[FOM] Sharp mathematical distinction between potential and actualinfinity?
Alexander Zenkin
alexzen at com2com.ru
Sun Sep 28 02:49:23 EDT 2003
Dear Tim,
A sharp definition of the potential infinity was given by Aristotle. Being
translated into the language of modern mathematics, the Aristotle definition
is a sharp definition of the infinity of the series 1,2,3, . . . by Peano's
axioms.
As regards the notion of actual infinity, a lot of philosophers (beginning
from Aristotle) and mathematicians (Gauss, Kronecker, Poincare, Brouwer,
etc.) expressed their negative (intuitive though) opinion about the notion:
"Infinitum Actu Non Datur" (Aristotle), "I must protest most vehemently
against <the> use of the infinite as something consummated, as this is never
permitted in mathematics" (Gauss), "There is no actual infinity; Cantorians
forgot that and fell into contradictions. Later generations will regard
Mengenlehre (set theory) as a disease from which one has recovered "
(Poincare), and so on.
Obviously, they, as outstanding logicians and mathematicians, had in mind
that the notion of actual infinity is a self-contradictory one.
Your desire to have a sharp mathematical distinction between potential and
actual infinity can display the "tragic" fact that the actual infinity is
self-contradictory not only at a philosophical and intuitive level, but at
the formal ZFC-level as well. It would be terrible.
I think that it is one of the main reasons why modern ZFC formalized and
axiomatized, by Bourbaki, "almost all mathematics", but hitherto has not a
sharp definitions of potential and actual infinity though, according to
S.Feferman, "some ZFC-axioms are justified by an assumption (!?) of actual
infinity" and, according to W.Hodges, "the ZFC-axioms include implicitly
(!?) actual infinity".
By the way, a version of "a sharp mathematical distinction between
potential and actual infinity" was presented to the FOM-community:
[FOM] As to strict definitions of potential and actual infinities.
Alexander Zenkin
http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html
and
FW: [FOM] As to strict definitions of potential and actual infinities.
Alexander Zenkin
at: http://www.cs.nyu.edu/pipermail/fom/2003-January/006137.html
Hope it will help to near to a solution of the main problem of Foundations
of Mathematics (not only of its part formalized in ZFC).
Alexander Zenkin
-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu]On Behalf Of
Timothy Y. Chow
Sent: Friday, September 26, 2003 6:50 PM
To: fom at cs.nyu.edu
Subject: [FOM] Sharp mathematical distinction between potential and
actualinfinity?
Years ago, Stephen Simpson proposed a list of "the most basic mathematical
concepts," which included number, set, function, etc. I wonder if
"infinity" belongs on this list?
It seems to me that at minimum, "potential infinity" of some sort is basic
to mathematics. Consider the following caricature of a debate that has
been rehearsed innumerable times on the FOM list and elsewhere:
P: The concept of N, the set of all natural numbers, is crystal-clear.
F: No it isn't. For example, first-order PA has nonstandard models.
P: Huh? By appealing to mathematical logic, you betray your belief
that the concept of a *rule* is clear, and N is just as clear as
the concept of a rule.
F: Nonsense. A rule is finite, or at most potentially infinite, but
N is a completed infinity.
I confess that my sympathies lie with P here; I tend to think that
if we take skepticism towards N seriously, then we also need to take
"Kripkensteinian" skepticism towards rules seriously, and this leads
quickly to a wildly, and in my opinion unacceptably, ultraskeptical
view of virtually all mathematics and logic.
However, I might change my mind if someone could demonstrate a sharp
distinction between potential and actual infinity. The distinction
seems to have pretty much evaporated in modern mathematics, and it
seems that only the philosophers still talk about it. Or am I just
underinformed? Is there, for example, a way of drawing a clear
mathematical distinction between potential and actual infinity that
blocks the move from skepticism-towards-N to skepticism-towards-rules?
Tim
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