Determinacy

Determinacy called in set theory a property of sets of real numbers.

A real number is here understood as a countably infinite sequence of natural numbers, for example. This is possible because of the continued fraction expansion, with the help of which any irrational number is still identified with such an effect.

A set of real numbers defines a game in the following manner: Two players alternately and select one each natural number. The game ends as soon as infinitely many numbers were chosen. Through this game A and B have now but a sequence of natural numbers, so thus produces a real number. If the generated real number now, so has won, otherwise.

Is called determined if there is a winning strategy for one of the two players. In this context we mean by a winning strategy for a player is a function which is on the set of game situations in which the game is not finished yet and he's just at the train defined. The range of values ​​of this function is the set of natural numbers, ie the " tells" the player which natural number it is to play in a given game situation.

In the standard axiom system ZFC of set theory itself determinacy can be proved for all Borel sets. As an additional axioms are projective determinacy (PD ) and full determinacy (AD ) was investigated. (PD ) here means that even all projective sets of reals are determined, (AD) that all sets of reals be it. However, the latter statement contradicts the axiom of choice, so that ZF AD is examined in this case axiom system.