We may define the set of "first projective" subsets of Rn to be the set of all subsets which are projections of Borel subsets of Rn+1; then we may define "second projective" sets as projections of first projective sets or complements thereof, and so on. A set is said to be projective if it belongs to some level of this hierarchy.
The axiom of projective determinacy states that for any Banach_Mazur game on the real numbers, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then the game has a winning strategy.
The axiom is undecidable in ZFC, unlike the full axiom, which contradicts the Axiom of Choice; it follows from certain large cardinal axioms, such as the existence of infinitely many Woodin cardinals.
In mathematical logic, projectivedeterminacy is the special case of the axiom of determinacy applying only to projective sets.
The axiom of projectivedeterminacy, abbreviated PD, states that for any two-player game of perfect information of length ω in which the players play natural numbers, if the victory set (for either player, since the projective sets are closed under complementation) is projective, then one player or the other has a winning strategy.
PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfect set property and the property of Baire.
Feeds |
Contact