DeterminacyDeterminacy is a subfield of set theory, a branch of mathematics, 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, "determinacy" is the property of a game whereby such a strategy exists. Determinacy was introduced by Gale and Stewart in 1950, under the name "determinateness". The games studied in set theory are usually Gale–Stewart games—two-player games of perfect information in which the players make an infinite sequence of moves and there are no draws.
PointclassIn the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of definability property; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.
Axiom of determinacyIn 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 of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers.
Descriptive set theoryIn mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic. Descriptive set theory begins with the study of Polish spaces and their Borel sets.
Arithmetical hierarchyIn mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946). The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.
