Determinacy and Regularity

The axiom of determinacy and its fragments provide the strongest known regularity results for definable sets of reals, resolving questions that are independent of ZFC.

Gale-Stewart Games

Definition9.4Gale-Stewart game

For a set $A \subseteq \omega^\omega$ , the Gale-Stewart game $G_A$ is played between two players who alternately choose natural numbers:

$\text{I}: a_0, \quad a_2, \quad a_4, \quad \ldots$ $\text{II}: \quad a_1, \quad a_3, \quad a_5, \quad \ldots$

Player I wins if $(a_0, a_1, a_2, \ldots) \in A$ ; otherwise player II wins. The set $A$ is determined if one of the players has a winning strategy.

Theorem9.4Gale-Stewart theorem

Every closed (or open) game is determined. More generally, every Borel game is determined (Borel determinacy, proved by Martin in 1975 within ZFC).

Consequences of Determinacy

Theorem9.5Determinacy implies regularity

If $\boldsymbol{\Gamma}$ is a pointclass (e.g., $\boldsymbol{\Sigma}^1_n$ ) and all sets in $\boldsymbol{\Gamma}$ are determined, then all sets in $\boldsymbol{\Gamma}$ have:

The Lebesgue measurability property.
The Baire property.
The perfect set property.

ExampleApplications of determinacy to regularity

Borel determinacy (ZFC): All Borel sets are determined, recovering their known regularity properties.
Analytic determinacy (from a measurable cardinal): All $\boldsymbol{\Sigma}^1_1$ sets are determined, giving the perfect set property for all analytic sets.
Projective determinacy (from Woodin cardinals): All projective sets are determined, resolving all regularity questions at the projective level.

The Axiom of Determinacy

Definition9.5Axiom of determinacy (AD)

The axiom of determinacy states that for every $A \subseteq \omega^\omega$ , the game $G_A$ is determined. AD contradicts the axiom of choice (since AC implies the existence of non-determined sets), but is consistent with ZF.

RemarkAD in L(R)

While AD contradicts AC, the theory ZFC $+$ "AD holds in $L(\mathbb{R})$ " is consistent (assuming large cardinals). Here $L(\mathbb{R})$ is the smallest inner model containing all reals. In this model:

Every set of reals is Lebesgue measurable.
Every set of reals has the Baire property and the perfect set property.
$\omega_1$ is measurable (as a cardinal in $L(\mathbb{R})$ !).
The theory of $L(\mathbb{R})$ is completely determined by large cardinals in $V$ .

The equiconsistency $\text{AD}^{L(\mathbb{R})} \equiv_{\text{con}} \text{infinitely many Woodin cardinals with a measurable above}$ is one of the deepest results in set theory.