The Martin-Steel Theorem on Projective Determinacy

The Martin-Steel theorem establishes the exact large cardinal hypothesis needed for projective determinacy, providing a deep bridge between large cardinals and the definability of sets of reals.

Statement

Theorem8.4Martin-Steel theorem

If there exist $n$ Woodin cardinals with a measurable cardinal above them, then every $\boldsymbol{\Sigma}^1_{n+1}$ set of reals is determined. In particular, infinitely many Woodin cardinals imply projective determinacy (PD): every projective set is determined.

The Projective Hierarchy

Definition8.8Projective sets

The projective hierarchy classifies subsets of $\omega^\omega$ (or $\mathbb{R}$ ):

$\boldsymbol{\Sigma}^1_0 = \boldsymbol{\Pi}^1_0$ = Borel sets.
$\boldsymbol{\Sigma}^1_1$ = analytic sets (continuous images of Borel sets).
$\boldsymbol{\Pi}^1_1$ = coanalytic sets (complements of analytic sets).
$\boldsymbol{\Sigma}^1_{n+1}$ = continuous images of $\boldsymbol{\Pi}^1_n$ sets.
$\boldsymbol{\Pi}^1_{n+1}$ = complements of $\boldsymbol{\Sigma}^1_{n+1}$ sets.

A set is projective if it belongs to $\bigcup_n (\boldsymbol{\Sigma}^1_n \cup \boldsymbol{\Pi}^1_n)$ .

Proof Outline

Proof

The proof is highly technical. We outline the key ideas for $\boldsymbol{\Sigma}^1_1$ determinacy (Martin, 1975, from a measurable cardinal) and indicate how Woodin cardinals handle higher levels.

$\boldsymbol{\Sigma}^1_1$ determinacy: Let $A$ be an analytic set. Using an iteration of the measurable cardinal's ultrafilter, one constructs a "tower" of measures that allows the translation of the game on $A$ into a game on a simpler (closed) set in an iterated ultrapower. Closed games are determined (Gale-Stewart), and the winning strategy transfers back.

Higher levels via Woodin cardinals: For $\boldsymbol{\Sigma}^1_{n+1}$ determinacy, one needs $n$ Woodin cardinals. The key innovation (Martin-Steel) is the use of iteration trees: sequences of ultrapowers that branch according to the structure of the projective formula defining $A$ . The Woodin cardinals provide enough "room" for these iterations.

The proof shows that if player I has no winning strategy in $G_A$ , then player II can use the iteration tree structure to construct a counter-strategy, leveraging the embedding properties guaranteed by the Woodin cardinals. $\blacksquare$

■

Converse Direction

RemarkNecessity of Woodin cardinals

Woodin proved the converse: PD implies the consistency of infinitely many Woodin cardinals. More precisely, if $\boldsymbol{\Pi}^1_{n+1}$ determinacy holds, then there is an inner model with $n$ Woodin cardinals. This establishes the equiconsistency:

$\text{PD} \equiv_{\text{con}} \text{infinitely many Woodin cardinals.}$

This is one of the most celebrated results in modern set theory.

ExampleApplications of projective determinacy

Under PD:

Every projective set is Lebesgue measurable and has the Baire property.
The uniformization theorem holds for all projective sets: every projective relation can be uniformized by a projective function.
$\boldsymbol{\Sigma}^1_{2n+1}$ sets have the scale property, giving a detailed structural analysis.
The theory of projective sets under PD is as "complete" and well-behaved as the theory of Borel sets.