Infinity in Torah and Mathematics

PART 1

The Interaction between Torah and Science

The Lubavitcher Rebbe, Rabbi Menachem M. Schneerson wrote1 that, starting in 1840, there will be great advances in the secular sciences and Chasidism (Jewish mystical knowledge)2 in order to ready the world for the Messianic Era. The Rebbe states that the deepest level of positive interaction between secular knowledge (science and mathematics) and Chasidism occurs when secular knowledge is used to explain and illuminate deep concepts in Chasidism. In this paper, we will demonstrate this type of deep interaction between the modern theory of mathematical infinity and the concept of Infinity as discussed in Torah and Chasidism.

The Zohar teaches that in the six-hundredth year of the sixth millennium after Creation [this corresponds to the year 1840] there will be great advances in Chasidism and in the secular sciences, in order to ready the world for the advent of the Messianic Era.

The Rebbe asked: why is scientific advance related to the coming of the Moshiach, and why does the Zohar link advances in Chasidism and science to the coming of Moshiach?

This can be understood from three perspectives:

With G‑d’s help, and the Rebbe’s inspiration, we will follow this program for the concept of infinity, that is, to use the theory of mathematical infinity to explain and illuminate the concept of infinity as discussed in Torah and Chasidism.

Different Levels of Infinity in Torah and Chasidism

The concept of Infinity is used in Torah and Chasidism in several fundamentally different ways. First, G‑d is referred to as Ein Sof,3 literally, Without End, or Infinite. On the other hand, the Torah has revealed the fact that the creation is quantitatively infinite. For example, G‑d created infinitely many troops of hosts that serve Him4 and infinitely many spiritual worlds.5 Now surely the concept of infinity as it measures and describes the creation is vastly inferior to the concept of infinity as it applies to G‑d Himself. In fact, all of G‑d’s infinitely many troops are considered as absolutely nothing before Him. These insights lead to the question of: How can infinity be limited, and how can there be different levels of infinity?

PART 2

Introduction to Mathematical Infinity

As a brief introduction to the subject of mathematical infinity, let’s pose a simple question. You are making a party and are serving sandwiches. You don't know how many people are coming to the party and you don't know how many sandwiches you are making. You request that everybody take one sandwich. That's the rule and everybody follows it. At the end of the party, there are no sandwiches left. Everybody took exactly one sandwich. Now what inference can we make? We do not know how many people came to the party, nor do we know how many sandwiches you made.

An inference can be made, however, that the number of people who came to the party and the number of sandwiches are the same. This is true because there is a one-to-one correspondence between the people and the sandwiches, defined by the relationship that each person took exactly one sandwich, and each sandwich was taken by exactly one person.

This notion of one-to-one correspondence was used by Georg Cantor in the 1870’s to formulate the modern theory of mathematical infinity. The basic definition is that two sets of entities have the same number of elements if there is a one-to-one correspondence between the entities of the two sets, like in the example of the people and the sandwiches. For finite sets, this definition conforms to what we would expect, but for infinite sets, there are many surprises, some of which I would like to discuss.

The most important and simplest infinite set is the set of positive integers 1, 2, 3, ... . The ‘...’ means ad infinitum. Consider also the set of squares of integers 1, 4, 9, ... . The set of squares is obviously smaller than the set of positive integers, because all squares are positive integers, but all of the non-squares like 2, 3, 5 etc. are left out, as in the following diagram:

Diagram 1

But look at what happens if we associate each positive integer with its square, so 1 is paired with 1, 2 is paired with 4, 3 is paired with 9, 4 with 16, n with n^2, and so on.

Diagram 2

Each positive integer has exactly one square, and each square has exactly one positive integer which is its square root, very similar to the people and the sandwiches in our earlier example. Thus the set of positive integers and the set of squares have the same number of elements, which should surprise you, since there seem to be far fewer squares. In fact, this is characteristic of infinite sets, namely, any infinite set has the same number of elements (via a one-to-one correspondence) as some of its subsets which are distinctly smaller than itself.

The one-to-one correspondence between the positive integers and the squares was first observed by Galileo in the early 1600’s. Galileo was quite confused by his discovery, and, therefore, he rejected the notion of infinite numbers. Cantor, on the other hand, explored the notion of one-to-one correspondence with fresh insight, and in the 1870’s, developed the theory of infinite numbers. Note that over 250 years passed from the initial discovery to development of a formal, coherent theory, which is an enormous length of time, especially considering the vast scientific progress during this period. It is noteworthy that the Theory of Infinite Numbers was developed after the date of 1840 which was specified by the Zohar.

Before Cantor, mathematical infinity was regarded as "infinity as a potential." For example, the set of positive integers 1, 2, 3 … is infinite because there is no last number. But it was infinity as a potential because there was no perspective or claim that the set of all the positive integers actually exists as a complete object. Cantor’s basic contribution was that mathematically infinite objects, such as the set of positive integers, can be considered to be well-defined objects that actually exist and can be manipulated in many of the same ways as finite objects.6

Cantor made a major breakthrough when he showed that not all infinite sets have the same number of elements, by proving that the set of positive integers has less elements than the set of all numbers between 0 and 1. Remember our example of the people and the sandwiches. If every person had taken exactly one sandwich, and there were sandwiches left over, then we would have concluded that there were more sandwiches than people. Similarly, Cantor explored all possible one-to-one correspondences between the set of positive integers and the set of numbers between 0 and 1. He proved that for any such correspondence, there is some number between 0 and 1 which is not paired with any positive integer. Therefore, there are more numbers between 0 and 1 than there are positive integers. In addition to these two different mathematical infinities, there are vastly more levels of mathematical infinity which are subject to the laws of mathematical infinite numbers. However, it can also be proven that there are levels of mathematical infinity which are too great to be subject to the same laws. We shall now sketch the proof that the Totality of everything that exists is above mathematical infinity, in the sense that it is not a set, and cannot be assigned a (mathematically infinite) number.

One of the basic axiomatic assumptions of mathematical infinity is that for any set X, there exists a set called the power set of X, denoted by P(X), which is the set of all subsets of the original set. It can be proven that the number of elements in the power set P(X) is distinctly greater than the number of elements in the original set X. More precisely, if X has N elements, then P(X) has 2^N elements, and 2^N is greater than N. Thus for any set, there is always a bigger set. However, since the Totality already includes everything that exists, it is impossible that there is a bigger set. Thus the Totality is not a set, and is not subject to the confines of mathematical infinity. Then of course neither can G‑d, Who is certainly much greater than the Totality of everything that exists.

Thus mathematical infinity gives us a framework to deal with limited infinities which model the quantitatively infinite aspects of the creation, and acknowledges that there are unlimited infinities which we call Absolute Infinity which are beyond its realm.

PART 3

Theological Discussion of Mathematical Infinity

Let's start with Cantor. Cantor was a deeply religious person who had an intimate relationship with G‑d. He insisted that actual mathematical infinity (called "transfinite" by Cantor) exists because G‑d exists and that mathematical infinity is a limited form of infinity which acts as a bridge between us finite humans and an absolutely infinite G‑d.7

Cantor writes:8

Cantor is saying that since G‑d can create infinity, then He did. Moreover9 Cantor believed in the absolute truth of his set theory because it had been revealed to him from G‑d directly, as he once told his colleague and supporter Mittag-Leffler. Dauben writes extensively about the fundamental link between Cantor's deep religious convictions and his perception of mathematics. He also expresses surprise that these religious convictions have received so little attention in other academic discussions of his development of set theory.

I would now like to discuss some concepts from Chasidic philosophy that have a direct bearing on the existence of actual mathematical infinity. It is a pleasure to express my thanks to Rabbi Simon Jacobson for helping me with this analysis and in showing me and studying with me some of the referenced works. Several Lubavitch rabbis, including Rabbi Jacobson, have asked me about a certain passage in the works of Rebbe Menachem Mendel, usually referred to as the Tsemach Tsedek, the third Lubavitcher Rebbe, which seems to contradict the existence of actual mathematical infinity. The Tsemach Tsedek wrote10 that it is impossible that many finite individuals should join together to form an actual infinity. This would seem to contradict the existence of actual infinity, in which the actual infinite set {1, 2, 3, … } is composed of (infinitely many) individual numbers. However, as discussed before, there are numerous references in the Chasidic literature which refer to G‑d having created infinitely many worlds. It is also stated in the Talmud (Chagiga 13b) that G‑d created infinitely many troops of hosts to serve Him, where each troop is limited. These sources would seem to contradict the statement of the Tsemach Tsedek, because each world or troop is limited, but there are infinitely many of them.

To resolve this seeming contradiction, the present Lubavitcher Rebbe, Rabbi Menachem Mendel Schneerson, writes11 that according to human logic, it is impossible that an actual infinity composed of finite individuals should exist. The Tsemach Tsedek’s statement is made with the implicit assumption that G‑d prefers to create the world so as to conform to human logic, and thus his statement that an actual infinity of limited entities is impossible is valid. However, G‑d is above all limitations and contradictions, and when He so chooses, He can and does create the world with qualities that contradict human logic. In the case of infinitely many worlds or troops, which our Sages have revealed to us, He used His unlimited supra-rational power to create infinitely many of them. Then for this particular case, the limitations of logic can no longer veto the existence of an actual infinity of limited entities.

I believe that this insight of the Rebbe can be used to explain how great geniuses could disagree about whether or not the world is infinite. Only a person such as Cantor, who had an intimate relationship with G‑d, could perceive that for this issue, G‑d overruled the normal principal that the world is created according to human logic, and created an infinite world. Other geniuses, such as Aristotle and Gauss, who did not have such an intimate relationship with G‑d, came to the conclusion that the world must be finite. Since they were limited to their human understanding, the conclusion that the world is finite was, according to the statement of the Tsemach Tsedek as explained by the Rebbe, the only conclusion that they could reach. The discovery and realization that the creation is infinite could only have been made by a person whose deep connection to G‑d would enable him to rise above the limitations of human intellect and connect with the G‑dly intellect.

Conclusion

In this paper, we have demonstrated the Rebbe’s vision of the deep interaction between secular knowledge and Chasidism, as prophesized in the Zohar, by using the modern theory of mathematical infinity to clarify and shed light on the concept of Infinity as discussed in Chasidism. May it be G‑d’s will, that now that prophecy of the Zohar has been fulfilled in its entirety, that the conclusion of the prophecy be fulfilled, namely that Moshiach should come and usher in the redemption, and "the earth will be filled with the knowledge of G‑d, as the waters cover the ocean bed."

Biographical note:
Tsvi-Yehuda ("Victor") Saks completed a PhD in mathematics in 1972 at Wesleyan University and an MS in computer science at the State University of New York at Buffalo. After teaching on a professorial level at various universities, Saks conducted research and development in applications of artificial intelligence to solve real world problems at Carnegie Group in Pittsburgh, Pennsylvania. An internationally reputed topologist, Saks's specialty in abstract infinite space prepared him well for learning and teaching Kabbala and Chasidic philosophy, which he did for adult groups until his untimely passing in 2005. The annual "Dr. Tzvi Yehuda Saks Memorial Lecture on Torah and Science" takes place at Dartmouth College.

[Reprinted with permission from B’Or HaTorah.]