Re: Poetential infinity


"Chris Menzel" <cmenzel@xxxxxxxxxxxxxxxxxxxx> wrote in message
news:slrndtct35.ls.cmenzel@xxxxxxxxxxxxxxxxxxxx
> On Tue, 24 Jan 2006 12:02:30 GMT, Stephen Harris
> <cyberguard1048-usenet@xxxxxxxxx> said:
>>
>> "Bill Taylor" <w.taylor@xxxxxxxxxxxxxxxxxxxxx> wrote in message
>> news:1138078424.348829.99120@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>>> There was once a lot of debate, and still occasionally is, about the
>>> difference between "actual infinity" and "potential infinity".
>>>
>>> It always struck me that, whatever you thought about them, at least
>>> the defintion of actual inf was reasonably clear, but that the same
>>> couldn't really be said of potential inf.
>>>
>>> So, it occurred to me that a better name for "potential infinity"
>>> would be "unboundedly finite".
>>>
>>> Would this cover the case, do y'all think?
>>>
>>
>> The common term in use is "finitely unbounded". This comes up in
>> describing the length of the TM Turing tape and is preferred to
>> describing the TM Turing tape as "infinite".
>
> On what basis do you say it is preferred? TM tapes are most commonly
> referred to as (one- or two-way) infinite, and they are commonly
> represented more formally as discrete infinite sequences. Granted, the
> little machine metaphors that are often used in describing TMs sometimes
> characterize the tape at any given moment as finite, and postulate some
> mechanism for extending the tape indefinitely to accommodate the machine
> as it does its work, and that might be thought of as a representation of
> the idea of "finitely unbounded". But that is only a heuristic device,
> introduced to give the appearance of a physically possible machine, and
> doesn't strike me as sufficient basis for saying that "finitely
> unbounded" is "preferred" in general for describing TM tapes (unless
> perhaps heuristic value is all you have in mind).
>

Hello Chris,

I learned the description of a TM tape using the term "potentially
infinite".
If you search Google with terms "potentially infinite" turing machine tap
548 English pages will be returned, mostly with academic credentials.

So I was reluctant to give up the term "potentially infinite" in favor of
the
term "finitely unbounded". However, I kept reading descriptions by
respectable theorists who thought "finitely unbounded" was a better term.

My investigation started with Pi, which is computable, but the computation
of Pi never ends. It is an algorithmic-like process except that the
definition
for an algorithm has a finiteness requirement. Then I came across Knuth's
term, "computational method" which abandons the finiteness requirement:
-------------------------------------------------------------------

Knuth, Vol. 1, Sec. 1.1:

"The modern meaning for algorithm is quite similar to that of
_recipe_,_process_, _method_, _technique_, _procedure_, _routine_,
_rigmarole_, except that the word "algorithm" connotes something
just a little different. Besides merely being a finite set of
rules that gives a sequence of operations for solving a specific
type of problems, an algorithm has five important features:


1) Finitness. An algorithm must always terminate after a finite
number of steps. [...] (A procedure that has all of the
characteristics of an algorithm except that it possibly lacks
_finitness_ may be called a _computational method_. [...])


(A procedure which has all the characteristics of an algorithm
except that it possibly lacks finitness may be called a
"computational method". Besides his algorithm for the greatest
common divisor of two integers, Euclid also gave a geometrical
construction that is essentially equivalent to algorithm E,
except that it is procedure for obtaining the ``greatest common
measure'' of the lengths of two line segments; this is a
computational method that does not terminate if the given
lengths are ``incommensurate'')."

--------------------------------------------------------------------

SH: I don't think the earlier description or requirement by Turing, that

1.. The method consists of a finite set of simple and precise instructions
that are described with a finite number of symbols.

is removed because the restriction to finitely terminate is removed. To me,
there is a finite process of computation that appends a finite result in an
unending sequence of outputs. As I understand it, each decimal numeral in
the neverending computing generation of the Pi expansion of digits. I don't
think that if an infinite number of steps were allowed to create an
infinitely long output that the method/process would still be computable.
The computational method works AFAIK, by appending a digit to a previously
generated finitely computed sequence of Pi, which is the basis of the next
finite step which builds sequentially towards a never reached last digit.

Using terms like endless or indefinite are not attempts for a precise
description. Those terms are not at the same categorical level as when
"potentially infinite" is compared to "finitely unbounded". I read a few
papers awhile ago that liked "finitely bounded" and I think this term has
correspondences to the build process which is finitely stuctured even though
the output never ends. Potentially infinite is not any longer in terms of
time or digit production of Pi, than finitely unbounded. Potentially
infinite is more in use now because it originated earlier maths
historically.

Regards, Stephen





.

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