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Dense ideals

Monroe Eskew, Yair Hayut

Abstract

In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a $σ$-complete, $\aleph_1$-dense ideal on $\aleph_{n+1}$ for every $n < ω$, answering a question of Foreman. Using this construction we show the consistency of the existence of various irregular ultrafilters on $ω_n$, the consistency of the Foreman-Laver reflection property for the chromatic number of graphs for all possible pairs of cardinals below $\aleph_ω$, and the simultaneous consistency of the partition hypotheses $\mathrm{PH}_n(ω_m)$ for $n < m$.

Dense ideals

Abstract

In this paper, we obtain the consistency, relative to large cardinals, of the existence of dense ideals on every successor of a regular cardinal simultaneously. Using a consequent transfer principle, we show that in this model there is a -complete, -dense ideal on for every , answering a question of Foreman. Using this construction we show the consistency of the existence of various irregular ultrafilters on , the consistency of the Foreman-Laver reflection property for the chromatic number of graphs for all possible pairs of cardinals below , and the simultaneous consistency of the partition hypotheses for .

Paper Structure

This paper contains 17 sections, 60 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complete $\kappa$-closure
  4. Inverse limits
  5. Reasonable collapses
  6. Dual Shioya forcing
  7. Uniformization
  8. Dense ideals on $\omega_1$ and $\omega_2$
  9. More dense ideals
  10. Applications
  11. GCH, diamond, and reflection
  12. Boolean transfer
  13. Irregular ultrafilters
  14. The Foreman-Kunen spectrum
  15. Graph colorings
  16. ...and 2 more sections

Key Result

Theorem 1

It is consistent, relative to a huge cardinal, that for every regular cardinal $\mu$, there is a normal ideal $I$ on $\mu^+$ such that

Theorems & Definitions (126)

  • Theorem : Main Theorem
  • Corollary : Main Corollary
  • Definition 1
  • Lemma 2
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6: Folklore
  • proof
  • ...and 116 more