Canonical structure in the universe of set theory: part one

Abstract. We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of "singular cardinal compactness" for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterize the tight structures in terms of PCF theory and use our characterisation to prove some covering results for tight structures, along with some results on tightness and stationary reflection. Finally we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals.

• Type: Journal
• Year: 2004
• Where: APAL
• Authors: James Cummings, Matthew Foreman, Menachem Magidor
• Keywords: APAL 2004, Approachable Points, Singular Cardinal Compactness, Tight Structures