On 2010-06-19, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: > Well, there are lots of definitions of a computable Real, I will use > Wikipedia's most intuitive definition: "computable reals, are the > real numbers that can be computed to within any desired precision by > a finite, terminating algorithm."
Most intuitive, and also least mathematically useful. However, we can work with that. The input to the algorithm is the desired precision and the output is a suitable approximant, correct?
> 1. Given a list of computable Reals, we can identify the nth > computable Real on the list by simply counting down to the nth > entry.
That is only a "finite, terminating algorithm" if the list is finite. Since the only input to the algorithm is the desired precision, your recipe only works if you can embed the list into the algorithm.
The rest of your argument depends upon this non-algorithmic step, and so is snipped.
However, it occurs to me that your misunderstanding may be deeper than I expected. Let's consider a simpler case: "lists" of length 1. Suppose x is a real number, say Chaitin's Omega. Is x+1 computable?
If so, why?
If not, how does this differ from "suppose L is a list of real numbers. Is antidiag(L) computable?"