Infinity is Real!
Zvi Yehuda Saks, Pittsburgh
For many mathematicians including myself, modern mathematics
begins with the revolutionary work of Georg Cantor who formulated the
theory of mathematical infinity in the 1870s. Before Cantor, mathematical
infinity was "infinity as a potential." For example, the sequence of positive
integers 1,2,3,... is infinite because there is no last number. But it
was only potential infinity because there was no concept of reaching the
end of the sequence. Cantor's contribution was to show that mathematically
infinite objects can be considered to be actual objects which are well
defined and manipulable in many of the same ways as finite objects. For
example, the set of positive integers {1,2,3,...} is an actual object.
This can be formulated as an axiom.
AXIOM OF INFINITY: There exists an infinite set, or
more precisely, {1,2,3,...} is a set, that is, an actual object.
Joseph Dauben, in his book on Cantor, writes extensively
about the fundamental link between Cantor's deep religious convictions
and his perception of mathematics. He expresses surprise that this belief
and faith have received so little attention in discussions of Cantor's
development of set theory.
Several rabbis have asked me about a certain passage
in the works of the third Lubavitcher Rebbe, "the Tzemach Tzedek":
"It is impossible that many finite individual entities should join together
to form an actual infinity" [Derech Mitzvosecha p.113].
Not only does this seem to contradict the axiom of the existence of actual
mathematical infinity, in which the infinite set {1,2,3,...} is composed
of infinitely many individual numbers, there are also several references
in classic Chassidic literature that refer to Gd having created infinitely
many finite worlds. To resolve this seeming contradiction, the present
Lubavitcher Rebbe writes [Likutei Sichos vol X, 1789]
that there is no contradiction because Gd's power is above all limitations
and contradictions. The statement of the Tzemach Tzedek is true according
to logic, and will apply in any normal situation. But since Gd used His
unlimited suprarational power to create infinitely many worlds, logic
no longer applies for this particular case.
My claim is that since Gd has created infinitely
many worlds, then mathematics has the right and the ability to postulate
the existence of actual infinity, because it really does exist. The collection
of worlds as created by Gd is infinite and so the axiom of infinity is
true.
In a fascinating book entitled "The Mathematical Experience,"
[Davis and Hersh], I found:
Mathematical axioms have the reputation of being selfevident,
but it might seem that the axioms of infinity and that of Gd have the
same character as far as selfevidence is concerned. Thus, ...'Axiom of
Infinity: An infinite set exists.' Axiom of Gd: (Maimonides: Mishneh
Torah, beginning):
The basic principle of all basic principles and the
pillar of all the sciences is to realize that there is a First Being who
brought every existing thing into being.
Which is mathematics and which is theology? Does this
lead us to the idea that an axiom is merely a dialectical position on
which to base further argumentation, the opening move of a game without
which the game cannot get started?"
While I agree with this perspective, I take a positive
stance. I believe, or rather I know, that Gd exists and that actual infinity
exists because Gd created it. The main new idea here is that although we
finite humans could not construct actual infinity, Gd created an actual
infinity of worlds, and that therefore when mathematics asserts the existence
of actual infinity, this assertion is true.
(An excerpt from Dr. Saks' paper delivered at the
second Torah and Science conference in 1990. )
