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Aatu Koskensilta wrote:
> R. Srinivasan wrote:
> > Note that in nonstandard analysis, Nelson's "idealization principle"
> > goes something like
> >
> > (For all standard x) (There exists y) y > x <=> (There exists y) (For
> > all standard x) y > x.
> >
> > Here the y on the right hand side is a non-standard object (integer,
> > real number, etc.). Of course even non-standard analysis rejects the
> > inference in the form with "standard" removed from the above formula.
> > But still, look at the intuition behind this assertion. The above
> > equivalence tells us that if the TM computes the digits of Pi
> > corresponding to every "standard" position for cells on the tape, there
> > could still be infinitely many cells at "nonstandard" positions that
> > will not be accessed by the TM.
>
> There are no non-standard positions on the tape of an ordinary TM.
Yes of course. And there are no non-standard integers at all as far as
NAFL is concerned.
>
> > Forget the "standard" and "nonstandard"
> > terminologies and ask yourself, what exactly is the logical intuition
> > that allows for such a result? This basically seems to be saying that
> > if every finite integer is exceeded by some integer, then all finite
> > integers are exceeded by some integer, where we use "finite" in the
> > intuitive ("external") sense.
>
> Well, I certainly lack any such intuition.
>
> > This is precisely why I believe that the "one-digit-at-a-time" computability
> > of Pi by a non-halting TM is at the very least paradoxical.
>
> If you say so. I'm afraid I can't really understand what you're after
> with all these non-standard integers and mappings. I wish you good luck
> in your endeavours nevertheless.
>
To put it in a nutshell:
(1) If you want to consider one-at-a-time counting process as
"completable", non-standard analysis tells us that you will necessarily
end up at non-standard (infinitley large) integers after exhausting the
standard integers. This is not acceptable in NAFL and there are no
non-standard integers in NAFL.
(2) The second example seems to illustrate that the one-at-a-time
counting process is not completable and is an ill-defined process even
within the naturals. This is what NAFL accepts. Therefore NAFL needs a
different model of computation from the classical one and I have made a
start in this direction.
Regards, RS
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