The Banach-Tarski Paradox
How a sphere can be decomposed and reassembled into two spheres
Summary
A mathematical system begins with axioms that define what exists and what follows. The Banach-Tarski paradox demonstrates how accepting the Axiom of Choice leads to the existence of non-measurable sets, allowing a sphere to be decomposed into finitely many pieces that can be reassembled into two identical spheres using only rigid motions.
Key Concepts: Axiom of Choice, measure theory, free groups, non-measurable sets
The Axiom Foundation
A mathematical system begins with a specification of its basic elements: which objects exist, which operations are defined on them, and which relations must hold. These specifications are encoded in the system’s axioms. An axiom is a formal assumption that serves as the foundation for the system. Within that system, no statement can be derived unless it is implied by the axioms in conjunction with the rules of logical inference.
Once a set of axioms is fixed, all further reasoning, including definitions, proofs, and theorems, must proceed within the structure they determine. The consistency and character of the system depend entirely on these initial choices.
Different axiomatic systems describe different mathematical worlds. In one system, every set may have a well-ordering. In another, it may not. In one geometry, parallel lines exist; in another, they do not. These are not contradictions. They are the consequences of working under distinct sets of assumptions. Axioms define what exists and what follows.
From Numbers to Sets
The axioms of Peano Arithmetic define the system of the natural numbers, denoted $\mathbb{N}$. This set includes $0$, $1$, $2$, $3$, and so on without end. It is not treated as a completed totality but built from a starting element and a rule for generating new ones. The system introduces a constant symbol $0$ and a unary function $S$, called the successor function, which assigns to each number the number that comes immediately after it.
Peano Arithmetic can itself be formulated within a more general framework: axiomatic set theory. In that setting, the natural numbers are modeled as specific sets, and their properties follow from axioms governing set formation, membership, and equality. The objects of arithmetic — numbers, functions, sequences — are all encoded as sets, and reasoning about them is carried out using the axioms of the underlying set theory.
Set theory defines and relates entire collections of structures. New axioms allow statements not just about numbers, but about functions between functions, collections of collections, and hierarchies of infinities.
The Axiom of Choice
The Axiom of Choice is one such axiom. It asserts that for any collection of non-empty sets, there exists a function that selects exactly one element from each set. In finite cases, such a selection can usually be written down explicitly or proved to exist by simple arguments. In infinite settings, this is not always possible.
For example, consider an infinite collection of drawers, each containing a left and right shoe. A rule such as “choose the right shoe” provides a valid selection and does not require the Axiom of Choice. But if each drawer contains a pair of identical socks with no distinguishing features, then no explicit rule can be formulated. The existence of a function that selects one sock from each drawer in this case depends on accepting the Axiom of Choice.
Measure and Size
Measure theory formalizes how size is assigned to sets. A measure $\mu$ is a function that maps certain subsets of a space to non-negative numbers, subject to specific requirements. Chief among these is countable additivity: if a set is decomposed into countably many disjoint measurable parts ${A_i}{i=1}^\infty$, then $\mu\left( \bigcup{i=1}^\infty A_i \right) = \sum_{i=1}^\infty \mu(A_i)$.
In words: the measure of the union of non-overlapping parts equals the sum of their individual measures. This ensures that $\mu$ accumulates size coherently across disjoint components.
A further expectation is invariance under rigid motions: translating or rotating a measurable set leaves its measure $\mu$ unchanged. This reflects the principle that volume is intrinsic to the object and independent of its position or orientation.
However, the Axiom of Choice implies the existence of subsets for which these conditions fail. These are non-measurable sets: subsets of space to which no consistent volume can be assigned while preserving both countable additivity and invariance under rigid motions.
The Bacteria Game
The Banach–Tarski construction depends on a combinatorial game. Consider bacteria and antibiotics arranged as beads on a string. There are two types of bacteria, $A$ and $B$, and two corresponding antibiotics, $A’$ and $B’$. A bacterium dies if placed adjacent to its matching antibiotic: $A$ next to $A’$, or $B$ next to $B’$.
Under this rule, certain sequences survive while others collapse. The string $ABA’B$ remains intact because no bacterium sits next to its antibiotic. The string $AA’B$ reduces to $B$ because the $A$ and $A’$ cancel. The string $BBA’$ cannot be reduced further, leaving two bacteria and one antibiotic.
This game generates all possible finite sequences using the symbols $A$, $B$, $A’$, and $B’$, with adjacent cancellations as the only allowed reduction. Each surviving sequence is unique and irreducible. This system is the free group on two generators, $F_2 = \langle A, B \rangle$. The group contains no relations beyond the cancellation of adjacent inverses.
The Card Game Construction
Now transfer this game to geometry. Mark the north pole of a sphere. Define four rotation cards: $A$ rotates the sphere by some irrational multiple of $\pi$ degrees around one axis, $A’$ reverses this rotation, $B$ rotates by another irrational multiple around a different axis, and $B’$ reverses that rotation. The irrational angles ensure that no sequence of rotations ever returns the north pole to its original position unless the sequence reduces to the identity.
Two players sit in separate rooms, each with a sphere. Player One may start any sequence with card $A$. Player Two may start any sequence with card $B$. After the opening move, both players may use any card. Each player builds all possible sequences of any length following these rules.
Each player collects the set of points where the north pole lands after applying their sequences. Call these point clouds $S_A$ and $S_B$. Since Player One starts with $A$ and Player Two starts with $B$, their collections appear disjoint. Each player holds what seems to be at most half the possible positions.
The Paradoxical Transformation
Here comes the construction. Someone hands Player One the card $A’$ to place at the beginning of every sequence they built. This rotates their entire point cloud $S_A$ by the rotation $A’$. The cloud becomes $A’ \cdot S_A$. Player Two receives card $B’$ to place at the beginning of every sequence, transforming their cloud to $B’ \cdot S_B$.
After this transformation, both players possess identical collections. Player One now has access to sequences beginning with any card, because $A’$ followed by sequences starting with $A$ produces sequences starting with any symbol. Player Two gains the same universal access. Both $A’ \cdot S_A$ and $B’ \cdot S_B$ contain all possible sequences from the rotations.
Each player now holds a complete copy of the full set of reachable points. The original sphere has been decomposed into disjoint pieces $S_A$ and $S_B$, and rigid rotations have transformed these pieces into two complete spheres. No points are created or destroyed. The construction uses only rotations.
The Dependence on Choice
This outcome depends on the ability to define these non-measurable sets (the collections of points for which no volume assignment is possible that satisfies both countable additivity and invariance under movement). The existence of such sets requires the Axiom of Choice. Without it, the decomposition cannot be carried out. With it, the construction proceeds formally and without contradiction.
The Axiom of Choice plays a central role within the standard framework known as Zermelo–Fraenkel set theory with Choice, abbreviated ZFC. The Axiom of Choice is not derivable from the other axioms of this system. It is independent of the base theory ZF: both ZFC and ZF without Choice are consistent, provided that ZF itself is consistent.
Despite its independence, the Axiom of Choice is almost universally accepted. Many classical theorems in algebra, topology, analysis, and logic depend on it. The version needed for the Banach–Tarski paradox allows choices from certain well-structured families of non-empty sets. This weaker form still goes beyond what can be derived in constructive frameworks, but it is logically weaker than the full axiom (it requires only Hahn-Banach theorem).
Commentary
This chapter forces a distinction between mathematical and physical reasoning. The Banach–Tarski construction is not a paradox in the sense of contradiction or physical impossibility, but it demonstrates a dependence on foundational axioms. It isolates the consequences of adopting the Axiom of Choice, revealing that intuitive notions like volume are not preserved across all valid set decompositions.
The result is clean and formally sound, yet incompatible with empirical modeling. That gap — between internally consistent mathematics and physically grounded expectation — illustrates the epistemic boundaries explored throughout this book. As with other chapters that emphasize when simplifications fail (relativity in gold, curvature in gravity, topology in voting), this example shows that what appears insane may instead be a well-posed feature of a chosen formal system.
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“Axioms define what exists and what follows.”