Bounded forcing axioms and Baumgartner's conjecture

TL;DR

This paper explores the hierarchy of bounded forcing axioms related to $eta$-proper and Axiom A forcings, establishing their connections with weak club guessing principles and resolving Baumgartner's conjecture.

Contribution

It characterizes the class of forcings embeddable into $\sigma$-closed followed by ccc iterations using Baumgartner's Axiom A, confirming a longstanding conjecture.

Findings

Bounded forcing axioms are connected to weak club guessing principles.

Weak club guessing principles distinguish between different $eta$-proper axioms.

The paper proves Baumgartner's conjecture on forcings embeddable into $\sigma$-closed then ccc iterations.

Abstract

We study the spectrum of forcing notions between the iterations of $\sigma$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of $\alpha$-proper forcings and connect them to a hierarchy of weak club guessing principles. We show that they are, in a sense, dual to each other. In particular, these weak club guessing principles separate the bounded forcing axioms for distinct countable indecomposable ordinals. In the study of forcings completely embeddable into an iteration of $\sigma$-closed followed by ccc forcing, we present an equivalent characterization of this class in terms of Baumgartner's Axiom A. This resolves a well-known conjecture of Baumgartner from the 1980's.