AxiomIndependent of the axioms

Axiom of choice

Statement

For every family $(A_i)_{i\in I}$ of non-empty sets, there exists a choice function $f$ with $f(i)\in A_i$ for every $i\in I$ : one can simultaneously pick one element from each set in the family, even when $I$ is infinite and no explicit rule for choosing is given.

Why is it true?

For finitely many non-empty sets, picking one element from each is completely obvious. The axiom of choice extends this obvious-looking ability to arbitrary, even uncountable, families of sets, without requiring any uniform recipe for making the choices.

Proof sketch

Kurt Gödel (1938) built the constructible universe $L$ , a model of ZF in which AC holds, showing $\mathrm{ZF}+\mathrm{AC}$ is consistent relative to ZF. Paul Cohen (1963) invented the method of forcing to build models of $\mathrm{ZF}+\neg\mathrm{AC}$ , showing AC cannot be derived from ZF alone. Together the two results establish that AC is independent of ZF.

Proved by

proved 1963Kurt Gödel proved 1963Paul Cohen

Topics that use this theorem

The ZFC axioms

Related theorems

Zorn's lemma

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Kurt Gödel (1940). The Consistency of the Continuum Hypothesis · DOI:10.1515/9781400881635
Paul J. Cohen (1963). The Independence of the Continuum Hypothesis