The determinacy of long games


Itay Neeman



Table of contents


Backcover text:


In this volume the author develops and applies methods for proving,

from large cardinals, the determinacy of definable games of countable

length on natural numbers. The determinacy is ultimately derived

from iteration strategies, connecting games on natural numbers with

the specific iteration games that come up in the study of large cardinals.


The games considered in this text range in strength, from games of

fixed countable length, through games where the length is clocked

by natural numbers, to games in which a run is complete when its

length is uncountable in an inner model (or a pointclass) relative to

the run. More can be done using the methods developed here,

reaching determinacy for games of length $\omega_1$.


The book is largely self-contained. Only graduate level knowledge

of modern techniques in large cardinals and basic forcing is assumed.

Several exercises allow the reader to build on the results in the text,

for example connecting them with universally Baire and

homogeneously Suslin sets. Overall it is intended that the book

should be accessible both to specialists and to advanced graduate

students in set theory