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Two-cardinal derived topologies, indescribability and Ramseyness

Brent Cody, Chris Lambie-Hanson, Jing Zhang

Abstract

We introduce a natural two-cardinal version of Bagaria's sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Peter Holy and Philip White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

Two-cardinal derived topologies, indescribability and Ramseyness

Abstract

We introduce a natural two-cardinal version of Bagaria's sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Peter Holy and Philip White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.
Paper Structure (13 sections, 40 theorems, 89 equations)

This paper contains 13 sections, 40 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Strong stationarity and weak clubs
  3. Two-cardinal derived topologies and $\xi$-strong stationarity
  4. A generalization of the order topology to $P_\kappa X$
  5. Definitions of derived topologies and iterated stationarity in $P_\kappa X$
  6. The $\tau_1$ topology on $P_\kappa X$
  7. The $\tau_\xi$ topology on $P_\kappa X$ for $\xi \geq 2$
  8. Consequences of $\Pi^1_\xi$-indescribability
  9. Variations
  10. On Ramseyness and indescribability
  11. New results on Ramseyness and indescribability
  12. New results on two-cardinal Ramseyness
  13. Questions and ideas

Key Result

Proposition 2.2

If $\kappa$ is weakly inaccessible, $X \supseteq \kappa$ is a set of ordinals, and $C$ is a club in $P_\kappa X$, then $C$ is a weak club in $P_\kappa X$.

Theorems & Definitions (73)

  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 63 more