> On 5 Mar 2019, at 00:42, Bruce Kellett <bhkellet...@gmail.com> wrote:
>
> On Tue, Mar 5, 2019 at 10:25 AM Russell Standish <li...@hpcoders.com.au
> <mailto:li...@hpcoders.com.au>> wrote:
> On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote:
> > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal <marc...@ulb.ac.be
> > <mailto:marc...@ulb.ac.be>> wrote:
> >
> >
> > >> I don't follow you. If the 8000th BB number is unknowable then
> > it is
> > certainly uncomputable
> >
> >
> > > That is not true. All natural number n are computable. The program is
> > “output n”.
> >
> >
> > I think you're being silly. You're saying if you already know that the
> > answer
> > to a problem is n then you can write a program that will "compute" the
> > answer
> > with just a "print n" command. But that's not computing that's just
> > printing.
>
> OK, so what about the program "print X+1", where X is the expansion of
> the number BB(8000)-1?
>
> If that's not computing something, then I'm sure I can cook up
> something more complicated to compute.
>
> I think the trouble with that, or with variations of that idea, is that they
> render the notion of 'computability' vacuous. In order to write such a
> program, or concoct such an algorithm, you need to know the answer in
> advance. That is fine, if you just want a program to compute the number 'n',
> 'n' being given in advance. But that is no help in computing a number that
> can be defined, but is not known in advance.
>
> So what people are really looking for here is a constructive notion of
> computability -- anything else has a tendency to render the notion of
> 'computability' trivial.
Then the whole recursion theory (computability theory) should be trivial, when
on the contrary it is a mine of surprising counter-intuitive results.
There is not yet clear notions of “non computable” for the constructive notion
of computability, which are dependent of the subjectivity of the
mathematicians. We got them in the talk of the “… & p” modes of self-reference.
The first person mental space is intuitionist. Indeed, it says even “no” to the
doctor, a priori.
The function defined by
If the twin conjecture is true output 0, else output 1.
Is a well defined function and it is computable, although not constructively.
It is computable, because it can be proved (easily) that its code is in the set
{K0 K1} (the two constant function [x]0 and [x]1.
The function defined by
if phi_x(x) converges output 1, else output 0
is provably NOT computable. That illustrates that the Turing-Church’s notion of
computability is not trivial.
Bruno
>
> Bruce
>
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