Canonical structure in the universe of set theory: part two

James Cummings*, Matthew Foreman, Menachem Magidor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 (1-3) (2004) 211-243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness.

Original languageEnglish
Pages (from-to)55-75
Number of pages21
JournalAnnals of Pure and Applied Logic
Volume142
Issue number1-3
DOIs
StatePublished - Oct 2006

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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