{This is the 3rd article in a series in which, in a simple and informal way, I clarify a number of mathematical concepts of general interest that, unfortunately, are not usually included in school-level math curricula, creating basic cultural deficiencies. It is my hope that those who read them will share them, especially with children, as much as they can.}
I begin this article with some questions that the reader should try to answer. Doing so and then reviewing your responses at the end will help you better understand what follows.
- What does infinity mean to you in mathematics?
- Can you give three examples of infinity? Can you think of a concrete numerical or geometric model?
- Is it always true that the whole is greater than any of its parts?
- Where are there more points: on a line or on a segment of it?
- Do you believe there is just one infinity, or more than one?
I often hear mathematical terms used in popular speech, not always precisely, and I wonder whether the person using them clearly understands what they are saying. The term infinity is a good example of this. To illustrate this point, in this article, instead of using comments overheard on the street, I have decided to consult dictionaries of universal reputation.
The dictionary of the Real Academia Española (RAE), for example, gives us several definitions of infinity. It says: “that which has no end or limit,” and also “a value greater than any assignable quantity.” So far, so good — both expressions are correct. However, it also says “very numerous or enormous,” which, in my opinion, is incorrect. This is because, when we look up “numerous,” the same RAE dictionary defines it as “much larger than normal, or a multitude of people or things,” or “that which appears in great numbers.” It’s clear that neither something larger than normal, nor a multitude, are necessarily infinite — nor is a large number.
The entries in another excellent dictionary, María Moliner’s, are a bit better in my view. Still, part of the explanation includes “very large,” which is clearly wrong, and “the firmament populated by infinite stars,” which — unless used poetically — is not true, because the universe is finite. It is precisely the proven finiteness of the universe that prevents us from finding concrete models of infinity within it, meaning we can only speak of this concept in an abstract sense.
Hence, infinity can only be spoken of in the spiritual realm for believers — since God is infinite — or, as we will see, in mathematics.
But what do we call infinite in mathematics? It’s actually quite easy to explain — just count. Numerically, we even have the advantage of concrete examples. Remember that the cardinality of a set is the number of elements it contains. For example, the set of vowels A = {a, e, i, o, u} has cardinality 5, while any subset properly contained within it has fewer than 5 elements. For example, the subset of strong vowels F = {a, e, o} has 3 elements, and the subset of weak vowels D = {i, u} has 2.
Now then, is this property always true? That is, is it always true that, given a set, any subset properly contained within it has fewer elements?
Let’s analyze the cardinality of the set of natural numbers:
ℕ = {1, 2, 3, ⋯, n, n+1, ⋯}
If we try to count the number of elements in ℕ, we see that it never ends, because, as its definition shows, if n were the last number, we could simply add one to get n+1, which is a larger natural number.
In mathematics, the idea of having no end is expressed by saying that the cardinality of the set is infinite, and infinity is represented using the symbol ∞. It’s important to emphasize that infinity is not a number, but a symbol that denotes endlessness or without limit. The cardinality of ℕ is denoted by the symbol ℵ₀ (read aleph-naught or aleph-zero), and it is the first infinite number. Note that “Aleph” is the first letter of the Hebrew alphabet.
Another peculiarity that characterizes infinite sets is that one of their parts can be as large as the whole. Consider, for example, the subset of even numbers:
𝓟 = {2, 4, 6, ⋯, 2n, ⋯}
If we establish a correspondence between the natural numbers and the even numbers, we see that each natural number corresponds to exactly one even number and vice versa. Therefore, there are just as many natural numbers as there are even numbers!
ℕ = {1, 2, 3, 4, ⋯, n, ⋯}
⬍ ⬍ ⬍ ⬍ ⬍
𝓟 = {2, 4, 6, 8, ⋯, 2n, ⋯}
Figure 1
Also note that if, instead of the subset of even numbers 𝓟, we use the subset of odd numbers 𝓘 = {1, 3, 5, 7, ⋯, 2n+1, ⋯}, or the set of multiples of 5, say 𝒞 = {5, 10, 15, 20, ⋯, 5n, ⋯}, we again see that both are subsets properly contained in ℕ, and as we can observe below, there is a correspondence in which each element of these sets corresponds to a unique element of ℕ and vice versa. Therefore, 𝓘 and 𝒞 are proper subsets of ℕ that have the same number of elements as ℕ.
ℕ = {1, 2, 3, 4, ⋯, n, ⋯} ℕ = {1, 2, 3, ⋯, n, ⋯}
⬍ ⬍ ⬍ ⬍ ⬍ ⬍ ⬍ ⬍ ⬍
𝓘 = {1, 3, 5, 7, ⋯,2n+1, ⋯} 𝒞 = {5, 10, 15, ⋯, 5n, ⋯}
Figura 2
We therefore see that, in infinite numerical sets, the whole is not greater than any of its parts! It is this peculiarity— which distinguishes finite sets from infinite ones— that led Cantor to define an infinite set as one that contains proper subsets with the same cardinality as itself.
Is there more than one infinite number?
It is interesting to know that, mathematically, if a set S has cardinality n, then the set of all possible subsets of S has cardinality 2^n (2ⁿ). Therefore, if ℕ has cardinality ℵ₀, the set of all subsets of ℕ has cardinality 2^ℵ₀—a strictly larger infinite number than ℵ₀. And the set of subsets of that new set will have cardinality (2^ℵ₀)^ℵ₀, an infinite number larger than the previous one, and so on. This shows that there is an infinity of infinite numbers, each larger than the last!
Visualizing infinity with points and segments
It always helps to visualize any concept graphically, so we briefly consider infinity using points and line segments. Let’s look at two basic questions.
Can a segment contain infinitely many points?
Remembering that a point has no dimension, and that every real number corresponds to exactly one point on the number line, it is easy to see that any segment contains infinitely many points. For example, take the segment [0, 1]. Observe that the set of reciprocals of the natural numbers {1, 1/2, 1/3, …} is infinite and that all these points lie between 0 and 1; therefore, the points representing them lie on the segment [0, 1]. Geometrically, one can show similarly that this holds for any segment.
Can two segments of different lengths have the same number of points?
We consider two segments AB and CD of different lengths. As shown in Figure 3, we form triangle EFG, with base EF = AB and height FG = CD, and we consider an arbitrary point P on EF, as seen in Figure 3. A perpendicular line to EF by point P intersects side EG, in a corresponding point H. If we then draw a line through H parallel to the base of the triangle, it intersects side FG at a point Q. It is easy to observe that if we move point P to the right (or left), the corresponding point Q moves upward (or downward).
Thus, we see that for every point P on EF, there is a unique corresponding point Q on FG. Therefore, the segments EF = AB and FG = CD have the same number of points.
Despite differing in length, both segments contain the same cardinality of points—namely, continuum many—because one can establish a one‑to‑one correspondence between the points of one segment and those of the other.
Let us remember that any segment has finite length, while a line extends indefinitely, and therefore has infinite length. Keeping this in mind, what do you think—does a segment or a full line have more points?
Let us consider segment AB and the vertical line passing through its center O, known as the y-axis. Figure 4 illustrates that, following the same technique used in Figure 3, any pair of points C and D from segment AB correspond to points C” and D” on the vertical axis, and vice versa. Therefore, segment AB and the vertical line have the same number of points!
