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Concept
Axiom of determinacy
Summary
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 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. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research. Assuming the existence of some uncountable cardinal numbers analogous to , they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).
The axiom of determinacy refers to games of the following specific form:
Consider a subset A of the Baire space ωω of all infinite sequences of natural numbers. Two players, I and II, alternately pick natural numbers
n0, n1, n2, n3, ...
After infinitely many moves, a sequence is generated. Player I wins the game if and only if the sequence generated is an element of A. The axiom of determinacy is the statement that all such games are determined.
Not all games require the axiom of determinacy to prove them determined. If the set A is clopen, the game is essentially a finite game, and is therefore determined. Similarly, if A is a closed set, then the game is determined. It was shown in 1975 by Donald A. Martin that games whose winning set is a Borel set are determined.
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