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