No, each tape position cannot... (was Re: GUT and P=NP)

[Scott Collins]

  >Could I not let each position on the tape represent a real value in
  >[0...1]?

You could try!  But you would always omit values.  You can demonstrate this
with the `diagonal rule' or similar proofs.  Here's a simple one:

Take any two adjacent `positions' on the tape; Write out the decimal (or
binary) notation for the real values they represent (note, the
representations may be infinite).  Given two such strings that are not
identical, you can always find a string numerically `between' them (even if
both are infinite) as long as they are not identical.  E.g.,

      "0.12345"
  --->"0.123455"<----
      "0.12346"

Such a string is a real value you omitted.  Your tape, even if it is
infinite, is not the right order of infinity to model the Real numbers.

Scott Collins     | "Invention, my dear friends, is 93% perspiration,
                  |  6% electricity, 4% evaporation, and 2% butter-
  collins at acm.org |  scotch ripple."                   -- Willy Wonka
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