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@acm.org | scotch ripple." -- Willy Wonka ..................|.................................................. Apple Computer, Inc. 5 Infinite Loop, MS 305-2D Cupertino, CA 95014 408.862.0540 fax:974.6094 R254(IL5-2N) collins@newton.apple.com ..................................................................... 408.257.1746 1024:669687 catalyst@netcom.com