There’s a specific moment in learning advanced mathematics where the ground shifts under you a little. For me, it was learning about the axiom of choice and realizing that mathematicians can prove sets exist — with total logical rigor — that you could never actually construct, describe, or even coherently assign a size to. These are what mathematicians call non-measurable sets, and understanding how the axiom of choice creates them opens a window into one of the strangest, most debated corners of modern mathematics.
First, What Does “Size” Even Mean for a Set?
Before I can explain “sizeless” sets, I need to explain what “size” (technically called measure) means in the first place. In everyday geometry, measuring a shape’s size — its length, area, or volume — feels obvious. A line segment from 0 to 1 has length 1. A square with side length 2 has area 4.
Mathematicians formalized this intuitive idea into something called measure theory, developed largely by Henri Lebesgue in the early 1900s. Lebesgue measure extends the idea of length, area, and volume to a huge range of sets, even very complicated, scattered, infinite ones, not just simple shapes like intervals and rectangles.
$$ \mu([a,b]) = b – a $$
This equation says the measure (length) of an interval from a to b is just b minus a — nothing surprising there. But Lebesgue’s framework can also assign consistent measures to far stranger sets, like the set of all rational numbers between 0 and 1 (which, despite being infinite, has measure zero, since rational numbers are “sparse” in a precise mathematical sense compared to the full continuum of real numbers).
The Properties We Expect Any Reasonable Measure to Have
For a measure to behave sensibly, mathematicians expect it to satisfy a few basic, intuitive properties:
- Non-negativity: No set should have a negative size.
- Countable additivity: If you break a set into a countable number of non-overlapping pieces, the total measure should equal the sum of the measures of the individual pieces.
- Translation invariance: Moving a set around in space (shifting it left, right, up, or down) shouldn’t change its size.
$$ \mu\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mu(A_i) \quad \text{for disjoint } A_i $$
These properties seem so obviously true that it’s hard to imagine any reasonable set failing to satisfy them. And yet, some sets do — and the axiom of choice is exactly what’s responsible for their existence.
What the Axiom of Choice Actually Says
The axiom of choice is a foundational rule in set theory. Informally, it states: given any collection of non-empty sets, no matter how large or how strange, it’s possible to select exactly one element from each set simultaneously, forming a new combined set of chosen representatives — even when there’s no specific rule, formula, or procedure describing how to make each individual choice.
For finite collections, or even simple infinite collections with an obvious pattern (like choosing the smallest number from each set of natural numbers), this feels completely uncontroversial. But the axiom of choice asserts this is possible in general, for absolutely any collection of non-empty sets, including collections that are so unstructured, so lacking in any natural ordering, that no explicit selection rule could ever be written down.
This is subtle but important: the axiom doesn’t tell you how to make the choices. It simply asserts that a valid, complete set of choices exists, as an act of pure logical postulation rather than actual construction.
Building a Non-Measurable Set: The Vitali Set
The clearest, most classic example of a “sizeless” set built using the axiom of choice is called a Vitali set, named after Italian mathematician Giuseppe Vitali, who constructed it in 1905, remarkably early in this whole discussion.
Here’s the construction, simplified conceptually. Consider all real numbers between 0 and 1. Define two numbers as “equivalent” if their difference is a rational number. This equivalence relation splits the entire interval from 0 to 1 into an enormous number of separate groups, called equivalence classes, where every number within a given group differs from every other number in that same group by some rational amount.
$$ x \sim y \iff x – y \in \mathbb{Q} $$
There are uncountably many of these equivalence classes (since rational numbers are countable, but real numbers are uncountable, so there must be uncountably many distinct classes to cover the whole interval). Now, using the axiom of choice, select exactly one representative real number from each of these uncountably many equivalence classes, forming a new set — this is the Vitali set.
Why the Vitali Set Can’t Have a Sensible Size
Here’s where the paradox emerges. Take the Vitali set, call it V, and shift it by every possible rational number between -1 and 1, creating countably many shifted copies of V (since there are countably many rational numbers in that range).
$$ V_q = {v + q : v \in V}, \quad q \in \mathbb{Q} \cap [-1,1] $$
By construction, these shifted copies are all disjoint from one another (no overlaps), and their union covers a range of real numbers between roughly -1 and 2, entirely containing the original interval from 0 to 1.
$$ [0,1] \subseteq \bigcup_{q} V_q \subseteq [-1,2] $$
Now here’s the contradiction. If the Vitali set V had some well-defined measure, then by translation invariance, every shifted copy $V_q$ would have to have the exact same measure as V. And by countable additivity, the total measure of the union of all these shifted copies would have to equal the sum of an infinite (countably infinite) number of identical measures.
$$ \mu\left(\bigcup_q V_q\right) = \sum_q \mu(V_q) = \sum_q \mu(V) $$
But this creates an impossible situation. If $\mu(V) = 0$, then the total sum is 0, which can’t be right, since the union contains the entire interval $[0,1]$, which has measure 1. But if $\mu(V)$ is any positive number, no matter how small, then summing that same positive number infinitely many times gives infinity, which also can’t be right, since the union is entirely contained within the interval $[-1,2]$, which has a finite measure of 3.
There is no value — zero, positive, or otherwise — that $\mu(V)$ could possibly take without producing a contradiction. The only way out is to conclude that the Vitali set simply doesn’t have a well-defined measure at all. It’s not that its size is zero, or infinite, or some strange in-between value — the very concept of “size” fails to apply to it. This is what mathematicians mean by a non-measurable set.
Where This Connects to Banach-Tarski
This same underlying mechanism — using the axiom of choice to build sets so pathological that ordinary measure simply doesn’t apply to them — is exactly what powers the famous Banach-Tarski paradox, which shows that a solid sphere can be split into a small number of pieces and reassembled, using only rotations and translations, into two spheres identical to the original.
$$ V(A \sqcup B) \neq V(A) + V(B) \quad \text{for non-measurable } A, B $$
The pieces involved in Banach-Tarski aren’t ordinary chunks of a sphere — they’re non-measurable sets, constructed via the axiom of choice, much like the Vitali set, just embedded in three-dimensional space and organized using rotation groups rather than a simple rational-number equivalence relation. Because these pieces have no well-defined volume in the first place, the usual rule that volumes must add up correctly when you combine disjoint pieces simply doesn’t apply to them, which is exactly how the paradox manages to “create” extra volume without contradicting the ordinary mathematics of measurable, everyday shapes.
Why Mathematicians Didn’t Just Reject the Axiom of Choice
Given that it enables such strange, unsettling conclusions, it’s reasonable to ask why mathematicians didn’t simply throw the axiom of choice out. The honest answer is that the axiom of choice is also deeply useful, arguably indispensable, for proving many important, widely relied-upon results across mathematics — including certain foundational theorems in linear algebra (like the existence of a basis for every vector space, even infinite-dimensional ones), key results in topology (like Tychonoff’s theorem), and various results in analysis.
Mathematicians have studied alternative systems that reject or weaken the axiom of choice, and in some of these systems, results like the existence of non-measurable sets or the Banach-Tarski paradox genuinely don’t arise. But these alternative systems also lose access to other results many mathematicians consider essential, creating a genuine trade-off rather than a clean “right answer.” Most modern mathematics operates within a system called Zermelo-Fraenkel set theory with the axiom of choice, abbreviated ZFC, precisely because the practical benefits are judged to outweigh the philosophical discomfort of results like non-measurable sets.
Distinguishing the Rigorous From the Philosophical
- Rigorously proven, given ZFC axioms: The Vitali set construction and its non-measurability; the logical validity of the Banach-Tarski theorem; the general fact that the axiom of choice permits constructing sets lacking a well-defined Lebesgue measure.
- Philosophically debated: Whether the axiom of choice should be considered an intuitively “true” or acceptable mathematical assumption; whether non-constructive existence proofs (proving something exists without showing how to build it) represent legitimate mathematical knowledge in the same sense as constructive proofs; which axiomatic system best captures “true” mathematics, if such a thing can even be meaningfully said to exist.
Final Thoughts
Non-measurable sets like the Vitali set reveal something genuinely strange sitting quietly underneath huge swaths of ordinary, respected mathematics: the assumption that you can always make an infinite number of unstructured choices simultaneously, even without any rule for making them. That single, seemingly modest assumption is enough to conjure sets so wild, so disconnected from ordinary geometric intuition, that the basic concept of “size” simply breaks down when applied to them. It’s a powerful reminder that mathematics isn’t just about calculating with familiar objects — it’s about carefully examining which foundational assumptions we’re willing to accept, and following their consequences with total honesty, even when those consequences feel deeply uncomfortable.