Axiom of Determinacy

The axiom of determinacy is a proposed axiom of set theory that is consistent with Zermelo-Fraenkel set theory (ZF) but is inconsistent with the axiom of choice (and hence ZFC). It was proposed by Mycielski and Steinhaus in 1962 as a way to avoid some of the more unpleasant consequences of the axiom of choice. In particular, it implies that the subsets of the real numbers are well-behaved in certain precise ways; for example, the sets constructed in the Banach-Tarski paradox do not exist in ZFD, the Zermelo-Fraenkel set theory with the axiom of determinacy.

The axiom is concerned with the outcomes of certain infinite combinatorial games. Namely, let \( {\mathbb N}^{\infty}\) be the set of sequences \[ n_1,n_2,n_3,\ldots \] of natural numbers.

Alice and Bob choose a set \( S \subseteq {\mathbb N}^\infty\) and play a game as follows: they take turns choosing natural numbers, Alice choosing \( n_1,\) Bob choosing \(n_2,\) and so on. In this way they generate an infinite sequence \( (n_1,n_2,n_3,\ldots) \in {\mathbb N}^\infty.\) Alice wins if the sequence is in \( S;\) Bob wins if it is not.

The axiom of determinacy states that for every subset \( S\) of \({\mathbb N}^\infty,\) this game is determined; that is, either Alice or Bob has a winning strategy.

The axiom of determinacy is inconsistent with the axiom of choice. The idea of the proof, roughly, is to use the axiom of choice to well-order the real numbers. Using this ordering and an enumeration of Alice and Bob's possible strategies, it is possible to construct a set \( S\) such that none of Alice and Bob's strategies can possibly be winning, via a diagonal argument.

On the other hand, the axiom of determinacy implies "countable choice"--that is, every countable collection of nonempty sets has a choice function. To some mathematicians, countable choice has more palatable consequences than the full axiom of choice, and in fact the axiom of determinacy does avoid some of the more counterintuitive consequences of the full axiom of choice.

For example, the Banach-Tarski paradox is no longer possible in a universe where the axiom of determinacy holds, because in such a universe every subset of the real numbers is measurable (i.e. has a well-defined volume), and the paradox requires non-measurable sets.

The axiom of determinacy also implies a weak form of the continuum hypothesis: namely, no subset of \(\mathbb R\) has cardinality strictly between \( \aleph_0\) and \(\aleph_1.\)