Layered Posets and Kunen’s Universal Collapse

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).

2 For any $n\in \omega$, the tree property at ${\omega }_{n+3}$ is consistent with Chang’s conjecture $({\omega }_{n+3},{\omega }_{n+1})\twoheadrightarrow ({\omega }_{n+1},{\omega }_n)$ (Theorem 9).