Games on Boolean algebras

Abstract

The method of forcing is widely used in set theory to obtain various consistency proofs. Complete Boolean algebras play the main role in applications of forcing. Therefore it is useful to define games on Boolean algebras that characterize their properties important for the method. The most investigated game is Jech’s distributivity game, such that the first player has the winning strategy iff the algebra is not (ω, 2)-distributive. We define another game characterizing the collapsing of the continuum to ω, prove several sufficient conditions for the second player to have a winning strategy, and obtain a Boolean algebra on which the game is undetermined.

Forsing je metod široko korišćen u teoriji skupova za dokaze konsistentnosti. Kompletne Bulove algebre igraju glavnu ulogu u primenama forsinga. Stoga je korisno definisati igre na Bulovim algebrama koje karakterišu njihove osobine od značaja za taj metod. Najbolje proučena je Jehova igra, koja ima osobinu da prvi igrač ima pobedničku strategiju akko algebra nije (ω, 2)-distributivna. U tezi definišemo još jednu igru, koja karakteriše kolaps kontinuuma na ω, dokazujemo nekoliko dovoljnih uslova da bi drugi igraš imao pobedničku strategiju, i konstruišemo Bulovu algebru na kojoj je igra neodređena.