An absoluteness principle for Borel sets

Abstract

The purpose of these notes is to describe an absoluteness principle due to Jacques Stern and discuss some applications to the general study of Borel sets. This paper will not be engaged in independence results, but in proving outright theorems about the Borel hierarchy.Roughly speaking, Stern's absoluteness principle states that if a certain set can be introduced into the universe by forcing, then it can be introduced by some small forcing notion.The notation , and so on, will be defined in Section 1; this gives a notational system for describing the complexity of Borel sets beyond Fσ or Gδ. The “universe” refers to the totality of all sets. “Forcing” refers to Paul Cohen's technique for, in some sense, changing this totality by the introduction of new sets. Here “small” means relatively small cardinality.The size of this small forcing notion is roughly the ath iteration of the power set operation. Just to get an idea of what this theorem might be saying, we can argue that under certain conditions, if a closed set can be introduced by forcing, then it exists already. There are a number of other qualifications that need to be made to this rough description, and we will come to them later.Unlike, say, Shoenfield absoluteness, Stern's absoluteness can only be made understood in the terminology of forcing. Since forcing is typically associated with the pursuit of independence results, we could easily assume that Stern's work has little relevance in proving positive theorems about the Borel hierarchy.However, this would be untrue. Using abstract and indirect metamathematical arguments, and availing ourselves of Stern's absoluteness principle, we will prove a string of ZFC theorems for which no direct proof is known.