On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote:
> On Tue, Mar 5, 2019 at 2:03 PM Russell Standish <li...@hpcoders.com.au> wrote:
>     On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote:
>     >
>     > My problem with your idea that the function: "(n-1)+1" is a valid
>     computational
>     > algorithm for n is that it makes all real numbers also computable, but
>     the
>     > notion of Turing computability applies only to the integers. We do not
>     want a
>     > definition of 'computable', that makes all reals computable.
>     You cannot represent n as a finite string for an arbitrary real number
>     n. But you can for an arbitrary integer n.
> Sure. But that was not part of your definition of a 'computation'. The
> algorithm f(x): (r-1)+1 works for all reals r as well as for finite strings n.
> Bruce

I don't think it's 'my definition'. The usual meaning of computable
integer is that there exists a program that outputs it. For real
numbers, this is changed to a program exists that outputs a sequence
of numbers that converges to the real number in question. One could
also consider "spigot" programs for this purpose too - a program that
outputs the decimal (or binary say) expansion of the real number. It
is clear that this more relaxed definition is equivalent to the former
in the integer case.

It is clear that all integers are computable according to the above,
and that nearly all real numbers aren't (by virtue of there only being
a countable number of programs).



Dr Russell Standish                    Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellow        hpco...@hpcoders.com.au
Economics, Kingston University         http://www.hpcoders.com.au

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