Layered posets and Kunen's universal collapse

TL;DR

This paper introduces layered posets to prove a new iteration theorem that simplifies chain condition arguments for universal Kunen iterations, with applications to saturated ideals and the Tree Property.

Contribution

It develops the theory of layered posets and proves an iteration theorem that generalizes chain condition arguments for universal Kunen iterations.

Findings

Universal Kunen iterations of c posets are c under certain conditions.

Various c posets can be absorbed by quotients of saturated ideals.

The Tree Property at c is consistent with Chang's Conjecture.

Abstract

We develop the theory of layered posets, and use the notion of layering to prove a new iteration theorem (Theorem 6): if $\kappa$ is weakly compact then any universal Kunen iteration of $\kappa$-cc posets (each possibly of size $\kappa$) is $\kappa$-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications: (1) For any $n \ge 1$, a wide variety of $<\omega_{n-1}$-closed, $\omega_{n+1}$-cc posets of size $\omega_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on $\omega_n$ (see Theorem 7 and Corollary 8); and (2) For any $n \in \omega$, the Tree Property at $\omega_{n+3}$ is consistent with the Chang's Conjecture $(\omega_{n+3}, \omega_{n+1}) \twoheadrightarrow (\omega_{n+1}, \omega_n)$ (Theorem 9).