AD+¶

Woodin’s strengthening of the Axiom of Determinacy that combines dependent choice for reals, ordinal determinacy below Theta, and infinity-Borel definability for every set of reals.

Version: v2 · 2026-09-06 · History
Domain-specific #: 1239
Origin domain: mathematics
Subdomain: set theory
Aliases: AD plus, Woodin's AD+

Core Idea¶

AD+ is W. Hugh Woodin’s strengthening of the Axiom of Determinacy for the study of sets of reals. In a standard formulation, it includes dependent choice for reals, ordinal determinacy, and the assertion that every set of reals is infinity-Borel. Ordinal determinacy says that for every ordinal \(\lambda<\Theta\), every continuous map \(\pi:\lambda^\omega\to\omega^\omega\), and every \(A\subseteq\omega^\omega\), the preimage \(\pi^{-1}[A]\) is determined.^[1]

The recognition invariant is determinacy strengthened by ordinal-coded games + universal infinity-Borel presentation + weak-choice background. The plus sign denotes this package, not an informal claim that “more determinacy” is assumed.

Structural Signature¶

Background set theory without full Choice, commonly ZF.
Dependent choice restricted to relations on the reals, \(\mathrm{DC}_{\mathbb R}\).
Baire space \(\omega^\omega\) representing the reals.
\(\Theta\), the supremum of ordinals surjected onto by the reals.
Ordinals \(\lambda<\Theta\).
Product spaces \(\lambda^\omega\) with the discrete topology on \(\lambda\).
Continuous maps from \(\lambda^\omega\) to Baire space.
Determinacy of every pulled-back payoff set.
Infinity-Borel codes using a formula and a set of ordinals.
Every set of reals receiving such a code.
A framework supporting fine structural analysis of natural determinacy models.
Explicit separation from AD, projective determinacy, and determinacy for games on reals.

What It Is Not¶

It is not ordinary AD written with decorative emphasis. AD asserts that every length-omega game on natural numbers with payoff set of reals is determined; AD+ adds definability/coding and ordinal-determinacy strength whose equivalence to AD is not simply assumed.^[2]

It is not \(\mathrm{AD}_{\mathbb R}\), which concerns games whose moves are reals, and it is not Projective Determinacy, which restricts payoff sets to the projective hierarchy. It is incompatible with full Choice in the usual setting because it entails AD.

Scope of Application¶

AD+ is used in descriptive inner model theory, the Wadge hierarchy, analysis of \(L(\mathbb R)\) and related models, Suslin cardinals, scales, uniformization, HOD analysis, and derived-model arguments. Its infinity-Borel clause gives every set of reals a definition from ordinal information with useful absoluteness properties.^[3]

Claims must name the ambient model. Saying that a universe “satisfies AD+” is a model-relative foundational assertion, not a theorem of ZFC and not a statement about arbitrary external subsets of its reals.

Clarity¶

An infinity-Borel presentation is not merely membership in the ordinary Borel sigma-algebra. A set \(A\subseteq\omega^\omega\) is presented using a formula and a set of ordinals so that membership can be read in a suitable constructible model containing the code and the real.

Ordinal determinacy quantifies over continuous preimages from \(\lambda^\omega\) for every \(\lambda<\Theta\). Omitting the bound, topology, continuity, or universal quantifiers changes the principle.

Manages Complexity¶

The axiom package makes very broad sets of reals accessible to coding, scale, and determinacy methods. Instead of treating each complicated payoff set as an isolated object, it supplies a uniform bridge from ordinal information to definitions and from continuous pullbacks to winning strategies.

That power also requires bookkeeping. Consequences can depend on the exact base theory, the ambient model, additional hypotheses such as \(V=L(\mathcal P(\mathbb R))\), and whether the statement assumes AD, AD+, or \(\mathrm{AD}_{\mathbb R}\).

Abstract Reasoning¶

State the ambient model and base theory.
Verify the intended formulation of \(\mathrm{DC}_{\mathbb R}\).
Define the represented reals, \(\Theta\), and the relevant pointclasses.
Separate the infinity-Borel and ordinal-determinacy clauses.
For an ordinal game, identify \(\lambda<\Theta\), its topology, \(\pi\), and the payoff set.
For a coding claim, exhibit the formula/set-of-ordinals form.
Track which consequence uses AD alone and which uses the plus clauses.
Avoid transferring an internal model statement to the ambient universe without justification.

Knowledge Transfer¶

The portable structure is an axiom package that strengthens a base principle by coupling coverage with a uniform coding discipline. The proposed immediate parent is Axiom.

Examples¶

Infinity-Borel consequence. Under AD+, every set of reals in the ambient model has an infinity-Borel code, enabling definability arguments from ordinal parameters.^[3]

Ordinal-determinacy instance. Fix \(\lambda<\Theta\), a continuous \(\pi:\lambda^\omega\to\omega^\omega\), and a payoff set \(A\); AD+ declares the pulled-back game determined.

Non-example. Borel determinacy in ZFC proves determinacy for Borel payoff sets but is not AD+.

Structural Tensions¶

Determinacy strength versus forms of Choice.
Universal coverage versus coded definability.
Internal model truth versus ambient-universe truth.
AD consequences versus genuinely plus-dependent consequences.
Game-theoretic formulation versus inner-model applications.
Compact axiom name versus formulation-sensitive base theory.
Strong regularity of sets of reals versus foundational consistency strength.

Structural–Framed Character¶

Axiomatic strengthening, closure, coding, and scope restriction are structural. Reals, Baire space, \(\Theta\), ordinal games, infinity-Borel codes, and determinacy models are set-theoretic frame.

Structural Core vs. Domain Accent¶

The portable core is a foundational package joining a behavioral coverage principle to a representation requirement. The constitutive domain accent is the exact determinacy, ordinal, topology, coding, and weak-choice apparatus. Abstracting those away would collapse AD+ into the generic Axiom prime.

Axiom is the proposed immediate parent. Determinacy, Formal System, Coding, Representation, Choice, and Consistency are related.

The prospective queue contains one strict edge to prime:axiom. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction AD+ Domain-specific

Parents (1) — more general patterns this builds on

AD+ is a kind of Axiom Prime

Axiom is the proposed immediate parent.

Hierarchy path (1) — routes to 1 parentless root

AD+ → Axiom → Epistemic Mode Of A Proposition

Neighborhood in Abstraction Space¶

AD+ sits in a sparse region of the domain-specific corpus (91^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Set Measures & Geometric Nullity (11 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

The Axiom of Determinacy (AD) alone.
\(\mathrm{AD}_{\mathbb R}\).
Projective Determinacy.
Borel determinacy.
The statement that every set of reals is ordinary Borel.
Full Axiom of Choice.
A theorem asserted without an ambient model.

References¶

[1] Paul B. Larson, Extensions of the Axiom of Determinacy, University Lecture Series 78, American Mathematical Society, 2023. registry ↩

[2] W. Hugh Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, de Gruyter, 1999. registry ↩

[3] William Chan, Stephen Jackson, and Nam Trang, “Infinity-Borel Codes in Natural Models of AD+”, research manuscript. registry ↩a ↩b

[4] Paul B. Larson, A Brief History of Determinacy, historical and technical survey. registry ↩