The Determinacy of Long Games
Suppose that two players play the following game. A subset C of the unit interval I is given and the players take turns in choosing digits 0 or 1 to produce a real number 0,x1 x2 x3… The objective of the first player is to keep the resulting real number in C, while the second one endeavours to avoid it. This is very much reminiscent of the plays of two persons with full information (only the play sequences are infinite) but the outcome can be radically different. While in “short games”, there is always a winning strategy (or, depending on the evaluation, a non-losing one) for one of the two players, there may be none in the “long games” case. Using the axiom of choice, it is not hard to produce a C such that the game is not determined; if, however, the target set C is “nice”, a winning strategy does exist. These two facts marked the two main areas in which the long games theory substantially influenced set theory and logic.
On the one hand, Mycielski and Steinhaus (1962) started studying set theories in which the axiom of choice (AC) is replaced by the assumption (contradicting AC, of course) that each infinite game of the type mentioned above is determined (the Axiom of Determinacy, or Mycielski Axiom; this has very interesting consequences: for instance, every subset of the real line is measurable). On the other hand, there is a flourishing theory keeping the choice and studying the nature of the set C. This led to very involved theories featuring rather advanced techniques (extenders, iteration trees, big cardinals). The book contains the author's recent results in this second branch of the long games theory. It is intended for well-informed specialists (the reader should follow the author's advice concerning prerequisites in the preface). However, there is an excellent extensive introduction presenting a view of the theory that would be profitable for a non-specialist as well.