Descriptive set theory and forcing; How to prove theorems about Borel sets the hard way

TL;DR

This paper provides lecture notes on descriptive set theory, focusing on Borel hierarchies, analytic sets, and forcing techniques, offering new results and simplified proofs in the context of set theory and forcing methods.

Contribution

It introduces new results on Borel hierarchy lengths in specific models and demonstrates how forcing can simplify proofs of classical theorems.

Findings

Unpublished theorem of Fremlin on Borel hierarchies and MA

New results on Borel hierarchy lengths in Cohen and random real models

Simplified proof of Louveau's theorem using forcing

Abstract

These are lecture notes from a course I gave at the University of Wisconsin during the Spring semester of 1993. Part 1 is concerned with Borel hierarchies. Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies and MA. Section 14 and 15 contain new results concerning the lengths of Borel hierarchies in the Cohen and random real model. Part 2 contains standard results on the theory of Analytic sets. Section 25 contains Harrington's Theorem that it is consistent to have $\Pi^1_2$ sets of arbitrary cardinality. Part 3 has the usual separation theorems. Part 4 gives some applications of Gandy forcing. We reverse the usual trend and use forcing arguments instead of Baire category. In particular, Louveau's Theorem on $\Pi^0_\alpha$ hyp-sets has a simpler proof using forcing.