axiom of determinacy
When doing descriptive set theory, it is conventional to use either or as your space of “reals” (these spaces are homeomorphic to the irrationals and the Cantor set, respectively). Throughout this article, I will use the term “reals” to refer to .
Let be given and consider the following game on played between two players, I and II: I starts by saying a natural number; II hears this number and replies with another (or possibly the same one); I hears this and replies with another; etc. The sequence of numbers said (in the order they were said) is a point in . I wins if this point is in , otherwise II wins.
A map is said to be a winning strategy for I if it has the following property: if, after the play has gone , I plays for each move, then I wins. A winning strategy for II is defined analogously.
The axiom of determinacy (AD) states that every such game is determined, that is either I or II has a winning strategy.
Using choice, a non-determined game can be constructed directly: for , enumerate the uncountable closed subsets of the reals . Now construct two sequences and by choosing as distinct points from which are not in (this is possible as each uncountable closed set has cardinality ). Then the game on the set of all s is non-determined.
From ZF+AD, one may prove many nice facts about the reals, such as: any subset is Lebesgue measurable, any subset has a perfect subset and the continuum hypothesis. ZF+AD also proves the axiom of countable choice.
AD itself is not taken seriously by many set theorists as a genuine alternative to choice. However, there is a weakening of AD (the axiom of quasi-projective determinacy, or QPD, which states that all games in are determined) which is consistent with ZFC (in fact, it’s equiconsistent to a large cardinal axiom) which is a serious axiom candidate.
|Title||axiom of determinacy|
|Date of creation||2013-03-22 14:50:46|
|Last modified on||2013-03-22 14:50:46|
|Last modified by||CWoo (3771)|