How do you count things you cannot finish counting? This question, seemingly paradoxical, sits at the heart of one of mathematics' most profound discoveries: not all infinities are equal.

When we count finite sets, we establish a correspondence. Five apples match five fingers match five tally marks. Georg Cantor realized this simple idea—matching—could extend beyond the finite. Two sets have the same size if their elements can be paired up perfectly, with nothing left over on either side.

This insight revolutionized mathematics. It revealed that infinity is not a single, monolithic concept but a vast hierarchy of different sizes. The natural numbers represent one infinity. The real numbers represent a strictly larger one. And the ladder continues upward, forever. Let us construct this hierarchy step by step, with the logical precision that makes such claims not merely plausible but certain.

The foundation of cardinal arithmetic rests on a deceptively simple definition. Two sets A and B have the same cardinality if and only if there exists a bijection between them—a function that pairs each element of A with exactly one element of B, and vice versa, with no elements left unpaired.

For finite sets, this matches our intuition perfectly. A set of three cats has the same cardinality as a set of three hats because we can match each cat to a hat: cat₁ → hat₁, cat₂ → hat₂, cat₃ → hat₃. No cat goes hatless; no hat goes unclaimed.

But here is where infinity reveals its strange nature. Consider the natural numbers {0, 1, 2, 3, ...} and the even numbers {0, 2, 4, 6, ...}. Intuition suggests the naturals should be 'bigger'—the evens are merely a subset! Yet the function f(n) = 2n establishes a perfect bijection. Every natural number maps to exactly one even number, and every even number comes from exactly one natural. They have identical cardinality.

This is our first theorem about infinity: an infinite set can have the same cardinality as a proper subset of itself. In fact, this property characterizes infinite sets. Dedekind used it as a definition: a set is infinite precisely when it can be placed in bijection with a proper part of itself. The finite and infinite worlds operate by fundamentally different rules.

Takeaway

Size in the infinite realm is determined not by containment but by correspondence. Two sets are equally large when their elements can be perfectly paired, regardless of whether one contains the other.

We denote the cardinality of the natural numbers as ℵ₀ (aleph-null). Any set that can be placed in bijection with ℕ is called countably infinite. The integers are countable. The rationals, surprisingly, are countable too—Cantor's famous diagonal arrangement proves this. One might wonder: is every infinite set countable?

Cantor's diagonal argument answers with a resounding no. Consider the real numbers between 0 and 1, each expressible as an infinite decimal. Suppose, for contradiction, we could list them all: r₁, r₂, r₃, ... Now construct a new number d where the nth decimal digit of d differs from the nth decimal digit of rₙ. This number d cannot be rₙ for any n, since it differs from rₙ in the nth position. Therefore d is not in our supposedly complete list. Contradiction.

The reals are uncountable. Their cardinality, denoted 2^ℵ₀ or 𝔠 (the cardinality of the continuum), strictly exceeds ℵ₀. We write ℵ₀ < 2^ℵ₀. This is not merely a different infinity—it is a larger infinity in a precisely defined sense.

And the hierarchy continues. Cantor's theorem states that for any set S, the power set P(S)—the set of all subsets of S—has strictly greater cardinality than S itself. Thus 2^ℵ₀ < 2^(2^ℵ₀) < 2^(2^(2^ℵ₀)) < ... The tower of infinities rises without end, each level strictly larger than the last.

Takeaway

Infinity is not one thing but infinitely many things. Cantor's theorem guarantees that no matter how large an infinity you encounter, a strictly larger one always exists above it.

How do infinite cardinals combine under arithmetic operations? The rules, once proven, are elegant—and profoundly different from finite arithmetic.

Addition: The cardinal sum |A| + |B| is defined as the cardinality of the disjoint union A ∪ B. For infinite cardinals, addition collapses in a remarkable way: if κ and λ are cardinals with at least one infinite, then κ + λ = max(κ, λ). Thus ℵ₀ + ℵ₀ = ℵ₀, and ℵ₀ + 2^ℵ₀ = 2^ℵ₀. The larger infinity simply absorbs the smaller.

Multiplication: The product |A| × |B| is the cardinality of the Cartesian product A × B. Again, infinite cardinals absorb: κ × λ = max(κ, λ) when at least one is infinite. The rational numbers, expressible as pairs of integers, illustrate this: ℵ₀ × ℵ₀ = ℵ₀. The plane of integer pairs is no larger than the integer line.

Exponentiation is where genuine growth occurs. The cardinal κ^λ represents the cardinality of all functions from a set of size λ to a set of size κ. Here infinity does not collapse—2^ℵ₀ is strictly larger than ℵ₀, as we proved. This operation is the engine that drives the infinite hierarchy upward. Each application of the power set operation—equivalent to exponentiating 2 by the current cardinality—yields a strictly larger infinity, forever.

Takeaway

Infinite arithmetic follows its own logic: addition and multiplication are absorbed by the larger operand, but exponentiation creates genuine leaps. The power operation is the ladder by which we climb the infinite hierarchy.

We began with a simple question about counting and arrived at a universe of infinities, each larger than the last, governed by arithmetic rules that defy finite intuition.

This is the power of rigorous definition. By defining cardinality through bijection, Cantor transformed vague philosophical musings about infinity into precise mathematical objects we can compare, combine, and calculate with. The infinite became tamed—not by making it finite, but by extending our logical tools to operate where intuition fails.

The hierarchy of cardinals stands as one of mathematics' great monuments to careful reasoning. It demonstrates that logical construction, pursued with sufficient rigor, can illuminate even the most seemingly ineffable concepts.