R. Srinivasan wrote: > Gerry Myerson wrote: [...] > > You can (Norm says) work with functions by knowing what kind of thing > > is an input, what kind of thing is an output, and what the rule is, > > without ever thinking of the set of all inputs. If you can show that > > any rule that takes an integer as an input and gives a computable > > number as an output must omit some computable number, then you have > > a proof that computable numbers are not countable, and you've done it > > without using set-theoretic constructions. > [...] > > > "What kind of thing is an input", "what kind of thing is an output", > "takes an integer as input" all have to be precisely defined. For > example, what do you mean by "takes an integer as input"? Takes some > specific integer? Or do you mean "zero is taken as input and if the > n'th integer is taken as input (in some enumeration of the integers) > then the n+1 th integer is also taken as input"? The latter clearly > defines an infinite class of integers even if we don't explicity name > this class.
Correction -- should read "...(in some enumeration of the integers with zero as the first integer) ...".