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

v3 · 2026-09-19 · History

Domain-specific #

1239

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomain

Set Theory → Mathematics

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.

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.

How would you explain it like I'm…

 

No faithful explanation at this level. All three generators judged the rung impossible: any five-year-old picture reduces AD+ to "a rule that someone always wins, only more so", collapsing it into the informal "more determinacy" reading the definition explicitly rules out (and confusing determinacy with every play having a winner), when AD+ is a specific package of dependent choice for reals, ordinal determinacy and universal infinity-Borel presentation.

The Endless Games Bundle

Some mathematicians study imaginary games where two players take turns picking numbers forever, and a chosen collection of endless number lists decides who wins. A rule called the Axiom of Determinacy says that in every such game, one of the players has a sure way to win. AD+ is a particular bundle of assumptions built on that rule. It allows a limited kind of making choices one after another, it says games where players pick from much bigger supplies of counting numbers (ones that keep going past infinity) also always have a sure winner, and it says every collection of real numbers can be built by a very long step-by-step recipe. The "plus" is the name of that exact bundle, not just "more of the same rule."

Woodin's Determinacy Package

In set theory, the Axiom of Determinacy (AD) is an assumption about infinite two-player games in which players alternately pick natural numbers, producing an infinite sequence (a 'real'); AD says every such game, defined by any set of reals as the winning condition, is determined, meaning one player has a winning strategy. AD+ is W. Hugh Woodin's strengthening of AD, used to study sets of reals. In a standard formulation it bundles three things. First, dependent choice for reals, a weak form of the axiom of choice. Second, ordinal determinacy, which extends determinacy to games whose moves are coded by ordinals below a certain bound and then translated into reals by continuous maps. Third, the assertion that every set of reals is infinity-Borel, meaning it has a certain kind of generalized Borel description. The plus sign names this specific package, not a vague idea of 'extra determinacy'.

 

AD+ is W. Hugh Woodin's strengthening of the Axiom of Determinacy, formulated for the descriptive set theory of sets of reals. In a standard presentation it consists of three components. The first is dependent choice for reals, a weak choice principle compatible with determinacy. The second is ordinal determinacy: for every ordinal λ below Θ, every continuous map π from λ^ω to ω^ω, and every set A of reals, the preimage π⁻¹[A] is determined, so determinacy extends to games whose plays are ω-sequences of ordinals, pulled back through continuous maps. The third is that every set of reals is infinity-Borel, meaning it admits an infinitary Borel-style code. Together these give determinacy strengthened by ordinal-coded games, a universal infinity-Borel presentation, and a weak-choice background. The plus refers to exactly this package, and a theory lacking any of the three components is not AD+ even if it assumes strong determinacy.

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.

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

1. State the ambient model and base theory.
2. Verify the intended formulation of \(\mathrm{DC}_{\mathbb R}\).
3. Define the represented reals, \(\Theta\), and the relevant pointclasses.
4. Separate the infinity-Borel and ordinal-determinacy clauses.
5. For an ordinal game, identify \(\lambda<\Theta\), its topology, \(\pi\), and the payoff set.
6. For a coding claim, exhibit the formula/set-of-ordinals form.
7. Track which consequence uses AD alone and which uses the plus clauses.
8. 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.

Relationships to Other Abstractions

Current abstraction AD+ Domain-specific

Parents (1) — outgoing typed relations

• AD+ is a kind of Axiom Prime

  Axiom is the broader abstraction this entry instantiates.

Hierarchy path (1) — routes to 1 parentless root

• AD+ → is a kind of Axiom → is a kind of Epistemic Mode Of A Proposition

Neighborhood in Abstraction Space

AD+ sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Vector Bundles & Classifying Constructions (11 abstractions)

Nearest neighbors

• Cylinder Set Measure — 0.79
• Predual — 0.78
• A-paracompact Space — 0.78
• Daniell Integral — 0.78
• Property of Baire — 0.78

Computed from structural-signature embeddings · 2026-10-25