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

1994/01/06 by Arnold W. Miller, Miller, Arnold · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.math/9401202

openalex publication_date 1994/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Section 13 contains an unpublished theorem of Fremlin concerning Borel hierarchies and MA.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 Π¹₂ 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 Π⁰_α hyp-sets has a simpler proof using forcing.

Cited by

• Quantum metrics on the tensor product of a commutative C*-algebra and an AF C*-algebra

Related

• Generalized Borel Sets
• A Cheerful Introduction to Forcing and the Continuum Hypothesis
• Sufficient conditions for the forcing theorem, and turning proper\n classes into sets
• Classification problems from the descriptive set theoretical perspective
• Useful axioms