Logic Seminar

For a countable Borel equivalence relation E we consider the weak choice principle every countable sequence of E-classes has a choice function''. We establish a relationship between ergodicity of the equivalence relations and the study of these choice principles. We will separate these choice principles as follows: if E is F-ergodic (with respect to some measure) then there is a model of set theory in which choice for E classes'' fails yet choice for F classes'' holds. For example, choice for E_\infty classes'' is strictly stronger than choice for E_0 classes''. A key lemma in the proof is the following statement: if E is F-ergodic with respect to an E-quasi-invariant measure \mu then the countable power of E, E^\omega, is F-ergodic with respect to the product measure \mu^\omega. The proof relies on ideas from the study of weak choice principles.

In this talk we will go over some of the basic ideas behind forcing and explain how they were used to construct models of set theory without the axiom of choice. We will then establish the relationship with ergodicity and focus on proving the lemma mentioned above.