With the COVID-19 pandemic severely disrupting my typical research habits, I shall no longer tailor subsequent posts around the Smart Dust project and the entailed mathematics. While I still remain very much engaged with the project, it is only appropriate to revert to the original intention of this blog: the conveyance of my lifelong passion of teaching theoretical mathematics. Today, I shall write about infinity and its several variations.

Ever since the birth of numbers themselves, humans have struggled to explain the nature of infinity. What does it mean for something to “go on forever?” Does the universe extend infinitely in all directions? Can one break an object into infinitely small (infinitesimal) pieces? While cosmologists and chemists have universally agreed upon negative answers to each of the two respective questions, the theoretical nature of numbers legitimizes the concept in counting. Indeed, our decimal number system is constructed so that one is able to begin counting “1, 2, 3, 4, 5, 6,…” and continue indefinitely. Because each number can be assigned an ordinal position (1 is the “1st number,” 2 is the “2nd number,” 3 is the “3rd number,” etc.), we say that the natural numbers—or whole numbers if 0 is included—are countably infinite. That is, although it would take an eternity, one is hypothetically able to enumerate the entire set of natural (or whole) numbers in a list.

This raises the question, “Which is larger: the set of natural numbers or the set of even numbers?” One may be quick to assert that because the even numbers are simply alternate natural numbers (1, 2, 3, 4, 5, 6, 7, 8,…), there are half as many even numbers as natural numbers. However, notice how each even number can be assigned to an ordinal number just like the natural numbers:

• 2 is the “1st number”
• 4 is the “2nd number”
• 6 is the “3rd number”
• 8 is the “4th number,” etc.

Since the even numbers are also countably infinite, it is only reasonable to conclude that the even numbers and the natural numbers have the same amount of elements (technically, are equivalent in cardinality). Assigning ordinal positions in more complex ways, one can deduce that the following sets have the same cardinality—i.e., are equal in size—as the natural numbers, and thus are also countably infinite:

• Integers (natural numbers plus 0 and negatives)
• Rational numbers (positive and negative ratios of natural numbers)
• Algebraic numbers (numbers that are solutions to algebraic equations—e.g., the square root of 2 is the solution to x^2 – 2 = 0)

The above observations are attributed to the German mathematician Georg Cantor (1845-1918).

Image Courtesy of Wikipedia

In determining whether the real numbers (all rational and irrational numbers) are also countably infinite, Cantor established his diagonal method, which takes the form of a reductio ad absurdum proof (proof by contradiction) and is illustrated below:

Cantor’s Diagonal Method

Suppose that one has compiled a comprehensive enumeration of all real numbers and has listed some below (recall that irrational numbers are non-terminating, non-repeating decimals):

• 0.738972491301701703724… is the “1st number”
• 0.218390218432041809123… is the “2nd number”
• 0.182043928417132908121… is the “3rd number”
• 0.213897412912010901010… is the “4th number”
• 0.631913630329878119937… is the “5th number,” etc.

Now, we form a number by taking the nth decimal digit of the nth number. Thus, the first digit of the number is the first decimal digit of the first enumerated number; the second digit of the new number is the second decimal digit of the second enumerated number; and so on. There is a “diagonal” of sorts that is constituted:

• 0.738972491301701703724…
• 0.218390218432041809123…
• 0.182043928417132908121…
• 0.213897412912010901010…
• 0.631913630329878119937…

Using this process, we deduce the decimal number 0.71281…. Let us now form a new number by adding 1 to each decimal digit of the current number: we obtain 0.82392….

Can 0.82392… be in our list? It certainly cannot be the first number, for the 8 in the tenths place differs from the 7 in the first number in the list. It cannot be the second number; the 2 in the hundredths place differs from the 1 in the second number. In fact, the nth digit in the newly formed number differs from the nth digit in the nth enumerated number. Therefore, our seemingly comprehensive list of all real numbers is incomplete, for a new number can always be added by the method above. As a result, the real numbers exhibit an entirely different level of infinity—i.e., they are uncountably infinite.

Such “levels” of infinity are quantified in the transfinite numbers symbolized by the Hebrew letter aleph. Aleph-null signifies the countable infinity of the natural numbers (and additional sets like the rational and algebraic numbers). “Uncountable infinity” is expressed as aleph-1. In fact, there are an infinity of transfinite numbers (aleph-2, aleph-3, etc.); there exist infinities beyond that of the real numbers, but they encompass a higher mathematics that is much beyond the scope and demands of this blog.