Re: Penrose vs the Robot


Stephen Harris wrote:

<snip>
>
> > Because points on the lattice would be finite,
> > then the behaviour of what goes on is necessarily computable (assuming
> > the usual axioms).
>
> This is news to me! An algorithm is defined as a terminating process so
> it is finite. But in the other direction, where is the rule or axiom which
> says
> that any finite process must be computable or computably representable?

I was handwaving in what I thought was the spirit of your claim that
any continuous process can be simulated by a discrete process. If a
set is finite, then it is obviously recursive. If one supposes that
every behaviour in the Universe can be represented by a finite set,
then in some trivial sense it is computable; but not in any interesting
sense (which I think is your point as well?).

<a big snip>

>
> > Strictly speaking you are correct. We can grid the four-dimensional
> > Universe (or at least the Universe to some large finite limit) in a
> > very fine way, so finely that for all pratical purposes, our
> > perceptions are unable to distinguish the difference between the grid
> > and continuous reality. Because points on the lattice would be finite,
> > then the behaviour of what goes on is necessarily computable (assuming
> > the usual axioms).
>
> SH: So I am interested in how finiteness and axioms actually apply.
> I want to know what axioms you mean below.

I was thinking in the context of the Peano Axioms and specifically
about the Successor Axiom (there is always a next natural number),
which is what gives the natural numbers their infinite nature. If you
don't assume the Successor Axiom you have to be more careful - but I'd
rather not turn this thread towards this direction, if I may.

<and another big snip>

.

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