Canonical structure in the universe of set theory: Part one

James Cummings*, Matthew Foreman, Menachem Magidor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

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 characterise 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.

Original languageEnglish
Pages (from-to)211-243
Number of pages33
JournalAnnals of Pure and Applied Logic
Volume129
Issue number1-3
DOIs
StatePublished - Oct 2004
Externally publishedYes

Keywords

  • Approachable ordinal
  • Covering properties
  • Good ordinal
  • Internally approachable structure
  • Mutual stationarity
  • PCF theory
  • Precipitous ideal
  • Square sequence
  • Stationary reflection
  • The ideal I[λ]
  • Tight structure

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