An Introduction to Balanced Forcing.

Solovay’s model in which all sets of reals are Lebesgue measurable is built by collapsing a strongly inaccessible cardinal to be the first uncountable cardinal, and then considering the class of all sets which are hereditarily ordinal definable from a real number and a set from the ground model. Balanced forcing is a technique for forcing over a Solovay model to recover some forms of the Axiom of Choice while preserving the failure of others. A partial order is balanced if it carries a dense set of balanced virtual conditions, and forms of the Axiom of Choice are classified by their amalgamation properties over families of models of ZFC. The examples considered so far include nonprincipal ultrafilters on the integers, selectors for Borel equivalence relations, discontinuous homomorphisms, Hamel bases and countable colorings for Borel graphs, among many others. We will present some of the fundamental results in this area, and discuss some open problems.