Characterizing large cardinals in terms of layered posets

TL;DR

This paper characterizes large cardinal properties, such as weak compactness and inaccessibility, using the concept of layered partial orders, linking combinatorial properties of posets to fundamental set-theoretic cardinals.

Contribution

It establishes new characterizations of weakly compact and inaccessible cardinals via layered posets, connecting combinatorial properties to large cardinal axioms.

Findings

Weak compactness characterized by $orall$ partial orders with $ ext{kappa}$-chain condition are $ ext{kappa}$-stationarily layered.

Similar characterization for strongly inaccessible cardinals.

The statement that all $ ext{kappa}$-Knaster partial orders are $ ext{kappa}$-stationarily layered implies $ ext{kappa}$ is Mahlo and stationary subsets reflect.

Abstract

Given an uncountable regular cardinal $\kappa$, a partial order is $\kappa$-stationarily layered if the collection of regular suborders of $\mathbb{P}$ of cardinality less than $\kappa$ is stationary in $\mathcal{P}_\kappa(\mathbb{P})$. We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal $\kappa$ is weakly compact if and only if every partial order satisfying the $\kappa$-chain condition is $\kappa$-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all $\kappa$-Knaster partial orders are $\kappa$-stationarily layered implies that $\kappa$ is a Mahlo cardinal and every stationary subset of $\kappa$ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.