"Tim Little" <email@example.com> wrote in message news:firstname.lastname@example.org... > 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. >
I can't see how the move from addition of "computable" in front of "Reals" requires any additional substitution in my "definition of a list". A list is simply a mapping from N. Yes, I want to know the first item, the second item etc; that is what a list is.
> 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? >
Sort of. You are saying that you could effectively not specify what the first computable Real on the list is, because it depends upon some uncomputable function based on Chaitans. I can see why that is less of a problem to Cantor than it is to me, because he cares less about what it really was. But I still can't accept that you are providing a list of computable Reals when you aren't even telling me what the first one actually is. That's cheating.