In
mathematical logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, projective determinacy is the special case of the
axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game o ...
applying only to
projective set
In the mathematical field of descriptive set theory, a subset A of a Polish space X is projective if it is \boldsymbol^1_n for some positive integer n. Here A is
* \boldsymbol^1_1 if A is analytic
* \boldsymbol^1_n if the complement of A, X\se ...
s.
The axiom of projective determinacy, abbreviated PD, states that for any two-player infinite game of
perfect information
Perfect information is a concept in game theory and economics that describes a situation where all players in a game or all participants in a market have knowledge of all relevant information in the system. This is different than complete informat ...
of length
ω in which the players play
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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
Determinacy is a subfield of game theory and set theory that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and similarly, "dete ...
.
The axiom is not a theorem of
ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD), which contradicts the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
, it is not known to be inconsistent with ZFC. PD follows from certain
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
axioms, such as the existence of infinitely many
Woodin cardinal
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions f : \lambda \to \lambda, there exists a cardinal \kappa < \lambda with and an Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coin ...
(in fact,
universally measurable In mathematics, a subset A of a Polish space X is universally measurable if it is measurable with respect to every complete probability measure on X that measures all Borel subsets of X. In particular, a universally measurable set of reals is ne ...
) and have the
perfect set property In the mathematical field of 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 (Kechris 1995, p. 150). Note that having the perfect set property is ...
and the
property of Baire
A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such tha ...
. It also implies that every projective
binary relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
may be
uniformized by a projective set.
PD implies that for all positive integers
, there is a largest countable
set.
[Donald A. Martin, "The largest countable this, that, and the other". Cabal seminar 79–81, Proceedings, Caltech-UCLA Logic Seminar 1979–81, edited by A. S. Kechris, D. A. Martin, and Y. N. Moschovakis, Lecture notes in mathematics, vol. 1019, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983, pp. 97–106.]
References
*
*
Citations
Axioms of set theory
Descriptive set theory
Determinacy
Large cardinals
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