[FOM] Sharp mathematical distinction between potential and actual infinity?

[Dean Buckner]

My  impression is that mathematicians dissolve into complete incoherence
whenever they stray into philosophy.  (Ditto for philosophers & mathematics,
btw).  Especially when on the subject of "potential" versus "actual".
Hilbert says that we have an actual infinity when "we consider the totality
of the numbers 1,2,3,4,... itself as a completed entity, or when we regard
the points of a line segment as a totality of objects that is actually given
and complete".  What does all that mean?  How do mathematical facts depend
on "considering" or "regarding" those facts in a certain way?

Stephen Simpson writes (on  his website) that " A potential infinity is more
like a finite but indefinitely long, unending series of events".  What's
that?  If the series of events is finite, how can it be unending?  What does
"indefinitely long" mean?  It would ordinarily mean, not that the length is
indefinite (surely all lengths as such are absolutely definite), but that
the information we have about the series does not include its length.

 Tim Chow writes

> ... A rule is finite, or at most potentially infinite, but
>    N is a completed infinity.

In what sense is a rule "potentially infinite"?  I understand that a rule
must be expressed in a finite number of words.  But then how can it be
potentially infinite?  (Chow may be offering a parody of some view here, but
I don't think so).  And Chow asks " if someone could demonstrate a sharp
distinction between potential and actual infinity."  The distinction comes
from Aristotle, in many places, one of which is :

". it is always possible to think of a larger number: for the number of
times a magnitude can be bisected is infinite.  Hence the infinite is
potential, never actual; the number of parts that can be taken always
surpasses any assigned number.  " [Aristotle Physics 207b8]

Notice two subtly different things going on here.  One is the idea that
whatever "assigned number" we alight upon, there is always a number that CAN
be taken, that surpasses it.  Whence the expression "potential", which is a
modal word meaning, well, "CAN".  The distinction between potential & actual
is traditional and distinguishes that which could or can exist, from that
which is, i.e. that which "actually" exists.  The other is the idea that
whatever number we quantify over, there IS (not just can be) a larger one.
This was an idea taken up by the medieval philosophers, such as Ockham, who
writes

"Every continuum is ACTUALLY existent. Therefore any of its parts is really
existent in nature. But the parts of the continuum are infinite because
there are not so many that there are not more, and therefore the infinite
parts are ACTUALLY existent."   (I'm grateful to Professor Richard Arthur
for this translation & other research which has illuminated my
undersdtanding of the medieval view).

This is the view that there is no largest number, whatever number there is,
there is one larger.  This is sometimes referred to as the idea of an
"incomplete" infinity, that there is no "totality" of all numbers, but the
word "totality" is obscure.  Ockham's formulation by contrast is perfectly
clear.

This is closer (in a way) to the standard set-theoretical view.  Zermelo's
axiom of infinity  states that

"There exists ... at least one set Z that contains the null set as an
element and is so constituted that that to each of its elements a there
corresponds a further element of the form {a}, in other words, that with
each of its elements a it also contains the corresponding set {a} as
element". *

Thus to every assigned member of the object Z, there corresponds a "larger"
one that contains it, thus there is no largest member of Z.  To parody
Zermelo "There is no member of Z that is so large, that there be not another
one that contains it".

The difference of course is in the idea of a set Z that contains all of
these sets.  but I don't see that as moving from a "incomplete" or potential
infinity, to a "completed" one.  The series of members of Z is still
incomplete, it can never end.  Obviously not.

Dean



* Zermelo, Ernst. "Untersuchungen über die Grundlagen der Mengenlehre I".
Mathematische Annalen, 65:261- 281, 1908. English translation,
"Investigations in the foundations of set theory'' in [Heijenoort 1967],
pages 199-215.