## Why There Are Infinitely Many Different Levels of Infinity

Because a recent post about the German mathematician Georg Cantor (1845-1918) generated a great deal of interest, I decided to do two more posts about Cantor’s contributions to mathematics. Of course, I find Cantor’s mathematics fascinating, and apparently many readers found the content of that post intellectually stimulating as well. You might think that Cantor’s work amounts to a bunch of clever and interesting mathematical mind games, but this is not the case. His work helped lay the foundation upon which much of modern mathematical analysis rests. If you have not read my first Cantor post, Infinity Does Not Necessarily Equal Infinity, I strongly suggest that you read it before continuing. The concepts presented in this post heavily depend on an understanding of the concepts presented in that post.

The purpose of this post is to show how Cantor proved that there are infinitely many levels of infinity or there are infinitely many different infinite cardinal numbers! The purpose of the next Cantor post will be to give readers a general description of the Cantor ternary set which is counter-intuitive, preposterous, absurd, and astonishing. I will not delve deeply into formal abstract mathematics, because my understanding of Cantor’s work only scratches the surface of his deep mathematics.

To get started, I will do a quick review of some basic definitions and concepts in set theory.

• A set can be any collection of objects such as numbers, character symbols, cars, people, cats, etc.
• Set A is a subset of set B if and only if every element of set A is an element of set B.
• Let A equal any subset of B. A is a proper subset of B if and only if there is an element in B that is not in A.
• The null or empty set is a set that contains no elements. The symbols { } or Ø denote the empty set.
• Sets {0}, {Ø}, and {{ }} are not the empty set because each of the three sets contains an element.
• The null set is both a subset and proper subset of every set.
• Set A equals set B if and only if the sets are subsets of each other.
• In set theory and Boolean algebra, the word “or” means “one or the other and possibly both.” In contrast, when a parent uses the word “or” with a child, the parent means “one or the other, but not both.”
• In set theory and Boolean algebra, the word “and” means “both are in the set” or “both are true.”
• The union of sets A and B, denoted by AB, is the set of elements that are in A or B.
• The intersection of sets A and B, denoted by A ∩ B, is the set of elements that are in A and B.
The text box below uses sets of numbers to illustrate the set definitions above.

The power set of set A, denoted by P(A), equals the set of all possible distinct subsets of A. In other words, P(A) is just another set that contains all of the distinct subsets of set A. To get a better idea of what P(A) means, the text box below gives P(A) for different finite sets of counting numbers. Note that if set A has n elements, then P(A) has 2n elements.

To see why increasing the number of elements in a set by one causes the number of elements in the power set to double, consider how you could go about creating a list of all 32 subsets of the set {1, 2, 3, 4, 5}. The first 16 subsets of {1, 2, 3, 4, 5} are given by the power set of {1, 2, 3, 4}. The other 16 subsets of {1, 2, 3, 4, 5} can be obtained by forming the union of {5} with each of the 16 subsets of {1, 2, 3, 4}. You should later go ahead and list all 32 subsets of {1, 2, 3, 4, 5} and then all 64 subsets of {1, 2, 3, 4, 5, 6}. I’m very serious about this suggestion because it will help you learn to think in a different way and help you better understand the fundamental counting principle, permutations, and combinations. You may find this task tedious and boring, but you will be rewarded with a better understanding of fundamental counting concepts.

I can now explain the proof Cantor’s theorem which states that the cardinal number of P(A) is strictly greater than the cardinal number of A where A is any finite or infinite set. Cantor’s theorem can be used to show that there are infinitely many different infinite cardinal numbers. Recall from my first Cantor post that the cardinal number of a set equals the number of elements the set contains. Therefore the cardinal number of a google of water molecules equals 10100, and the cardinal number of a MLB active roster equals 25.  Also recall from my first Cantor post that the symbol for the cardinal number of the counting numbers is ℵ0, the symbol for the cardinal number of the real numbers is ℵ1, and ℵ0 < ℵ1.

For finite sets, Cantor’s theorem is obvious. If the cardinal number of finite set A equals n, then the cardinal number of P(A) equals 2n. For infinite sets Cantor’s theorem might seem obvious, but it’s much more difficult to prove. To make Cantor’s proof more comprehensible for infinite sets, I will first give a proof that shows that the cardinal number of P(C) is strictly greater than the cardinal number of C where C equals the set of counting numbers. Like many deep abstract mathematical proofs, Cantor’s proof uses the sophisticated technique of proof by contradiction. For set C, his proof goes something like this:

• Assume that there is a one-to-one function f with domain C and range P(C) that matches the counting numbers in C with all of the elements of P(C). The text box below shows one of the infinitely many possible ways that we could create a matching rule for f. The order in which domain and range elements are listed makes no difference. The important point is that there exists a one-to-one function f such that the domain of f equals set C and the range of the function, all function values, equals the set P(C).
• Construct a special subset M of the counting numbers as follows: (See the text box above.) Let set M equal all counting numbers n such that n is not contained in f(n). M is not the empty set because counting number j, such that f(j) = {   }, must be an element of M by definition.
• Set M raises a contradiction as follows: There must be a unique counting number k such that f(k) = M. Either k is contained in M or k is not contained in M. If k is contained in M, then by the definition of set M, k is not an element of M. If k is not contained in M, they by the definition of set M, k is an element of M. Therefore the function f can’t exist. Hence there is no counting number k that matches with M, and the cardinal number of set P(C) is greater than the cardinal number of set C. (The contradiction is somewhat like damned if you do and damned if you don’t.)

Now let’s see how we can prove that the cardinal number of the set of real numbers is strictly less than the cardinal number of the power set of real numbers. We need only to modify the last proof a little bit to give us our proof. The following algorithm describes how to create the modified proof:

• Let R equal the set of real numbers and the variable x equal any real number or element of R.
• Assume that there is a one-to-one function f with domain R and range P(R) that matches the real numbers in R with all of the elements of P(R).
• Construct a special subset M of the real numbers as follows:
1. Let set M equal all real numbers x such that x is not contained in f(x).
2. M is not the empty set because real number y, such that f(y) = {   }, must be an element of M by definition.
• Set M raises a contradiction as follows:
1. There must be a unique real number k such that f(k) = M. Either k is an element of M or k is not an element of M.
2. If k is an element of M, then by definition of set M, k is not an element of M.
3. If k is not an element of M, they by definition of set M, k is an element of M.
4. Therefore function f can’t exist. Hence there is no real number k that matches with M, and the cardinal number of set P(R) is greater than the cardinal number of set R.
Using the proofs described above as a model, it’s relatively easy to prove that the cardinal number of P(A) is strictly greater than the cardinal number of set A where A is any infinite set. By letting our imaginations run wild and considering set expressions such as P(P(A)) and P(P(P(P(A)))), we can create as many different infinite cardinal numbers we wish.

1. How will I ever apply set theory and cardinal numbers in my daily life? A: Probably never.
2. Do many engineers and scientists use set theory and cardinal numbers? A: Very few.
3. Name a math class that uses set theory? A: Venn diagrams to model probabilities of events in statistics.
4. Who uses set theory on a regular basis? A: People who design computer logic circuits use Boolean algebra.
5.  Is the study of set theory and cardinal numbers really just a mind game played by crackpot mathematicians? A: No – Cantor’s work helped create the foundation upon which much of modern mathematics rests.
6. Should very bright and curious high school math students be exposed to some of Cantor’s ideas? A: Yes.
I will close this post with a bit of personal information about Cantor. Cantor was a devout Lutheran who acknowledged the Absolute infinity of God. He believed that his theories about different levels of infinity were communicated to him by God. Some contemporary Christian theologians viewed Cantor’s work as a direct challenge to the idea that there is a unique infinity that only resides in God. For about the last 35 years of his life, Cantor suffered recurring bouts of depression. Most likely, the numerous vicious attacks on his work by many of his contemporaries contributed to his bouts of depression. Eventually Cantor’s work received praise and accolades from prominent contemporaries. In 1904, the Royal Academy awarded Cantor the Sylvester Medal which was the highest honor in mathematics. The brilliant mathematician David Hilbert said “No one can expel us from the Paradise that Cantor has created.” He spent the last year of his life in a sanatorium where he died on January 16, 1918.

## Infinity Does Not Necessarily Equal Infinity

A light year is about 6 trillion miles and the U.S. national debt reached 18 trillion dollars in 2015. Numbers of this magnitude are almost impossible to comprehend, but compared to infinity they are rather small. The German mathematician Georg Cantor (1845-1918) invented set theory and the mathematics of infinite numbers which in Cantor’s time was considered counter-intuitive, utter nonsense, and simply wrong. Many of Cantor’s contemporaries considered him to be nothing more than a charlatan. Set theory and the mathematics of infinite numbers are now part of mainstream mathematics.

To understand why Cantor upset so many mathematicians, I need to explain a basic concept of Cantor’s set theory. The cardinal number of a set or collection of objects equals the number of objects the set contains. Therefore the cardinal number of a gross of pencils equals 144 and the cardinal number of a mole of atoms is about 6.023 x 1023. For finite sets, the concept of cardinality is simple and straight forward, but for infinite sets the concept of cardinality can be counter-intuitive and utter nonsense. Cantor proved that the cardinal number of one infinite set can be greater than the cardinal number of another infinite set; infinity no longer necessarily equals infinity. Cantor also proved that there are infinitely many levels of infinity. In other words, there are infinitely many different infinite cardinal numbers!

What does Cantor mean when he says that two so seemly different infinite sets can have the same cardinality? Why are there as many real numbers between 0 and 1 as there are from –∞ to +∞? Why are there as many counting numbers as there are rational numbers? Why is the cardinality of the set of rational numbers less than the cardinality of the set of real numbers?  The purpose of this post is to provide answers to these questions without delving into formal abstract mathematics. If you want a mathematically rigorous discussion of cardinal numbers and set theory, take a graduate level course in set theory or point-set topology. My previous post Relations, Functions, and One-to-One Functions discussed concepts that will be used in this post and therefore readers may find it helpful.

To get started, I will do a quick review of the different types of real numbers. All real numbers are either rational or irrational. The set of rational numbers is composed of counting numbers, whole numbers, integers, and numbers that can be expressed as the ratio of two integers. The decimal expansion of all rational numbers starts to repeat in a pattern of fixed finite length at some point. The decimal expansion of an irrational number never starts to repeat in a pattern of fixed finite length. The text boxes below give examples of the different types of real numbers. Note that there is a pattern in the decimal expansion of n, but the length of the pattern increases. The term “real number” is unfortunate because it suggests that some numbers are valid and other numbers like the imaginary numbers are fake numbers.

Cantor uses the concept of cardinality to define when two sets have the same cardinality. Set A has the same cardinality as set B if and only if there is a one-to-one function that matches elements of A with elements of B such that the domain of the function is set A and the range of the function is set B. When both sets have a finite number of elements, this definition makes perfect intuitive sense. Example: When two bags of golf balls contain an equal number of golf balls, it’s easy to see how we can match the golf balls one-to-one, to show that the two bags of golf balls have the same cardinality. When both sets A and B have infinitely many elements, Cantor’s definition leads to a new and profound understanding of the nature of infinity. The matching rule for the one-to-one function in Cantor’s definition may be described by an equation or general algorithm that tells us how to match domain elements with range elements.

Using Cantor’s definition, let’s see why it makes sense to say that the set of real numbers between 0 and 1 has the same cardinality as the set of real numbers greater than 1. Initially, this seems preposterous. Two numbers are reciprocals if and only if the product of the two numbers equals 1. It’s a mathematical fact that every nonzero real number has a unique reciprocal and the reciprocals of two numbers are different if the numbers are different. If 0 < n < 1, then 1/n > 1. If n > 1, then 0 < 1/n < 1. Therefore the one-to-one function y = 1/x with domain equal to the open interval (0, 1) and range equal to the open interval (1, ∞) leads us to the conclusion that the open intervals (0, 1) and (1, ∞) have the same cardinality. Graph A below shows the graph of this function. If you think about it, the only practical way to show that two sets have the same cardinality is to show that there exists a one-to-one function that pair wise matches the elements of the two sets.

I will now demonstrate that the open interval (p, q) has the same cardinally as the open interval (-∞, +∞) for any pair of real numbers p and q such that q > p. Let the width of the interval w = q – p and midpoint of the interval m = (p + q)/2. The one-to-one function y = Tan(π/w(x – m) with domain (p, q) has a range equal to (-∞, +∞). From Cantor’s definition, it follows that the cardinality of the open interval (p, q) equals the cardinality of the open interval to (-∞, +∞). Graph B below illustrates that the open interval (0.75, 1.25) has the same cardinality as the open interval (-∞, +∞).

The next part of this post will demonstrate that the cardinality of the counting numbers equals the cardinality of the positive rational numbers. This will be accomplished by showing that there is a one-to-one function, f, that matches the counting numbers with the positive rational numbers. The matching rule of the one-to-one function is an algorithm that describes how we can systematically go about matching every counting number with a unique positive rational number in such a manner that every positive rational number gets matched with a counting number. Mathematicians say that the rational numbers are countable.

The algorithm for the matching rule of the one-to-one function is as follows:

1) Organize the positive rational numbers in a rectangular grid as shown below.

2) Start in the upper left corner of the grid. Set the counting number n = 1 and let f(n) = 1/1.

3) Continue moving from grid element to grid element forever as indicated in the diagram. If grid element p/q is not equivalent to a previous function value, then increase n by 1 and let f(n) = p/q. If grid element p/q is equivalent to a previous function value, then skip the grid element and go to the next grid element. (Note that skipped grid elements in the diagram are crossed out.)

Now for Cantor’s famous diagonal proof that the real numbers are not countable. His proof used the sophisticated technique of proof by contradiction which is commonly used by mathematicians to prove a theorem. The diagonal proof goes something like this.

• Assume that there is a one-to-one function f(n) that matches the counting numbers with all of the real numbers. The box below shows the start of one of the infinitely many possible matching rules for f(n) that matches the counting numbers with all of the real numbers. The real numbers in the range of the function are represented as strings of base 2 real number digits or binary digits (i.e. consisting only of zeros and ones).
• Now construct a real number p as follows: Let n equal any counting number and f(n) equal the corresponding function value.
• If the nth binary digit of f(n) = 0, then set the nth binary digit of p = 1.
• If the nth binary digit of f(n) = 1, then set the nth binary digit of p = 0. (See the text box below.) It’s clear that the real number p is not in the range of f(n) which in turn contradicts the original assumption about f(n). Therefore the cardinality of the real numbers is greater than and the cardinality of the counting numbers and the real numbers are not countable.

Some Comments Regarding Cardinal Numbers and Real Numbers:

• The symbol for the cardinal number of the counting numbers is ℵ0. (aleph naught)
• The symbol for the cardinal number of the real numbers is ℵ1.
• The continuum hypothesis states that there is no cardinal number between ℵ0 and ℵ1.
• If you can prove the continuum hypothesis, you will become world famous overnight.
• There are infinitely many rational numbers between any two real numbers.
• The rational numbers are an infinitely small fraction of the real numbers.
• To work with irrational numbers in practical applications, we use rational numbers to approximate irrational numbers. (3.1416 ≠ π)
• The points, lines and curves that we draw on a chalkboard or computer screen are just crude approximations of true mathematical points, lines, and curves.
• True mathematical points and curves are infinitely thin, and therefore they can’t reflect light which in turn tells us that we really can’t see true mathematical points and curves in the physical sense.
• Mathematical objects only exist in the mind of man and God.