Comparing forcing approaches to dense ideals

TL;DR

This paper analyzes forcing notions that generate dense ideals on successor cardinals, focusing on the Anonymous Collapse $\mathbb{A}(\omega_1,\kappa)$ and the Dual Shioya Collapse $\text{W}(\omega_1,\kappa)$. It proves a forcing-equivalence result for inaccessible $\kappa$ (i.e., $\mathbb{A}(\omega_1,\kappa)$ and $\text{W}(\omega_1,\kappa)$ are interchangeable in forcing arguments) and develops a canonical canonicity result for collapsing with $\sigma$-strategically closed posets, showing such forcings collapse to $\mathrm{Col}(\kappa,|D|)$ when a dense $D$ of size $|D|$ exists. The article then provides an alternative pathway to produce a dense ideal on $\omega_2$ by combining $\text{W}(\omega_1,\kappa)$ with collapses, culminating in a normal ideal $I$ on $\omega_2$ with $\mathcal{P}(\omega_2)/I \cong \mathrm{Col}(\omega_1,\omega_2)$. Overall, it clarifies the forcing-theoretic relationship between these a priori different constructions and shows how they yield dense/quotient-isomorphic structures, with implications for lifting almost-huge embeddings.

Abstract

We analyze some posets involved in forcing constructions for dense ideals, showing that the Anonymous Collapse and the Dual Shioya Collapse are equivalent for collapsing a large cardinal to $ω_2$. We also give a somewhat simplified construction of a normal ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I \sim \mathrm{Col}(ω_1,ω_2)$.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3: Ishiu-Yoshinobu iy
• Lemma 4: Ishiu-Yoshinobu iy
• Lemma 5
• proof
• Lemma 6
• Definition
• Definition
• Lemma 7