLY complicated
scheme of REPRESENTATION for numbers in terms of characters!
In later posts, you are going to say that 0 and 1 are the only things
in the alphabet. What kinds of "numbers" will this allow you to
represent? Real? Rational? Complex? Natural?

"The finite output string of a TM"
IS NOT ANY kind of definition of number, computable
or otherwise. You have to GIVE us a definition of "number"
BEFORE you start talking about computable numbers!

Basically, you have to say, for every finite string you
have just talked about above, what number it represents.

Then, the fact that you have only definied numerical representations
for FINITE strings will mean that ANY infinite string represens
an "uncomputable" number!

But that is just silly.

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