Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Walks along a weak square sequence and the non-semiproperness of Namba forcings

Kenta Tsukuura

TL;DR

The paper investigates how the semiproperness of two-cardinal Namba forcings $\mathrm{Nm}(\kappa,\lambda,F)$ interacts with square principles and two-cardinal combinatorics. It develops minimal walk techniques and a thin-list/ultrafilter framework to connect forcing properties with partition relations and stationary reflection, proving that if all such forcings are semiproper then $\square(\mu,{<}\aleph_1)$ fails for a broad range of $\mu$. A key outcome is a separation result: semiproperness of $\mathrm{Nm}(\kappa,\lambda,F)$ for every $F$ can imply stronger anti-compactness phenomena than semiproperness of $\mathrm{Nm}(\kappa,\lambda)$ alone, with implications for consistency strength, stationary reflection, and SCH. The work also provides model constructions showing both the preservation and failure of semiproperness under collapses and square forcings, illustrating the utility of $\mathrm{Nm}$-type forcings as a diagnostic for partial strong compactness.

Abstract

In this paper, we demonstrate that if, for every $κ$-complete fine filter $F$ over $\mathcal{P}_κλ$, the associated Namba forcing $\mathrm{Nm}(κ,λ,F)$ is semiproper, then $\square(μ,{<}\aleph_1)$ fails for all regular $μ\in [λ, 2^λ]$ under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of $\mathrm{Nm}(\aleph_2,F)$ for every $\aleph_2$-complete filter $F$ over $\aleph_2$ exceeds the strength of infinitely many Woodin cardinals. Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive $C$-sequences and show that the existence of non-reflecting stationary subsets implies $\mathcal{P}_κλ\not\to [I_{κλ}^{+}]^{3}_λ$.

Walks along a weak square sequence and the non-semiproperness of Namba forcings

TL;DR

The paper investigates how the semiproperness of two-cardinal Namba forcings interacts with square principles and two-cardinal combinatorics. It develops minimal walk techniques and a thin-list/ultrafilter framework to connect forcing properties with partition relations and stationary reflection, proving that if all such forcings are semiproper then fails for a broad range of . A key outcome is a separation result: semiproperness of for every can imply stronger anti-compactness phenomena than semiproperness of alone, with implications for consistency strength, stationary reflection, and SCH. The work also provides model constructions showing both the preservation and failure of semiproperness under collapses and square forcings, illustrating the utility of -type forcings as a diagnostic for partial strong compactness.

Abstract

In this paper, we demonstrate that if, for every -complete fine filter over , the associated Namba forcing is semiproper, then fails for all regular under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of for every -complete filter over exceeds the strength of infinitely many Woodin cardinals. Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive -sequences and show that the existence of non-reflecting stationary subsets implies .

Paper Structure

This paper contains 11 sections, 63 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Namba forcings
  4. Minimal walk methods
  5. Two-Cardinal Combinatorics
  6. A filter associated with a thin list
  7. Walks along a weak square sequence
  8. Theorems \ref{['maintheorem']} and \ref{['maintheorem:model']}
  9. Stationary reflections and $\mathrm{SCH}$
  10. Two-cardinal walks and its applications
  11. observations and questions

Key Result

Theorem 1.1

For regular cardinals $\kappa \leq \lambda$, the following are equivalent: In particular, if every $\kappa$-complete filter over $\lambda$ can be extended to a $\kappa$-complete ultrafilter, then $\square(\mu,{<}\kappa)$ fails for all $\mu \in [\lambda,2^{\lambda}] \cap \mathrm{Reg}$.

Figures (2)

  • Figure 1: The "dashed" arrow and the "dotted" arrow are Theorems \ref{['hayut']} and \ref{['maintheorem']}, respectively.
  • Figure 2: The dissection of $(\dagger)$-principle focusing on $\kappa$. Dotted arrows were proved in this paper.

Theorems & Definitions (118)

  • Theorem 1.1: Hayut MR3959249
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Shelah MR1623206
  • Theorem 2.4
  • ...and 108 more