On 2010-06-20, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: > "Tim Little" <email@example.com> wrote in message >> If you intend your proof to be "exactly the same as Cantor's, but with >> the only difference being the word "computable" in front of the word >> "Real"", it must start with a conditional introduction. In other >> words, something like: >> >> "Suppose that L is a list of computable Reals. That is, L is a >> function from N to R and for all n in N, there exists a Turing >> Machine T_n such that when provided with k as input, T_n halts and >> outputs the k'th decimal digit of L(n)." > > That's not how Cantor's proof starts.
Correct: It didn't feature any mention of computable reals (but as I recall it did feature the relevant properties of the definition of a real number).
I am presuming that you want *your* proof to substitute "computable real" for "real", and therefore you need to substitute the definition of computable real number for the definition of real number. That means you also need to make that substitution in the definition of a list of computable real numbers.
Are you starting to see now why it makes no sense to just drop "computable" in front of every occurrence of the word "real" in the proof?