AD+

{{Short description|Set theory axiom extension}}

{{DISPLAYTITLE: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.