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R. Srinivasan wrote:
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
The first example is not considered "completeable" by other people
either. Apparently you think some people make this claim. Both
cases are considered equivalent to each other. When you posit many
infinite TMs, they take the same non-existent time to complete
as a UTM which simulates the results of the many infinite TMs.
They both use incremental steps which complete, but the entire
process of computing Pi is never completed in either case.
Your beef apparently is with the Successor axiom, and your
NAFL will have the equivalent because it is an attribute of the
human with time having a preferred direction in a causal chain
of events. UTMs can be assigned a natural number to "label"
each TM that corresponds to a particular digit value of Pi as
it is computed, and both are countably infinite.
You haven't presented any case why counting with natural numbers
is inadequate or that a UTM becomes choked as it searches for
new digits of the infinite expansion of Pi. I think you are
introducing a non-existent element of time into your conception
of completion which does not exist for a Turing machine. There
is *no* infinite task that can of course not be completed unless
the task is done for an infinite time. No such concept because
there is no completion of computing Pi. No idea that because
individual digits of Pi are incremented, that this process
reaches closer to the end of Pi approaching closer and closer
to a final end limit. You appear to think that NAFL corrects
some misunderstanding of computability. It is already the
case that the computation of an infinite computable number
does not terminate although finding one digit of Pi ends
before the algorithmic/computational proceeds to find the
next digit in an unending string of digits to be computed.
In neither case you provided is Pi considered to "have been
computed" in a sense of total completion, although individual
digits of Pi "have been computed". There is no logical
inference that one meaning implies the other.
RS: "Logically (I am talking about classical logic here) if
*any* digit of Pi has been computed, *all* digits of Pi
have been computed. ...Ask a logician."
I haven't seen any logician agree with this. Maybe because
what you meant has very little to do with the meaning of
what you wrote.
Regards,
Stephen
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