Paradoxical Consequences of the Axiom of Choice

The axiom of choice implies several results that are counterintuitive or "paradoxical," demonstrating the power and subtlety of non-constructive existence proofs.

Non-Measurable Sets

Theorem6.3Vitali's theorem

Assuming AC, there exists a subset of $[0,1]$ that is not Lebesgue measurable.

ExampleConstruction of a Vitali set

Define an equivalence relation on $[0,1]$ : $x \sim y$ iff $x - y \in \mathbb{Q}$ . By AC, choose one representative from each equivalence class to form a set $V \subseteq [0,1]$ .

If $V$ were measurable with measure $m$ , then $[0,1] \subseteq \bigcup_{q \in \mathbb{Q} \cap [-1,1]} (V + q) \subseteq [-1,2]$ . By countable additivity and translation invariance: $1 \leq \sum_{q} m(V+q) = \sum_q m \leq 3$ . But if $m = 0$ , the sum is $0 < 1$ ; if $m > 0$ , the sum is $\infty > 3$ . Contradiction.

The Banach-Tarski Paradox

Theorem6.4Banach-Tarski paradox

Assuming AC, the closed unit ball $B^3 \subset \mathbb{R}^3$ can be partitioned into finitely many pieces which, after rotations and translations, reassemble into two copies of $B^3$ . More precisely, there exist disjoint sets $A_1, \ldots, A_k, B_1, \ldots, B_l$ with $B^3 = A_1 \sqcup \cdots \sqcup A_k \sqcup B_1 \sqcup \cdots \sqcup B_l$ , and rigid motions $\sigma_i, \tau_j$ such that $\sigma_1(A_1) \sqcup \cdots \sqcup \sigma_k(A_k) = B^3$ and $\tau_1(B_1) \sqcup \cdots \sqcup \tau_l(B_l) = B^3$ .

RemarkKey ingredients

The Banach-Tarski paradox relies on:

The axiom of choice (to select representatives from orbits).
The non-amenability of the free group $F_2$ (which can be embedded in $SO(3)$ via rotations).
Dimension $\geq 3$ : In dimensions 1 and 2, finitely additive translation-invariant measures exist on all bounded sets (Banach), so Banach-Tarski fails. The paradox is specific to $n \geq 3$ because $SO(n)$ contains free subgroups for $n \geq 3$ .

Other Counterintuitive Consequences

ExampleFurther consequences of AC

Every infinite set has a countably infinite subset (using DC, which follows from AC).
Every surjection has a right inverse: given $f: A \twoheadrightarrow B$ , AC provides $g: B \to A$ with $f(g(b)) = b$ .
The existence of well-orderings of $\mathbb{R}$ : No explicit well-ordering of $\mathbb{R}$ is known or definable, yet AC guarantees one exists.
Non-principal ultrafilters on $\mathbb{N}$ exist: These are used throughout logic and combinatorics but cannot be explicitly constructed.

RemarkConsistency and independence

Godel (1938) proved that AC is consistent with ZF (if ZF is consistent): he showed AC holds in the constructible universe $L$ . Cohen (1963) proved that $\neg$ AC is also consistent with ZF using the method of forcing. Thus AC is independent of ZF, and mathematicians may choose whether to adopt it. Most mainstream mathematics assumes ZFC (ZF + AC).