Axiom of Choice & Consequences — choice.plnt.earth

Abstract

The Axiom of Choice (AC) asserts: for any family of nonempty sets, there exists a choice function selecting one element from each. AC is independent of ZF but accepted in ZFC. Its equivalents include the Well-Ordering Theorem and Zorn’s Lemma. Its consequences include Hamel bases, algebraic closures, non-measurable sets, and the Banach–Tarski paradox. Symbols are HTML-escaped (∀, &exists;, ∈, ⊆, ∅, ≠, ℵ).

1. Statement of AC

Axiom of Choice. If &mathcal;F is a family of nonempty sets (i.e. ∀A ∈ &mathcal;F, A ≠ ∅), then ∃ a function c: &mathcal;F → ⋃&mathcal;F with c(A) ∈ A for all A ∈ &mathcal;F.

2. Equivalents

Well-Ordering Theorem. Every set can be well-ordered.

Zorn’s Lemma. Every nonempty poset in which every chain has an upper bound contains a maximal element.

Hausdorff Maximal Principle. Every poset has a maximal chain.

Tychonoff’s Theorem. The product of compact spaces is compact (full generality ≡ AC).

Ultrafilter Lemma. Every filter extends to an ultrafilter (weaker than AC).

3. Consequences

Every vector space has a Hamel basis.
Every ring has a maximal ideal (via Zorn).
Existence of non-measurable sets (e.g. Vitali sets).
Paradoxical decompositions (e.g. Banach–Tarski in ℝ³).

4. Independence & Fragments

AC is independent of ZF: cannot be proved or refuted.
Dependent Choice (DC) is weaker but sufficient for analysis.
Countable Choice (CC) is weaker still.
Ultrafilter Lemma lies strictly between AC and ZF.

5. Banach–Tarski Paradox (ℝ³)

Theorem (Banach–Tarski, ZFC). In ℝ³, a ball can be partitioned into finitely many disjoint pieces and reassembled into two balls congruent to the original, using only rotations and translations.

Sketch. The construction uses a free subgroup of SO(3) to generate paradoxical decompositions. Creating orbit representatives requires arbitrary selections — exactly where AC is used.

Pieces are non-measurable; Lebesgue measure doesn’t apply.
Works in 3D and higher; fails in 2D (group is amenable).

Avoiding BT

Work in ZF without AC (or with DC only) → no BT.
In Solovay’s model (ZF + every set of reals measurable), BT fails.
Restrict to measurable/Borel sets or amenable groups.

Thus: accepting AC enables Banach–Tarski; rejecting AC avoids it.

6. Controversies

AC is widely accepted for its utility, but it allows nonconstructive existence and paradoxical outcomes. Constructivist and intuitionist schools often reject it or use weaker fragments.

7. Next Problems & Frontiers

Some of the deepest unsolved problems are independent of ZFC, meaning they cannot be decided by the current axioms. The “solutions” involve proposing new axioms or alternative frameworks.

Continuum Hypothesis (CH). Is there a set with |ℕ| < |X| < |ℝ|? Gödel and Cohen proved CH is independent of ZFC. Undecidable in ZFC

Figure 1 — CH asks whether a “middle rung” exists between countable and continuum cardinalities.

Definable Well-Order of ℝ. AC ensures a well-order exists, but no explicit definable one is known in ZFC. Open in ZFC

Figure 2 — A “searchlight” seeks an explicit definable well-order on ℝ.

Determinacy Axioms (AD). Replace AC by AD (“every infinite game is determined”) and the world of sets of reals changes: all sets become measurable; Banach–Tarski fails. Alternative to AC

Figure 3 — Determinacy reframes sets of reals via infinite games.

Large Cardinals. Strong axioms of infinity (measurable, supercompact, etc.) extend ZFC and calibrate consistency strength. Beyond ZFC

Figure 4 — A schematic “tower” of large cardinal axioms by strength.

Why it matters

In ZFC: Banach–Tarski holds; CH undecidable; no definable well-order of ℝ known.
In ZF+DC+AD: CH false; all sets measurable; BT fails.
With large cardinals: Hierarchies of infinity extend beyond Cantor’s ladder and guide new results.

The frontier is not a proof, but a choice: which axioms to adopt.

References

Cantor, G. (1895–97). Beiträge zur Begründung der transfiniten Mengenlehre.
Zermelo, E. (1904). Beweis, dass jede Menge wohlgeordnet werden kann.
Zorn, M. (1935). A remark on method in transfinite algebra.
Banach & Tarski (1924). Sur la décomposition des ensembles de points.
Gödel, K. (1940). Consistency of AC and GCH.
Cohen, P. (1963). Independence of CH.
Jech, T. Set Theory; Kanamori, A. The Higher Infinite.