Computability is Nonsense?

[magfrump]

Start with a Godel-numbering of Turing machines whose outputs are real numbers and which can be described with a finite amount of information. In an ideal world, build a Turing-complete programming language, and take a Godel-numbering of programs which take as input a natural number (n) and return as output the first n digits of a real number (let'

Comments

[anonymous]

What gets in your way is the so-called halting problem. The computable numbers are those that are approximated by computer programs that *always halt.* However, when you're enumerating all the computer programs to run your construction, you can't distinguish between those that halt and those that run forever. The distinction is important, since a program doesn't approximate a number at all unless it actually halts and produces output.

[magfrump]

Okay, so let's make the distinction by saying that for every n, the program has to halt. I don't think we can get a better definition of halting than this, since some numbers such as pi which are clearly computable must be approximated by a finite number of decimal places, since just outputting all of pi would be a process that doesn't halt