Axiom of projective determinacy
Encyclopedia
In mathematical logic
, projective determinacy is the special case of the axiom of determinacy
applying only to projective sets.
The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect information of length ω in which the players play natural number
s, 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.
The axiom is independent of ZFC (assuming that it is consistent with ZFC), unlike the full axiom of determinacy (AD), which contradicts the Axiom of Choice. PD follows from certain large cardinal axioms, such as the existence of infinitely many Woodin cardinal
s.
PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfect set property
and the property of Baire. It also implies that every projective binary relation
may be uniformized
by a projective set.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, projective determinacy is the special case of 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...
applying only to projective sets.
The axiom of projective determinacy, abbreviated PD, states that for any two-player game of perfect information of length ω in which the players play natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, 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.
The axiom is independent of ZFC (assuming that it is consistent with ZFC), unlike the full axiom of determinacy (AD), which contradicts the Axiom of Choice. PD follows from certain large cardinal axioms, such as the existence of infinitely many Woodin cardinal
Woodin cardinal
In set theory, a Woodin cardinal is a cardinal number λ such that for all functionsthere exists a cardinal κ In set theory, a Woodin cardinal is a cardinal number λ such that for all functions...
s.
PD implies that all projective sets are Lebesgue measurable (in fact, universally measurable) and have the perfect set property
Perfect set property
In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset...
and the property of Baire. It also implies that every projective binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
may be uniformized
Uniformization (set theory)
In set theory, the axiom of uniformization, a weak form of the axiom of choice, states that if R is a subset of X\times Y, where X and Y are Polish spaces,...
by a projective set.