|
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. The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies that all subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property. The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the same cardinality as the full set of reals). Furthermore, AD implies the consistency of Zermelo–Fraenkel set theory (ZF). Hence, as a consequence of the incompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF. ==Types of game that are determined== The axiom of determinacy refers to games of the following specific form: Consider a subset ''A'' of Baire space, the set of all infinite sequences of natural numbers. Two players, I and II, alternately pick natural numbers :''n''0, ''n''1, ''n''2, ''n''3, ... 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. It follows from the existence of sufficiently large cardinals that all games with winning set a projective set are determined (see Projective determinacy), and that AD holds in L(R). Equivalent to the axiom of determinacy is the statement that for every subspace ''X'' of the real numbers, the Banach-Mazur game ''BM''(''X'') is determined. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Axiom of determinacy」の詳細全文を読む スポンサード リンク
|