Torah and Science

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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 G-d having created infinitely many finite worlds. To resolve this seeming contradiction, the present Lubavitcher Rebbe writes [Likutei Sichos vol X, 178-9] that there is no contradiction because G-d'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 G-d used His unlimited supra-rational power to create infinitely many worlds, logic no longer applies for this particular case.

My claim is that since G-d 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 G-d 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 self-evident, but it might seem that the axioms of infinity and that of G-d have the same character as far as self-evidence is concerned. Thus, ...'Axiom of Infinity: An infinite set exists.' Axiom of G-d: (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 G-d exists and that actual infinity exists because G-d created it. The main new idea here is that although we finite humans could not construct actual infinity, G-d 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. ) 



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