AD plus
Encyclopedia
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 DCR (the axiom of dependent choice
for real number
s), states two things:
The second clause by itself is referred to as ordinal determinacy.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, AD+ is an extension, proposed by W. Hugh Woodin
W. Hugh Woodin
William Hugh Woodin is an American mathematician and set theorist at University of California, Berkeley. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinal, the Woodin cardinal, bears his name.-Biography:Born in Tucson, Arizona, Woodin...
, to the axiom of determinacy
Axiom of determinacy
The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
. The axiom, which is to be understood in the context of ZF plus DCR (the axiom of dependent choice
Axiom of dependent choice
In mathematics, the axiom of dependent choices, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis...
for real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s), states two things:
- Every set of reals is ∞-Borel.
- For any ordinal λ less than Θ, any subset A of ωω, and any continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
π:λω→ωω, the preimage π-1[A] is determinedDeterminacyIn set theory, a branch of mathematics, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies.-Games:...
. (Here λω is to be given the product topologyProduct topologyIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
, starting with the discrete topology on λ.)
The second clause by itself is referred to as ordinal determinacy.