AD+

In set theory, AD⁺ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DC_R (the axiom of dependent choice for real numbers), states two things:

1. Every set of real numbers is ∞-Borel.
2. For any ordinal λ < Θ, any A ⊆ ω^ω, and any continuous function π: λ^ω → ω^ω, the preimage π⁻¹[A] is determined. (Here, λ^ω is to be given the product topology, starting with the discrete topology on λ.)

The second clause by itself is referred to as ordinal determinacy.

See also

• Axiom of projective determinacy
• Axiom of real determinacy
• Suslin's problem
• Topological game

References

• Larson, Paul B. (2023). Extensions of the Axiom of Determinacy (1st ed.). Providence, RI: American Mathematical Society. p. 165. ISBN 978-1-4704-7210-8.
• Woodin, W. Hugh (1999). The axiom of determinacy, forcing axioms, and the nonstationary ideal (1st ed.). Berlin: W. de Gruyter. p. 618. ISBN 311015708X.