Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The directedness of the Rudin-Keisler order at measurable cardinals

Yair Hayut, Alejandro Poveda

TL;DR

This work identifies a bridge between the directedness of the Rudin-Keisler order on $\kappa$-complete ultrafilters and a Gluing Property (GP) constructed via a non-stationary Prikry-type forcing. By developing the Gluing Poset and a Coding Lemma, the authors produce models where $\kappa$—measurable with $o(\kappa)=\lambda^+$ or $o(\kappa)=\kappa$—exhibits strong compactness-like directedness for $\mathfrak{U}_{\kappa}$ at a much lower consistency strength than a full $\kappa$-compact cardinal. The main theorem shows equiconsistency between the RK-directedness of $\mathfrak{U}_{\kappa}$ and large-cardinal data about $o(\kappa)$, thereby offering the first compactness-type property at the measurable level with relatively weak assumptions. The Coding Lemma yields a complete map of $\kappa$-complete ultrafilters in the gluing extension, enabling precise control of ultrafilter structures and a negative result in Gitik’s model (GP fails). Open problems point to the precise strength of various GP variants and their connections to strong cardinals.

Abstract

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $λ\leq κ$, the following theories are equiconsistent modulo ZFC: (1) $κ$ is a measurable cardinal with $o(κ)=λ^+$ (resp. $o(κ)=κ$). (2) The Rudin-Keisler order restricted to the set of $κ$-complete (non-principal) ultrafilters on $κ$ is $λ^+$-directed (resp. $κ^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $λ$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed.

The directedness of the Rudin-Keisler order at measurable cardinals

TL;DR

This work identifies a bridge between the directedness of the Rudin-Keisler order on -complete ultrafilters and a Gluing Property (GP) constructed via a non-stationary Prikry-type forcing. By developing the Gluing Poset and a Coding Lemma, the authors produce models where —measurable with or —exhibits strong compactness-like directedness for at a much lower consistency strength than a full -compact cardinal. The main theorem shows equiconsistency between the RK-directedness of and large-cardinal data about , thereby offering the first compactness-type property at the measurable level with relatively weak assumptions. The Coding Lemma yields a complete map of -complete ultrafilters in the gluing extension, enabling precise control of ultrafilter structures and a negative result in Gitik’s model (GP fails). Open problems point to the precise strength of various GP variants and their connections to strong cardinals.

Abstract

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals , the following theories are equiconsistent modulo ZFC: (1) is a measurable cardinal with (resp. ). (2) The Rudin-Keisler order restricted to the set of -complete (non-principal) ultrafilters on is -directed (resp. -directed). The theorem reported here is proved after bridging the directedness of the RK-order with the -Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the -Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be -directed.
Paper Structure (15 sections, 42 theorems, 188 equations, 1 figure)

This paper contains 15 sections, 42 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ultrafilters
  4. Forcing
  5. Inner model theory
  6. The Gluing Property
  7. The Gluing Property and the Rudin-Keisler order
  8. The Gluing Poset
  9. Characterizing ultrafilters in the gluing extension
  10. Preliminary comments and technical lemmas
  11. Proving the coding lemma
  12. Proving the Key claim
  13. The $\kappa$-directednnes of $\langle \mathfrak{U}_\kappa,\leq_{\mathrm{RK}}\rangle$ from optimal assumptions
  14. The failure of the Gluing Property in Gitik's model
  15. Open problems

Key Result

Theorem 1

Let $\kappa$ be a measurable cardinal and $\lambda\leq 2^\kappa$ be a cardinal. Then, the following statements are equivalent: Moreover, if $\lambda\leq \kappa$ then (1) and (2) above are, respectively, equivalent to

Figures (1)

  • Figure 1: $o^\ell(\alpha)=2.$

Theorems & Definitions (144)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2: Non-stationary support iteration, BenUng
  • Lemma 2.3: BenUng
  • Lemma 2.4: BenUng
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • Definition 2.7: Fast Function Forcing
  • ...and 134 more