My Infinity Is Bigger Than Your Infinity

This article is based on an outreach talk that I have given for various events at Queen Mary University of London for secondary school students. It is a fairly common topic for outreach, but still one I often like to use in pre-university settings, as it allows the audience to delve fairly deeply into an "exotic" topic with very minimal mathematical background.

A "Real World" Scenario

Imagine for the moment that you are a young child again. Imagine also yourself, as a child, arguing with a sibling or a friend. Now, the start of your argument might hypothetically go like this:

Even as a child, you know that your argument would be more convincing if you were only to include a quantitative element:

Of course, no one wants to be one-upped, and even less so in a quantitative manner! Thus, you are now forced to respond to your rival's provocation with an even larger number:

This numerical arms race continues, until someone finally breaks the barrier and invokes the mysterious "infinity":

Still not wanting to be outdone, you keep reaching for larger and larger quantities—even though you now have no idea what you are talking about—until all hell breaks loose:

Since you are merely young kids, at this point you probably became bored and moved on to the next game to play...

The point here is that, even as children, and not long after being introduced to numbers, we were aware—at least to some extent—of this concept of "infinity", of something mysterious that is larger than every number. What we would like to do now is to take the above story a bit too seriously. We would like to indulge our curiosity a bit, take a deep dive, and see if we can develop some precise understanding of this so-called "infinity". In particular, we would like to see if we can dig up some answers to the following questions:

What is "infinity" anyway? How can we try to make any sense of this?
Does "infinity" come in different sizes, or is there just one "infinity"? More specifically, how can we compare various measures of infinity?
What about expressions such as \( \infty + 1 \), \( \infty \times 2 \), \( \infty + \infty \), \( \infty \times \infty \)? Can we give any real meaning to these quantities?

One important thing that should be mentioned is that there are many different notions of infinity throughout mathematics, and which one you ultimately want to work with depends on what you want to do with it. For this article, we will focus on one particular manifestation of infinity that requires very little background to get into—infinity with regards to sizes of sets.

A Bit About Sets

Now, in order to discuss "infinite set sizes", we will first need some background on what "sets" are. If you are not familiar with mathematical sets, then worry not! All we really need to know, for the purposes of this article, is that a set can roughly be thought of as a "bag" that contains things inside. (Of course, the precise description of sets is a bit more complicated than that, but this informal idea will suffice for the upcoming discussion.)

As a basic example, we can consider the set \[ \{ 1, 2, 3, 4, 5 \} \text{.} \] The above notation simply means that the set can be thought of as bag containing the five numbers \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 5 \)—and nothing else. Of course, we need not necessarily be so systematic; for example, we can also consider sets such as \[ \{ 7, 12.5, -72, \pi^2 \} \text{,} \] which is a "bag" containing the four listed numbers.

Since we are in abstract mathematics land, and no longer bound by real world rules, we can also consider sets containing infinitely many elements—"magic bags" for which we can always stuff more and more things inside. One simple example is the set of natural numbers, \[ \mathbb{N} = \{ 1, 2, 3, 4, \dots \} \text{,} \] which is like a bottomless bag in which we have stuffed all of the (infinitely many!) whole, or counting, numbers.

At this moment, it would be useful to mention a few more common infinite sets of numbers:

The set \( \mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \} \) of integers. These consist of the natural numbers, as well as zero and the negative whole numbers.
The set \( \mathbb{Q} \) of fractions, or rational numbers. These are numbers that can be expressed as ratios, such as \( \frac{1}{2} \), \( 3 = \frac{3}{1} \), or \( - \frac{5}{12} \).
The set \( \mathbb{R} \) of real numbers, consisting of all the numbers on the real line. These include not only the integers and fractions, but also all the weird numbers in between, such as \( \pi \) and \( e \).

Now, as we have run through these basic examples, we have been using the term "infinite" in an informal and imprecise way. While which sets should be "finite" or "infinite" should be intuitively very obvious to you, to prepare for later on, when things get weird, it will be useful to address now the following basic question: What precisely makes a set "infinite" rather than "finite"?

Finite and Infinite Sets

To explore this, let us consider two finite sets, \( A \) and \( B \). The key bit of intuition to be aware of here is the following idea:

Definition. \( A \) and \( B \) have the same size (i.e. the same number of elements) if and only if every element of \( A \) can be uniquely matched with every element of \( B \).

In other words, we can characterize \( A \) and \( B \) having the same size by constructing a one-to-one correspondence between the elements of \( A \) and the elements of \( B \).

To cut through the word salad and make things a lot more clear, we can consider some concrete objects, for instance, \[ A = \{ 1, 2, 3, 4, 5 \} \text{,} \qquad B = \{ 2, 4, 6, 8, 10 \} \text{.} \] Intuitively, it's clear that both sets have the same number of elements—namely, five. In terms of the above characterization, we can see this through the following matching of the elements of \( A \) and \( B \):

Now the above idea tells us when two sets have the same number of elements, but it does not yet directly say anything about how many elements a set has. To deal with this, we can tweak the above characterization by taking note of just one simple observation: that the set \( \{ 1, 2, 3, \dots, n \} \) has precisely \( n \) elements.

Definition. \( A \) has \( n \) elements if and only if \( A \) and the set \( \{ 1, 2, 3, \dots, n \} \) have the same size.

Note, in particular, that according to this definition, the preceding matching example tells us that \( \{ 2, 4, 6, 8, 10 \} \) has \( 5 \) elements, since it has the same size as \( \{ 1, 2, 3, 4, 5 \} \).

Now, all of this gives a complete characterization of the sizes of finite sets, thus it is time to start talking about infinite sets. First, though, what exactly is an infinite set? Since the word "infinite" is used to mean "not finite", we can use the our characterization of finite sets to give precise meaning to "infinite":

Definition. An infinite set is one that has size larger than any natural number \( n \).

To confirm that the set \( \mathbb{N} \) of natural numbers is indeed infinite, we can try to match its elements to any finite set \( \{ 1, 2, \dots, n \} \). Note that any attempt at this will inevitably fail, because you will eventually run out of numbers from the finite set \( \{ 1, 2, \dots, n \} \) to match with the unending list of natural numbers. The illustration below shows what happens when one tries to match the elements of \( \mathbb{N} \) and \( \{ 1, 2, 3, 4, 5 \} \). Thus, by the above definition, we conclude that \( \mathbb{N} \) is indeed infinite.

Comparing Infinite Sets

If you have been following along so far, then all the intuitions and definitions from the previous section seem a bit unnecessary. After all, we can all just directly count the number of elements in a finite set. What is the point of taking this extra detour into matching games?

Note that while one can measure the size of finite sets by simply counting their elements, this process entirely breaks down as soon as we consider an infinite set, such as the natural numbers or integers. At this point, our new description of set size comes to the rescue, because the same matching games can be played with either finite or infinite sets! In other words, the clever bit of insight here is that we can extend our previous definition to infinite sets essentially without any modifications:

Definition. Sets \( A \) and \( B \) (finite or infinite!) have the same size if and only if every element of \( A \) can be uniquely matched with every element of \( B \).

The short answer is that this definition gives us a very precise and reasonable way to talk more deeply about sizes of infinite sets. (The definition is precise because it characterizes sizes in an unambiguous way, and it is reasonable since we saw it worked well for finite sets.) With this clever definition in hand, let us now play around a bit and see what it says about some of the most common infinite sets of numbers!

Natural Numbers vs Integers

Let us begin with the two simplest sets of numbers, the natural numbers and integers: \[ \mathbb{N} = \{ 1, 2, 3, \dots \} \text{,} \qquad \mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \} \text{.} \] Is one of these sets larger in size than the other, or are they the same size?

Now, at first glance, you might think that the integers should be larger than the natural numbers. After all, \( \mathbb{Z} \) contains already all the natural numbers, as well as zero and all the negative numbers on top of that. However, is this intuition accurate or misleading? To really find out, we have to consult our precise definition—that is, we have to examine this more closely by playing the same matching game as before.

There are many ways that one can match the natural numbers to the integers. One especially systematic and simple way to do this is given in the illustration below:

Here, the natural numbers are drawn in green, while the integers are drawn in orange. Beyond the right edge of the illustration, the same pattern continues—\( 8 \) is matched to \( 4 \), and \( 9 \) to \( -4 \), and so on. Observe in particular that with the above pattern, each natural number is paired with exactly one integer, and each integer is paired with exactly one natural number.

In other words, the illustration demonstrates a one-to-one correspondence between the natural numbers and the integers! Thus, by our definition of set sizes, we conclude that:

The sets \( \mathbb{N} \) and \( \mathbb{Z} \) have the same size!

Again, this may seem quite counterintuitive at first, and you may need a bit of time to wrap your mind around this. However, our conclusion comes directly from applying our precise definition. Therefore, by devising our clever definition of size comparison, and by applying it to the natural numbers and integers, we obtain a rather interesting, and surprising, conclusion!

Natural Numbers vs Rational Numbers

Next, let us play the same game, but now with the natural numbers and the rational numbers: \[ \mathbb{N} = \{ 1, 2, 3, \dots \} \text{,} \qquad \mathbb{Q} = \{ \text{fractions } \tfrac{m}{n} \} \text{.} \] Again, we ask whether one of these sets is larger in size than the other, or whether they are the same size.

Similar to the previous example, one might think, at first glance, that the rational numbers must be much larger in size, since every natural number \( n \) is also a fraction (namely, \( \frac{n}{1} \)), but what does the actual definition tell us? Here, we can once again play the matching game, but the matching that we construct is more complicated than before.

First, to simplify the exposition a bit, we compare the natural numbers to all the positive fractions instead. (In fact, using a similar argument as in the previous part, one can show that the set \( \mathbb{Q} \) of all fractions and the set of all positive fractions have the same size. We leave this as a challenge for the reader—try to derive this yourself!) One possible matching between the natural numbers and positive fractions is summarized in the illustration below:

Here, we arranged all the positive fractions (drawn in purple) in a table, with the rows representing the numerator values and the columns representing the denominator values. Notice that the fractions in purple are not uniquely represented; for example, \( \frac{1}{1} \), \( \frac{2}{2} \), and \( \frac{3}{3} \) all represent the same number. To get around this difficulty, we crossed out in red all the fractions in the table that are not in simplest form (since any such number is equal to its representation in simplest form, which is elsewhere in the table).

Now, the trick is to list all the positive fractions in the table in a specific order—starting from the bottom left corner, we traverse along each downward-and-rightward diagonal of the table, working our way up the table. These diagonals are drawn in orange on the table. If we list these positive fractions (but skipping those that are crossed out in red) in this order while traversing the orange diagonals, then we will hit every positive fraction exactly one time.

Finally, all we have to do is to match all the natural numbers to these positive fractions in the order indicated above. In particular:

The first several values of this matching is given at the bottom of the illustration. Observe, most importantly, that by continuing this pattern indefinitely, we have produced a one-to-one correspondence between the natural numbers and the positive fractions. Since the rational numbers have the same size as the set of positive fractions, we hence conclude:

The sets \( \mathbb{N} \) and \( \mathbb{Q} \) have the same size!

Once again, while the rational numbers may seem like a much larger collection than the natural numbers, this is in fact not the case, as both sets have the same size!

(Note: You may be wondering why we proceeded along diagonals in the above table, rather than along horizonal or vertical directions. The answer is that each row or column of the table already contains infinitely many numbers, so if you were to list all the positive fractions in this order, then you would never get past the first row or column of the table! Thus, the clever insight here is that we can enumerate all the fractions if we proceeded diagonally, rather than horizontally or vertically.)

Natural Numbers vs Real Numbers

So far, all the infinite sets we have considered have the same size. Though we have not considered very many sets yet, we can prematurely ask: do all infinite sets have the same size, or are there some infinite sets that are strictly larger than others?

To spoil the answer to this question, let us consider now the natural and the real numbers: \[ \mathbb{N} = \{ 1, 2, 3, \dots \} \text{,} \qquad \mathbb{R} = \{ \unicode{x201C}\text{numbers on the real number line"} \} \text{.} \] Once again, we ask whether these two sets have the same size, or one set contains more elements than the other?

(Note: There is the foundational question of what exactly is a "real number line", and by extension, a "real number". The full answer to this question is, unfortunately, beyond the scope of this article. However, for those who want an informal but usable description, you can think of real numbers as those that can be given by an infinite decimal expansion, e.g. \( 0.33333\dots \) or \( 3.14159\dots \). Informally, these include all the fractions, as well as many more numbers that lie between the fractions.)

The above question was first answered by Georg Cantor in 1894. Using a brilliant argument (which we will demonstrate below), Cantor showed the following:

The set \( \mathbb{R} \) has strictly larger size than \( \mathbb{N} \)!

In other words, there are different level of infinite sizes! First, there is the size of \( \mathbb{N} \), \( \mathbb{Z} \), and \( \mathbb{Q} \), which we call countable infinity. (The name comes from the fact that the elements of \( \mathbb{N} \)—as well as any other set that can be matched with \( \mathbb{N} \)—can be systematically listed, or "counted", one by one.) Then, there are uncountably infinite sizes, such as the size of \( \mathbb{R} \), which is strictly larger than that of \( \mathbb{N} \).

Cantor Diagonalization

As a bit of bonus content for those with a bit of extra mathematical background, let us now present the argument for why \( \mathbb{R} \) has larger size than \( \mathbb{N} \). While this material is a bit more challenging than other parts of this article, it is worth showing, because it is a short argument that requires relatively little background to understand, yet it highlights a special level of mathematical creativity.

To simplify things a bit, let us just consider the set \( I \) of real numbers between \( 0 \) and \( 1 \). In the following, we will prove that any matching from the natural numbers to the set \( I \) has to miss some number in \( I \). This would in particular imply that there is no way to have a one-to-one matching between the natural numbers and the elements of \( I \), so that by our definition of set sizes, \( \mathbb{N} \) and \( I \) cannot have the same size. Moreover, since the natural numbers can only match into some, but not all, of \( I \), the above would further imply that the size of \( I \) must in fact be greater than the size of \( \mathbb{N} \)! Since \( I \) is contained within the set \( \mathbb{R} \) of all real numbers, we hence conclude that the size of \( \mathbb{R} \) is strictly greater than the size of \( \mathbb{N} \), as desired.

Thus, it suffices to establish the bolded statement above, and we now focus exclusively on this. For this, let us consider any arbitrary matching of natural numbers to real numbers between \( 0 \) and \( 1 \). One concrete example of this is given by the following illustration:

In the illustration, each natural number (in orange) is matched to a real number in \( I \) (written as a decimal expansion in green). To show that there is some number in \( I \) that is not matched to, we must produce a real number in \( I \) that is not one of the green numbers that has been matched to.

To do this, let us look at the "diagonal digits"—that is, the first digit of the (green) number matched from \( 1 \), the second digit of the number matched from \( 2 \), the third digit of the number matched from \( 3 \), and so on. These digits are highlighted in pink. We can then craft a new number \( x \) from these pink digits; in the illustration, \( x = 0.379184\dots \), which is drawn in pink near the bottom.

Now, the number \( y \) we want is obtained by altering every digit of \( x \); in the illustrated example, one possible \( y = 0.480295\dots \) is indicated in grey at the bottom. We claim that \( y \) is different from every element of \( I \) that was matched to a natural number. To see why this is true, we simply observe that:

Hopefully, the pattern is become clear at this point. Continuing down the line, we observe that \( y \) must be different from the number matched to any natural number \( n \), since they differ by at least one digit (in particular, the \(n\)-th digit). As a result, we conclude that this arbitrary matching of natural numbers to numbers in \( I \) must have missed \( y \), and hence did not hit every number in \( I \). This proves the above bolded statement, and we have hence completed the proof that \( \mathbb{R} \) has strictly larger size than \( \mathbb{N} \).

Finally, observe that the crucial step in this argument was to construct this missing number \( y \) using the diagonal digits of the numbers that were mapped from \( \mathbb{N} \). Consequently, this argument is often nicknamed Cantor diagonalization in his honor.

(One extra pitfall here is that some numbers have more than one decimal expansion; for example, \( 0.9999\dots \) and \( 1.000\dots \) represent the same number. Thus, when we alter digits in the above process, we must take a bit of care to ensure that we produce a genuinely different number.)

Arithmetic of Infinite Sizes

Now that we have a basic conception of infinite sizes, as well as the existence of different infinite sizes, we turn to the last point in our agenda. Let us see if we can make sense of arithmetic (additional and multiplication) involving infinite sizes, e.g. expressions of the form "\( \infty + \infty \)" or "\( \infty \times \infty \)".

Similar to our previous discussions, the first step is to devise a precise and reasonable way to describe how to add and multiply infinite sizes. For this task, we once again turn first to finite sets, where all our intutions are very clear.

Arithmetic of Finite Sizes

Let us begin with addition. Here, the main idea is the following rather obvious observation for how we can capture addition through sets and set sizes:

Let \( A \) be a set containing \( m \) elements, and let \( B \) be a set containing \( n \) elements.

We let \( A \sqcup B \) be the set that combines all the elements of \( A \) and \( B \). Notice that \( A \sqcup B \) has precisely \( m + n \) elements.

To be a bit less abstract, consider the example in the illustration below, where \( A \) is the set, or bag, containing \( m = 3 \) green shapes, while \( B \) is the set containing \( n = 2 \) orange shapes. In this case, \( A \sqcup B \) is the bag containing both the green and orange shapes, and a quick bit of counting will convince you that \( A \sqcup B \) has \( 3 + 2 = 5 \) items inside.

Notice that the intuition from the above illustration continues to hold for abstract sets, even when \( A \) and \( B \) contain \( m \) and \( n \) elements for any arbitrary \( m, n \). In short, we have simply highlighted the simple observation that addition of set sizes can be captured by pooling together the elements of two sets. Not so impressive at the moment, but it will pay off shortly!

(Note: For those who have some familiarity with set theory, \( A \sqcup B \) is the "disjoint union". This differs from the usual union \( A \cup B \), since we want all the elements of \( A \) and \( B \) to be viewed as distinct from each other, even when \( A \) and \( B \) contain common elements.)

We can play a similarly simple game with multiplication. Here, we observe the following:

Let \( A \) be a set containing \( m \) elements, and let \( B \) be a set containing \( n \) elements.

We let \( A \times B \) be the set of all pairs of objects, with the first object in the pair being in \( A \), and the second being in \( B \). Notice that \( A \times B \) has precisely \( m \times n \) elements.

The idea is a lot simpler than the wordiness of the above definition may suggest. Consider the illustration below, where we have the same bags \( A \) and \( B \) of shapes as before. Then, \( A \times B \) is a bag containing pairs of shapes, with the first being a green shape and the second being an orange shape. Since there are \( m = 3 \) green shapes in \( A \) and \( n = 2 \) orange shapes in \( B \), then a quick bit of counting tells us that \( A \times B \) contains \( 3 \times 2 = 6 \) pairs of shapes.

Again, this intuition extends to abstract shapes. If \( A \) and \( B \) now contain \( m \) and \( n \) elements, respectively, then each of the \( m \) elements of \( A \) can be paired with exactly \( n \) elements of \( B \), resulting in \( m \times n \) total possible pairings, so that \( A \times B \) indeed contains \( m \times n \) elements.

(Note: For those having some background in set theory, \( A \times B \) is known as the "Cartesian product", which is a very commonly used construction in mathematics.)

Extension to Infinite Sizes

Now, you may again be wondering why we have taken such an extended detour just to add and multiply numbers. Once again, the point here is that these intuitions, though obvious for finite sets, carry over to directly infinite sets!

In particular, even when \( A \) and \( B \) are infinite, the constructions \( A \sqcup B \) and \( A \times B \) still makes sense. As a result, with this clever observation in hand, we can make sense of the sum and product of the sizes of \( A \) and \( B \), even when infinite, by measuring the sizes of \( A \sqcup B \) and \( A \times B \), respectively. This is precisely the content of our subsequent definition:

Let \( A \) and \( B \) be (finite or infinite) sets, and let \( m \) and \( n \) denote the (finite or infinite) sizes of \( A \) and \( B \), respectively.

We define the sum \( m + n \) to be the size of \( A \sqcup B \).
We define the product \( m \times n \) to be the size of \( A \times B \).

Note when \( A \) and \( B \) (and hence \( m \) and \( n \)) are finite, the above definitions give us the usual addition and multiplication of numbers you have seen since primary school. However, when \( A \) or \( B \) is infinite, then we have defined genuinely new kinds of "infinite addition" and "infinite multiplication", which we can now explore to our heart's content!

Some Common Examples

To make the writing a bit easier, let us give names for some common infinite sizes. In particular, we let \( \infty_N \) be the size of the natural numbers \( \mathbb{N} \), and we let \( \infty_R \) be the (strictly larger) size of the real numbers \( \mathbb{R} \).

Let us first see what \( \infty_N + \infty_N \) should be. By our definition, this is defined to be size of \( \mathbb{N} \sqcup \mathbb{N} \), which is essentially the size of "two copies of \( \mathbb{N} \)". Now, if we view these two copies of \( \mathbb{N} \) as the "positive whole numbers" and the "negative whole numbers", then we see that \( \mathbb{N} \sqcup \mathbb{N} \) is roughly like the integers (but not including \( 0 \), which makes no real difference here). Now, we know that the integers \( \mathbb{Z} \) has the same size as \( \mathbb{N} \), namely \( \infty_N \), so the above argument tells us that \[ \infty_N + \infty_N = \infty_N \text{.} \]

Next, let us consider \( \infty_N \times \infty_N \), which now amounts to measuring the size of \( \mathbb{N} \times \mathbb{N} \). In fact, the trick here is just an easier version of when we studied the size of the rationals \( \mathbb{Q} \)! The idea is that we can arrange the elements of \( \mathbb{N} \times \mathbb{N} \), i.e. the pairs of natural numbers, in a table, with rows corresponding to the first part of the pair, and the columns corresponding to the second part; see the illustration below. We can then enumerate all these pairs by traversing along the down-right diagonals of the table (in orange in the illustration). This gives us a one-to-one matching between the elements of \( \mathbb{N} \) and the elements of \( \mathbb{N} \times \mathbb{N} \). As a result, we conclude that \[ \infty_N \times \infty_N = \infty_N \text{.} \]

In short, when we add or multiply \( \infty_N \) to itself, we simply obtain the same infinite size \( \infty_N \), rather than a larger infinity.

One can, of course, consider sums and products of other infinite values, or some mix of finite and infinite values. Some of these—especially those involving \( \infty_R \)—can be a bit tricky to prove without developing further background. To keep this article from becoming much longer than it already is, here we just list the values of some other sums and products involving \( \infty_N \) and \( \infty_R \):

One way to summarize the above table—and many similar relations that we did not list—is that when we add or multiply infinite values, then what we get in the end is the larger of the infinite values.

As an "application", suppose you were to go back in time to your childhood and relive your quantitative arguments. (Or, you can just refer back to the drawings at the beginning of the article.) If you said "I'm better \( \mathop{\times} \infty \)", and your rival countered with "I'm better \( \mathop{\times} \infty \times \infty \)", then you can use your current knowledge and explain: "Well, \( \infty \times \infty \) is the same as \( \infty \), thus you have not one-upped me at all." Thus, you have notched a win! (The purpose for obtaining such a win is a deeper question that we will not explore here.)

A Largest Infinity?

At this point, we have established that there are different levels of infinite sizes, some bigger than others. As a final topic of discussion, we consider the following question: Is there a largest infinite size?

Given a set \( X \), we let \( \mathcal{P} ( X ) \) denote the set of all subsets of \( X \)—that is, the set of all sets that are contained in \( X \).

(In set theory, \( \mathcal{P} (X) \) is commonly called the "power set" of \( X \).)

Whew! If you have not seen this definition before, then it may look quite confusing. To make this a bit concrete, we can consider the simple finite set \[ X = \{ 1, 2, 3 \} \text{.} \] Then, the elements of \( \mathcal{P} (X) \) are all the sets that are contained within \( X \), that is, \[ \mathcal{P} ( X ) = \{ \{ \}, \{ 1 \}, \{ 2 \}, \{ 3 \}, \{ 1, 2 \}, \{ 1, 3 \}, \{ 2, 3 \}, \{ 1, 2, 3 \} \} \text{.} \] (In the above, "\( \{ \} \)" is the empty set containing no elements.)

One basic observation is that \( X \) in the above example has \( 3 \) elements, while \( \mathcal{P} (X) \) has \( 8 \) elements. The significance of \( 8 \)—and the connection between the sizes of \( X \) and \( \mathcal{P} (X) \)—is that \( 8 = 2^3 \). In fact, if \( X \) is a finite set containing \( n \) elements, then you can see without too much difficulty that \( \mathcal{P} (X) \) contains exactly \( 2^n \) elements.

You may be wondering why we just embarked on a rather confusing digression to these sets of subsets. Well, once again, one can make sense of \( \mathcal{P} (X) \) even when \( X \) is infinite. Moreover, Georg Cantor (the same person from before!) discovered, in 1891, discovered the following rather surprising fact:

For any set \( X \), finite or infinite, \( \mathcal{P} (X) \) always has strictly larger size than \( X \)!

The conclusion from Cantor's result is that there is no largest infinite size! In other words, I can give you a set \( X \) that has some really large infinite size, and you can always find a set that has an even larger infinite size. In fact, all you have to do is just take \( \mathcal{P} (X) \).

(Given what we know about finite sets, we can reasonably define the following—given a set \( X \) with size \( n \), finite or infinite, we define \( 2^n \) to be the size of \( \mathcal{P} (X) \). Then, Cantor's result above can be concisely summarized as \( 2^\infty > \infty \)!)

In terms of your childhood arguments, this means that these rhetorical battles could in theory never end! Indeed, if you claim you are "better \( \mathop{\times} \infty \)", then your mortal enemy can always claim to be "better \( \times \) a larger \( \infty \)". Thus, the argument can go on indefinitely, at least if both competitors have infinite patience!

A Proof of Cantor's Result

As additional "bonus, bonus content" for those with some experience in set theory, here we show why \( \mathcal{P} (X) \) has larger size than \( X \). In fact, while it may not seem so at first glance, the main idea here is the same as the proof that \( \mathbb{R} \) has larger size than \( \mathbb{N} \)—this proof is essentially a "Cantor diagonalization" in disguise!

To make the proof a bit easier to digest, one can consider the illustration above. Again, the idea is to consider any matching of every element \( x \) of \( X \) to an element \( S_x \) of \( \mathcal{P} (X) \) (that is, a subset \( S_x \) of \( X \)), as shown in the illustration. The objective is then construct a new subset \( \tilde{S} \) of \( X \) that has not been matched to, i.e. that is different from every \( S_x \). This would then imply that no one-to-one matching of \( X \) and \( \mathcal{P} (X) \) exists, and hence \( \mathcal{P} (X) \) has strictly larger size than \( X \).

We can generate our desired subset \( \tilde{S} \) as follows, \[ \tilde{S} = \{ x \in X \mid x \not\in S_x \} \text{.} \] For those who may not be familiar with set notations, the above simply says that \( \tilde{S} \) contains exactly all the elements \( x \) of \( X \) such that \( x \) does not lie in \( S_x \). To show that this \( \tilde{S} \) has not been matched to, we must show that \( \tilde{S} \) is different from every \( S_x \).

To see this, let us fix any element \( x \) of \( X \). Then, by definition, if \( x \) lies in \( \tilde{S} \), then \( x \) does not lie in \( S_x \). Moreover, if \( x \) does not lie in \( \tilde{S} \), then \( x \) must lie in \( S_x \). The above shows in particular that \( \tilde{S} \) and \( S_x \) cannot be the same set, since the two sets differ, at the very least, by whether \( x \) lies in them!

As a result, we have shown that \( \tilde{S} \) differs from every \( S_x \), so that \( \tilde{S} \) has not been matched to. This concludes the proof that \( \mathcal{P} (X) \) indeed has strictly larger size than \( X \).

Conclusions

What we have discussed is but a glimpse of the study of infinite sizes. There are many more topics that one can discuss in this area, and you can explore these ideas in more detail in a university-level set theory or mathematical foundations course.

Moreover, "infinite sizes of sets" is but one notion of "infinity" that one encounters in mathematics. You will naturally come across other conceptions of "infinity" in university mathematics—e.g. in calculus, complex analysis, or geometry—that have different applications.

At a more general level, hopefully this discussion of infinity served as a demonstration of how one may explore strange and new mathematical ideas. Note that although we came across many mysterious and mind-bending statements, there was absolutely no black magic involved—every conclusion that we derived came about as a result of very precise and logical reasoning. In particular, this means that anyone, with a requisite amount of patience and grit, can learn and understand new mathematics, even if it first comes across as difficult!

In addition, this discussion hopefully highlighted the creativity inherent to mathematical exploration. For instance, in order to study infinite sizes in an interesting way, we had to first devise a viable way to work with infinite sizes. For this, the idea of matching elements of sets to each other provided a different spin to measuring finite sets that naturally extended to infinite sets. This then allowed us to study the sizes of natural numbers, real numbers, and so on—allowing us to draw some rather surprising conclusions.

All in all, if you continue studying mathematics, whether at the university, postgraduate, or research levels, then you will encounter many more instances of creative explorations and problem solving along the way!