From: Patrick Juola <juola@suod.cs.colorado.edu>
Date: Tue, 26 Jul 94 09:54:05 PDT
To: cypherpunks@toad.com
Subject: Re: GUT and P=NP
Message-ID: <199407261653.KAA22721@suod.cs.colorado.edu>
MIME-Version: 1.0
Content-Type: text/plain


  > > One last word on this.  Try and represnet a continum of states by an 
  > > infinite turing machene.  Go ahead, I dare you.  You can't.<=big period.
  
  >Could I not let each position on the tape represent a real value in
  >[0...1]?
  
  Cantor demonstrated, near the turn of the century, that no such system
  can represent all reals in [0,1].  Boring technical explanation follows.
  ['cept it doesn't 'cause I edited it out.]

Perhaps I misunderstood the original poster, but I assumed that
s/he was suggesting simply encoding a particular real number into
the infinite tape using standard binary encoding.  There's no
cardinality problem there at all -- but of course the tape will only
hold a single real number.  Standard tape compression techniques
will buy you enough space for a countably infinite set of reals.

Of course, the dead hand of Cantor forbids a continuous set of reals
encoded onto a TM, but we all know that he's just a Dead White Male
and therefore can be ignored.... 8-)

	- kitten