“What comes after one?” sounds like a question a toddler asks. But if you actually chase it seriously — not “two,” but philosophically and mathematically, what does it even mean for something to come “after” something else — you fall down one of the richest rabbit holes in mathematics. I want to take you through that rabbit hole: from simple counting, to different sizes of infinity, to the strange axiom that quietly underlies huge parts of modern mathematics, and to a paradox so counterintuitive it seems to break the rules of physical space itself.
The Deceptively Simple Idea of “Coming After”
In basic arithmetic, “after one” obviously means two, because natural numbers have a built-in successor function: every number n has a next number, n+1.
$$ S(n) = n + 1 $$
This works cleanly for the natural numbers: 1, 2, 3, 4, and so on forever. But the moment you step outside natural numbers into other kinds of ordered structures, “what comes after” stops being obvious. What comes after all the natural numbers? What comes after every fraction between zero and one? These aren’t trick questions — mathematicians have built entire formal systems to answer them, and the answers reshape how we think about infinity itself.
Ordinal Numbers: Counting Past Infinity
This is where ordinal numbers come in. Ordinals extend the idea of “counting order” beyond the finite numbers. After every natural number 1, 2, 3, 4, and so on, mathematicians define a new number called omega, written $\omega$, which represents the order type of the entire sequence of natural numbers taken together.
$$ \omega = {1, 2, 3, 4, \dots} $$
And then — this is the part that still amazes me — you can keep going. There’s a number after omega:
$$ \omega + 1, \quad \omega + 2, \quad \dots, \quad \omega \cdot 2, \quad \dots, \quad \omega^2, \quad \dots, \quad \omega^\omega, \dots $$
Ordinals describe order, not size. $\omega + 1$ isn’t “more numbers” than $\omega$ in a size sense — it’s the same infinite set of naturals, plus one additional element tacked onto the end, restructuring the order. This distinction between order (ordinals) and size (cardinals) turns out to be one of the most important conceptual splits in the mathematics of infinity.
Cardinal Numbers: Different Sizes of Infinity
This brings us to cardinality, which measures the size of a set rather than its order. And here’s the mind-bending part: not all infinities are the same size.
The smallest infinite cardinal, representing the size of the natural numbers, is called aleph-null, written $\aleph_0$. Any set that can be put into a one-to-one correspondence with the natural numbers — meaning you can pair up every element of one set with exactly one element of the other, with none left over — is called “countably infinite” and has size $\aleph_0$. Surprisingly, this includes sets that seem intuitively “bigger” than the natural numbers, like all the integers (including negatives) and even all the rational numbers (fractions).
$$ |\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| = \aleph_0 $$
Cantor’s Diagonal Argument: Proving Some Infinities Are Bigger
The real numbers — which include all the irrational numbers like $\pi$ and $\sqrt{2}$ in addition to the rationals — turn out to be a strictly larger infinity than the natural numbers. Georg Cantor proved this with an elegant technique called the diagonal argument.
Here’s the core idea, simplified: imagine you tried to list every real number between 0 and 1 in an infinite list, each written as an infinite decimal expansion. Cantor showed that no matter how you construct this list, you can always build a new number that differs from every single number on the list in at least one decimal place — just by going down the diagonal of the list and changing each digit. That new number can’t be on the list, which means the list was never complete in the first place. No such complete list can exist, which proves the real numbers cannot be put into one-to-one correspondence with the natural numbers. There are more real numbers than natural numbers, even though both sets are infinite.
$$ |\mathbb{R}| = 2^{\aleph_0} $$
This larger infinity is called the cardinality of the continuum. Whether this cardinality equals the next infinite cardinal after $\aleph_0$, often written $\aleph_1$, is a question called the continuum hypothesis, and it turns out to be formally unprovable within standard mathematical axioms — not merely unsolved, but genuinely undecidable, a discovery that itself reshaped twentieth-century mathematics.
Enter the Axiom of Choice
Once you’re working with infinite sets, ordinary intuitive reasoning starts to break down, and mathematicians needed formal rules, called axioms, to decide what operations on infinite sets are actually allowed. One of the most famous and controversial of these is the axiom of choice.
In plain language, the axiom of choice says: given any collection of non-empty sets, even an infinite collection, it’s possible to select exactly one element from each set simultaneously, forming a new set of chosen representatives — even if there’s no explicit rule describing how to make each choice.
For finite collections, this seems trivially obvious; of course you can pick one item from each of five boxes. But for infinite collections, especially ones without a natural ordering, the axiom of choice asserts something that can’t actually be constructed step by step — it simply asserts that such a selection exists.
Mathematicians have found the axiom of choice both indispensable and deeply strange. It’s needed to prove many important results across analysis, algebra, and topology, yet it also leads to conclusions so counterintuitive that some mathematicians historically viewed it with real suspicion.
The Banach-Tarski Paradox: Choice Taken to Its Extreme
This is where things get genuinely bizarre. The Banach-Tarski paradox, proven in 1924 by mathematicians Stefan Banach and Alfred Tarski, uses the axiom of choice to show something that sounds physically impossible: it’s mathematically possible to decompose a solid sphere into a finite number of pieces (as few as five), and then reassemble those pieces, using only rotations and translations, into two solid spheres, each identical in size to the original.
This isn’t a magic trick or a rounding error — it’s a rigorously proven mathematical theorem. The catch, and it’s an enormous catch, is that the pieces involved are not “solid chunks” in any normal sense. They are wildly, infinitely complex, non-measurable sets of points, constructed using the axiom of choice in a way that makes them impossible to actually build, visualize, or even meaningfully describe with a formula. They have no well-defined volume at all, which is precisely how the theorem manages to sidestep the ordinary conservation of volume that governs everyday reassembly of physical objects.
$$ V(A \sqcup B) \neq V(A) + V(B) \quad \text{for non-measurable } A, B $$
This equation captures the core weirdness: for ordinary, well-behaved (measurable) sets, volumes add up exactly as you’d expect when you combine disjoint pieces. But the Banach-Tarski construction relies on sets so pathological that the very concept of “volume” simply doesn’t apply to them, which is exactly why they can seem to violate conservation of volume without actually breaking any law of physics.
Why This Doesn’t Contradict Physics
It’s important to be precise here: the Banach-Tarski paradox is a statement about abstract mathematical sets of points in idealized three-dimensional Euclidean space, not about physical matter. Physical objects are made of atoms, which are countable, discrete, finite-in-number building blocks, not infinitely divisible continuous sets of points. You cannot actually perform this decomposition on a real orange or a real sphere of gold, because real matter doesn’t have the necessary mathematical structure. The paradox lives entirely within pure mathematics; it says something profound about the strange consequences of the axiom of choice, not something achievable in a lab.
Why Mathematicians Still Debate the Axiom of Choice
Given results like Banach-Tarski, some mathematicians have historically been uneasy about fully accepting the axiom of choice, precisely because it enables conclusions that feel so at odds with physical or geometric intuition. However, the vast majority of modern mathematics operates comfortably within Zermelo-Fraenkel set theory with the axiom of choice, often abbreviated ZFC, because rejecting the axiom would also mean giving up many results mathematicians consider essential and useful, from certain foundational theorems in linear algebra to key results in topology and analysis.
There are alternative systems of mathematics that reject or weaken the axiom of choice, and in those systems, Banach-Tarski-style paradoxes don’t necessarily arise. This reveals something important: mathematics isn’t one single fixed, discovered truth waiting to be found, but rather a structure built on chosen foundational axioms, and different reasonable choices of axioms can lead to genuinely different, internally consistent mathematical universes.
What’s Settled Versus What’s Philosophical
- Well established, rigorously proven: The existence of different infinite cardinalities; Cantor’s diagonal argument; the logical validity of the Banach-Tarski theorem given ZFC axioms; the fact that Banach-Tarski’s “pieces” are non-measurable and cannot correspond to physical objects.
- Open or philosophical: Whether the continuum hypothesis should be considered “true” (it’s independent of standard axioms, meaning it can neither be proven nor disproven within them); whether the axiom of choice should be considered intuitively acceptable; which system of foundational axioms best “matches” mathematical reality, a question that verges into philosophy of mathematics rather than pure calculation.
Coming Back Down to Earth
So, what comes after one? In ordinary counting, obviously two. But chase the question further and you end up walking through ordinal numbers marching past infinity, cardinal numbers revealing that infinity itself comes in different sizes, an axiom that lets you make impossible-sounding infinite choices, and a paradox that can seemingly double a sphere out of nothing using nothing but rotations. None of this breaks mathematics — it’s all rigorously consistent — but it reveals just how far “counting” can stretch once you’re willing to follow the logic past where everyday intuition gives out.