Note: Dr. Saks, a mathematics professor, computer software engineer, topologist and popular teacher of Chassidic mysticism, died one month ago at his home in Pittsburgh, at the age of 60. The following paper he delivered at the Yisroel Aryeh Leib Conference for Science and Torah.

As a brief introduction to the subject of mathematical infinity, let me pose a very simple question. Suppose you are making a feast. Everybody has to wash for the feast so you make sandwiches. You really don't know how many people are coming and you don't really know how many sandwiches you are making. But you make a rule that everybody takes exactly one sandwich. At the end of the party there are no sandwiches left. Everybody took exactly one sandwich.

Now what inference can we make? We know that everyone took exactly one sandwich and every sandwich was taken by exactly one person. Do we know how many people came to the party? We certainly do not. There is something that we do know, however. That is, we know that the number of people that came to the party and the number of sandwiches are the same. Because we have what is called a one-to-one correspondence.

This notion of one-to-one correspondence was used by George 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 natural numbers, i.e. the positive integers, 1, 2, 3, ... . The ‘...’ means ad infinitum.

The way we generate the natural numbers is to start out with 1. Then there is a principle or a rule that allows us to generate new numbers from the old numbers. The rule is that you can always add 1 to a number.  So you start out with 1, then add 1 to it and you get 2.  Now you add one to the 2 you get 3, and so on.  For any number n, you add 1 to it and you get n + 1. What I want to do, though, is to utilize the concept of one to one correspondence to show you a very strange example involving mathematical infinity.

First, consider the set of squares 1, 4, 9,… It is obviously smaller than the set of positive integers, because the set of squares is part of (a subset of) the set of positive integers, but with many numbers - all of the non-squares like 2, 3, 5 etc. – left  out.  Now, let’s set up a correspondence so that each number is associated with its square. The number 1 will correspond to 1, 2 will correspond to 4, 3 to 9, … In general, n corresponds to n².  What we now have is a one to one correspondence between the sequence "1, 2, 3 ..." and the sequence "1, 4, 9, ..."

 It is clear from the diagram that the set of squares is the same size as the set of positive integers since they can be matched up in a one to one correspondence.

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. The Zohar states that in the six-hundredth year of the sixth millennium after Creation (corresponding to the year 1840), there will be great advances in the inner spiritual wisdom and in the secular sciences, in order to ready the world for the advent of the Messianic Era.

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. He used the symbol א0 to represent the positive integers, and the letter c (for continuum) to the set of numbers between 0 and 1, and he was able to prove that א0  is less than c.

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. Here is the proof:  

Suppose the following is an arbitrary one-to-one correspondence between the set of positive integers and the set of number between 0 and 1, expressed as decimal expansions. 

Cantor now defines a number between 0 and 1 which is not paired with any positive integer. Let’s call this number j = .j1j2 … jn … . If x11 is not 4, we will define j1 to be 4; if x11 is 4, then we will define j1 to be 2. Note that we already know that j is not equal to x1, since they differ in the first decimal place.

For every positive integer n, we similarly define jn. If x1n is not 4, we define jn to be 4; if x1n is 4, then we define jn to be 2. Clearly as before, j is not equal to xn, since they differ in the n-th decimal place. Inductively we have a defined a number j = .j1j2 … jn …which is between 0 and 1 and is unequal to xn for every n. We have thus proved that for any one-to-one correspondence between the set of positive integers and the set of numbers between 0 and 1, there is some number between 0 and 1 which is not paired with any positive integer. Therefore, the set of numbers between 0 and 1 is bigger than the set positive integers, or in the language of mathematical infinity, c > א0.

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. Thus mathematical infinity is limited

It is implicit in the fact that mathematical infinity is limited, that since G-d is Absolutely Infinite, He is outside of the realm of mathematical infinity.  This can in fact be proven.  To begin with, let's refer to everything that exists as the Totality.  We will prove that the Totality 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 2N elements, and 2N 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. But 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.

Chapter Two

A Model for Creation

(In ch. 1, I explained the basic mathematics of infinity, introducing א0, the mathematical symbol for an infinite quantity. Here, in Part 2, I will develops the mathematics of infinity further and shows how it can be used as a model for the creation of the universe.)

The Creation as discussed in Chassidic mysticism is not that G-d just said, "Poof" and the universe came into being.  Rather, He created a an infinite panorama of spiritual worlds.  There is a concept of an infinite descent of worlds. In each world there is a concealment of G-dliness giving rise to the next lowest world in which there is more concealment and in which the world and spiritual beings that inhabit that world have more sense of self, more self definition and more individuality.  The culmination of this process of infinite descent of spiritual worlds is the existence of the physical world.

This description of the creation of the world by G-d is very abstract and complex, with many references to infinity. I believe that the mathematics of infinity can help us to understand and visualize some of the processes through which G-d created the world.  This should not be surprising. One of the basic tenets of this conference is that not only can there be no conceivable contradiction between true science and Torah, but that science can be helpful in understanding Torah.  True science is an investigation into the world, which is the creation of G-d. Torah is the blueprint that G-d used for creation, meaning that He looked into the Torah (which preceded the world) and created the world according to the Torah.  With that view, we see that science can help us reveal the creation and thus help us understand Torah. 

Since G-d used infinite processes when He created the world, as will be discussed at length, it is logical that the science of infinite processes, namely the mathematics of infinity, would be useful to help us understand those portions of the Torah which deal with the creation.

A mathematician develops an intimate relationship with his mathematical work and with the mathematical objects with which he works. Thus, although the concepts discussed are quite abstract, to me they are very real. The personal context for this paper is that I am a ba'al teshuvah, a Jew who was not raised in a Torah-observant home but made a personal decision as an adult to observe Torah and mitzvot. At that time I was already a mature professional mathematician.

My field of expertise is topology, and the best way that I have of describing my work is to say that I did research into abstract infinite space.  In fact, I used to say that my doctoral thesis and the research that followed it had no known relationship with the physical world.  Many people used to ask me to speculate on what possible application there might be for this work, and I had no meaningful answer. Thus it came as a great surprise to me when I began to study Torah, and in particular Chassidic philosophy, and specifically the description of the creation of the world by G-d, that I felt that this was indeed the actual application of the mathematical work which I had done.

Mathematics is perhaps the purest and most rigorous science. The propositions that one is assuming must be clearly specified.  In addition, the acceptable methods for constructing new objects and proving new propositions are clearly defined.  Thus mathematics is as objective as any science can be.  Nonetheless, the truth of any mathematical system depends on the truth of its assumptions, and hence can be said to be true only relative to its assumptions.

When I began to study Torah, the great difference between the relative truth of mathematics and the absolute truth of Torah, which puts us in contact with G-d Who is the Ultimate Truth, became very clear to me.  My view of the relationship between mathematics and Torah is that mathematics allows us to construct abstract models which can be useful in helping us to understand and visualize some of the processes through which G-d Almighty created the world.

Where Do I Come From?

The term used to characterize creation in Jewish Philosophy is yesh me’ayin - creation of something from nothing (ex nihilo). What does something from nothing mean?  And how is this creative process described?  When G-d Al-mighty created the world he did it by putting a spiritual life force in every created being which is the source of the existence of that being and continues to keep it in existence after it was originally created. In other words, he continuously recreates it. Thus everything that exists as a creation has G- dliness in it which is the source of its existence. So the question is asked in Chassidism, why is it called "something from nothing?" The physical universe, which has only a dependent existence, should be called, "nothing."

G-dliness, which is the source, is a true existence and should be called “something.”  So creation should be called, "nothing from something." Why is it called "something from nothing?"

Chassidus explains that its called "something from nothing" because the created being thinks its a something because it is unaware of the fact that it has a source. It is unaware of the G-dliness within it which created it and keeps it in existence. So the concept of yesh me'ayin, of something from nothing, is that the something cannot trace its existence back to its source. 

This is the critical condition for creation yesh me’ayin that I am focusing on here: that the created being cannot trace itself back to its source, that it is unaware of its dependence on the source for its existence, and perhaps even unaware that it has a source which is outside of itself.  Note that this is strongly contrasted with the concept of creation of "something from something, " like a work of art or a building.  A building is made from an architect's plans with physical materials.  No matter how complex the building may be, we can trace the construction back to the raw materials that were used and the plans that were followed.

My claim here is that the mathematical model that we described in Part 1 is a model for this because the number א0 is similar to yesh me'ayin. It cannot trace itself back to its source.  Its source is the entire infinite process which is coming into it.  To explain this, let’s introduce the concepts of predecessor and successor. The sequence of positive integers 1,2,3,... is generated by starting with the number 1, and repeatedly adding 1 to get the succeeding numbers. Thus each number n has an immediate successor which is n + 1, the number following it. For example, the successor of 2 is 3; the successor of 3 is 4, etc. Also, each number n, except for 1, has an immediate predecessor which is n – 1, the number preceding it. In this process, each number comes from a definite source, its immediate predecessor: 3 comes from 2, 4 comes from 3 and n comes from n – 1. But א0 has no immediate predecessor. (Any number n, no matter how big, will be the predecessor of n + 1, not of א0.) It does not come from any specific source. It requires the entire infinite sequence to generate א0. Thus the whole infinite sequence is its source.       

From the perspective of א0, if it looks back and tries to see where it came from, it is going to have a very hard time realizing that there is an infinite sequence which comes into it and generates it. It is completely unaware of this infinite process and thus cannot trace itself back to its source. Since it doesn't have a source that it can observe, it says, "I am independent." 

Similarly, we see in the physical world that people who don't believe in G-d and are not aware of the concept of the creation see themselves as being independent because they don't have any way to relate to the infinite process which is coming into them at every moment and continuously recreating them.  Even though we are talking about conscious human life, where intellectual abstraction is most pronounced, and daily life revolves around abstract realities, a person may still not recognize the complex spiritual structure through which his life is created and guided.

So we have a basic model for creation which models the infinite descent of spiritual worlds culminating in a physical world, the yesh, which is not aware of its origin which is therefore called ayin – nothing. א0 represents this physical world.

Infinite Gaps

This is a very simple model for the creation and in fact its too simple for a number of reasons. For example, we know that there are four major spiritual worlds preceeding the physical world.  There's Atzilus, the world of emanation; B’riah, the world of creation; Yetzira, the world of formation and Asiyah, the world of action. There are infinite gaps between each of these worlds. Atzilus is infinitely higher then B’riah; B’riah is infinitely higher than Yetzira and Yetzira is infinitely higher than Asiya. But we don't have that here in our model because in the progression of the numbers of the sequence, each time we attain a new object – a successor – there is a uniform gap of 1. There are no infinite gaps along the way.

Thus we need to expand our model into a more complex one which will include in it the notion of an infinite gap. The simplest way to expand our model is to look at א0 not only as the limit point of the sequence of positive integers, but also as the beginning of a new sequence. In order to do that, I need to talk about the concept of ordinal number. 

Ordinal numbers are numbers which are based on an ordering as opposed to cardinal numbers which indicate only quantity. For finite numbers, they are basically the same. But for infinite numbers there is a remarkable distinction. The ordinal numbers are much more subtle then the cardinal numbers in the following sense:  If we add 1 quantitatively to א0, the first infinite number, then its still א0:א0  + 1 = א0.

In fact the Tzemach Tzedek makes exactly this point in Derech Mitzvosecha (Ha’Amonas Elokus, Ch. 11, based on Tanya Ch.48 – from the Chitas of the coming days). He says that an infinite quantity will not increase by adding a finite quantity to it. The above equation is a mathematical representation of that concept.

Ordinal numbers, however, are different. With ordinals we can add 1 to א0  and get something new because we are considering the way that the number in question relates to what came before. א0 is the end point of the whole infinite sequence of numbers 1, 2, 3,…and, as we explained above, it has no immediate predecessor. But א0  + 1 has an immediate predecessor, namely א0.  So from the point of view of ordinality these two are very different despite the fact that quantitatively they are identical. 

(To express this qualitative difference between א0 and  א0 + 1 in more technical terms:  א0  is identified with the sequence 1, 2, 3,…which leads up to it. This is a sequence without a limit point. But א0 + 1 is identified with the sequence 1, 2, 3, …, א0 which leads up to it, and this sequence has a limit point, namely א0 .)

Let’s continue this process and add 1 repeatedly to א0. We will get the following new sequence: א0, א0 + 1, א0 + 2,…, א0 +n, …, א0 + א0 = 2א0 

We can repeat this process and add 1 infinitely many times to 2א0 until we get 2א0 + א0 = 3א0

Similarly, we can get 4א0, 5א0, etc. by constructing a new sequence each time. This gives rise to a sequence of multiples of א0 whose endpoint is א0·א0: א0 , 2א0 , 3א0 ,…, א0·א0` 

While each of these numbers is different from the others as ordinal numbers (since each has a different predecessor), they all represent the same infinite quantity so they are equal as cardinal numbers. But eventually we will get to something which is quantitatively bigger then all of the others. This is called א1.

Now, let’s put all this together into one long sequence:

1, 2, 3,…,א0, א0 + 1, א0 + 2,…2א0, 2א0 + 1,… ,3א0, 3א0 + 1, …,א0· א0 ,…א1

This is an infinite sequence that also has infinite gaps in it before א0 (which has no predecessor), before 2א0 (which also has no predecessor) and similarly before all the multiples of א0 . Recall that this is precisely what was missing in our first model. We needed the infinite gaps to represent the infinite transition (contraction) between Atzilus and B’riah, between Briah and Yitzira and between Yitzira and Asiya. The final endpoint א1 represents the physical universe. So this is a much better model for the creation as it is explained in Chassidus.

A question of the Tzemach Tzedek

Until now we have been working from the bottom up i.e. starting with Mathematics, then using it to help us understand something about G-d. Now I want to go the other way, starting off with the Chassidism and using it to answer an important question in mathematics and philosophy.

Earlier in this paper, we quoted from the Tzemach Tzedek’s discussion of infinity in Chapter 11 of the section on Ha’amonas Elokus (Belief in G-d) in his book Derech Mitzvosecha. Over the years, I've been asked a question by many students based on a statement that the Tzemach Tzedek makes there. He says that it is impossible that many finite, limited individuals should join together to form an actual infinity. Infinity cannot be composed of finite individuals. So the question is, how can there be mathematical infinity? Is this not precisely what we have done to define mathematical infinity? We said that you can take these finite individuals - the positive integers - one at a time, and join them together to form the infinite set {1, 2, 3, ...}. But the Tzemach Tzedek says that this cannot be. This seems to say that mathematical infinity does not exist.

Actually, however, the nature of mathematical infinity has been a matter of debate among the mathematicians themselves with some, such as Gauss, one of the greatest mathematicians of all time, saying that there cannot be actual infinity, and that mathematical infinity is just a potential infinity i.e. the sequence 1, 2, 3,… continues indefinitely but is never actually infinite. Other mathematicians, including Cantor, on the other hand, held that the infinite sets were infinite in actuality. Does it follow from what the Tzemach Tzedek says that Cantor was wrong?

One of those who had asked me this question was Rabbi Simon Jacobson and he was the one who helped me work out the answer. He showed me certain letters of the Lubavitcher Rebbe and learned them with me and the answer is based on those letters.

It turns out that a similar question had been asked of the Rebbe based on sources in Torah. There are numerous references in Chassidism which refer to G-d having created infinitely many worlds. It is also stated in the Talmud (Hagiga 13b) that G-d created infinitely many “regiments” of “troops” (i.e. angels) to serve Him, where each regiment is finite. These sources would seem to contradict the statement of the Tzemach Tzedek, because each world or troop is finite, yet there are infinitely many of them. How could this infinity possibly be attained?

To resolve this seeming contradiction, the Rebbe writes that according to human logic, it is impossible that an actual infinity composed of finite individuals should exist. The Tzemach Tzedek’s statement is made based on the fundamental principle that G-d prefers to create the world so as to conform to human logic. (In fact, Chassidus explains that the reason that G-d created the universe through a succession of worlds, as we explained above, rather than by just saying, “Poof!” is so that the creation could be understood by the human mind.) The Tzemach Tzedek rejects the notion of an actual infinity composed of finite entities because it contradicts human logic. 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 the infinitely many worlds and regiments, which Chassidism and the Talmud have revealed to us, G-d used His unlimited supra-rational power to create infinitely many of them. So for these particular cases, the limitations of logic can no longer veto the existence of an actual infinity of limited entities. The point is that the Torah would have to tell us explicitly that G-d is creating in this exceptional manner. Otherwise we would assume that he is creating in the usual way i.e. in a manner understandable by human logic.

I believe that this explanation of the Rebbe can be used to explain how great geniuses could disagree about whether or not the world is infinite. Cantor, who had a strong belief in G-d, understood that G-d could create an infinite world. Other geniuses, such as Aristotle and Gauss, who did not have a strong belief in 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 the only conclusion that they could reach.

I think this is the best example that I have of Torah and illuminating mathematics and philosophy.

Chapter Three: An Infinite Experience  

(In the first two parts, I discussed the concept of mathematical infinity, which is represented by the symbol À0; the concept of infinity in Chassidism and the application of mathematical infinity to model creation.

In this part, I will give a personal account of the events leading up to his discovery of the applicability of mathematical infinity to creation, and some personal issues I dealt with as a Chassidic mathematician.)

I would now like to talk about the ideas presented earlier (see Parts 1 and 2) on a personal level and explain how I came to all this because it was a big deal in my life.  My Ph.D. thesis and the subsequent research that I did had literally no relationship to the physical world. No one knew of anything that the mathematics I did would model and I always wondered, like anyone who did such work would wonder, what application could there possibly be for this mathematics. 

To put this in context, let me explain how I came to be observant and a Lubavitcher Hassid.

Before I became observant, I was a product of the 60's -- a so called “spiritual seeker.” Several people I knew from the old days who had become observant before I did, told me that any kernel of truth that I may have thought I had found in any of the spiritual disciplines that I studied can  be found, in an even profounder way, in the Torah and the Kabbalah. I decided to go and study Judaism.

I left my position as a professor in a university and went to learn for three years, in Hadar HaTorah in Brooklyn, New York. While sitting in yeshivah, I started to learn in Tanya and Chassidism about the descent of the worlds from the spiritual to the physical and I was just overcome by the realization that this was the actual application of the mathematical work that I had worked on for so many years that had no other application.

To find that the actual application of the mathematics I did was in Torah - that was something that I was totally unprepared for. I could never have imagined that and it was a truly awesome and inspiring experience.

So I became involved in applying mathematics to Chassidism. I spoke at some B’Or HaTorah conferences in Miami which gave me a chance to develop these ideas that were germinating inside of me. It took a long time to get to the point where I could take the ideas that I had which were very, very abstract and very difficult and bring them down to a point where I could actually talk about them more or less in English to a non-mathematical audience. 

A Mathematical Ma’amar

Then something happened in 1989. The Lubavitcher Rebbe published a Chassidic discourse for 22 Shevat, the yartzeit of his wife. In that discourse he discusses the concept of “the superiority of light over darkness,” not just that light is better then darkness but that there is a certain quality of goodness that comes into being when the light is the result of a process of the transformation of darkness into light. The light that comes from the darkness is a superior light.  The Rebbe mentioned this in connection with Jethro, the father in law of Moses, who came to the awareness that the Jewish G-d is greater than all of the gods and powers. Jethro was in fact a great scholar in philosophy and science. In making his declaration that the Jewish G-d is greater then all the other gods, he brought about a refinement of the external wisdom that he had mastered and a transformation of darkness into light. This was the final critical thing that had to happen before G-d could give the Torah to the Jewish people. 

But the Rebbe goes even further, referring to King Solomon’s statement, “I saw that there is an advantage to wisdom over nonsense like the advantage of light over darkness.” The Zohar asks why we need King Solomon, the wisest of all men, to tell us this? Doesn’t everyone know that wisdom is better than nonsense. The Rebbe explains that King Solomon is referring to the superior light that is attained by transforming darkness, and saying that in the case of wisdom vs. nonsense something even greater happens. “Nonsense” in this context refers to the sciences which are nonsense relative to the deep wisdom of the Torah. But when this nonsense – the sciences – is transformed into light, not only does a superior light come into being but the sciences themselves can actually become part of Torah.

This can happen in two ways, or on two levels.  The lower level is where one uses the scientific knowledge which he has to increase his knowledge of Torah. In this case the scientific knowledge becomes an intermediary to aid in his understanding of the Torah. This extends the influence of Torah into the world and. Then the Rebbe ays that there is an even higher level where the science actually becomes part of Torah such as when Rambam uses Greek mathematics and astronomy to explain issues in the Laws of Kiddush HaChodesh (declaring the new months and years). Rambam uses the actual calculations of the Greek mathematicians to decide laws in Torah. He thereby refined these Greek sciences, elevated them and transformed them into Torah. The mathematics is no longer an intermediary through which our understanding of Torah is enhanced but actually becomes a part of Torah. They are now part of Rambam’s book on Jewish Law, Mishneh Torah.

The possibility to transform external wisdom into Torah was the result of Jethro’s own declaration, by which all of his knowledge of philosophy and science was transformed. In fact, his name “Yisro”, meaning “superior”, refers to the superior revelation in Torah brought about by this transformation.

When this discourse of the Rebbe was published I had already been thinking that doing this work of using mathematical models for Chassidism would bring about an elevation of the mathematics itself.  That was clear to me. But that it would bring out a new revelation in the Torah - that was something that I could never have conceived of. That had a tremendous effect on pushing me harder to work on these things and to think of it more and make time for it,

Personal Lessons

The spiritual implications of the mathematical infinity that I have been discussing here are many. The concept expressed by the verses in Tehillim "How numerous are your creations, Oh G-d" and "How great are your creations, Oh G-d" is a fundamental principle that we have to meditate on.

I want to comment specifically on the existence of actual infinity that we discussed earlier (see Part 1).  This fact and the fact that for any level of mathematical infinity there is always a greater one, makes more concrete the concept that however far we have progressed in our own personal service of G-d, we can always go farther.  We can always go infinitely farther.  Even in the basic sequence of infinite numbers that we presented (see Part 2) we can see that there are infinitely many different levels of how infinitely far you can go.  We can always reach a new level which is incomparably higher than where we were before.

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