Subcomplete forcing principles and definable well-orders

TL;DR

This paper explores how certain forcing axioms and maximality principles in set theory imply the existence of definable well-orders of the power set of ω₁, connecting forcing, inner models, and definability.

Contribution

It introduces new implications of maximality principles for subcomplete forcing, establishing conditions under which definable well-orders of al(\u03c0) exist, and develops stronger bounded forcing axioms.

Findings

Maximality principles imply definable well-orders of al() under certain assumptions.

Absence of inner models with inaccessible limits of measurable cardinals leads to definability results.

Enhanced bounded forcing axioms can replicate the implications of maximality principles.

Abstract

It is shown that the boldface maximality principle for subcomplete forcing, together with the assumption that the universe has only set-many grounds, implies the existence of a (parameter-free) definable well-ordering of $\mathcal{P}(\omega_1)$. The same conclusion follows from the boldface maximality for subcomplete forcing, assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that $x^\#$ does not exist, for some $x\subseteq\omega_1$, implies the existence of a well-order of $\mathcal{P}(\omega_1)$ which is $\Delta_1$-definable without parameters, and $\Delta_1(H_{\omega_2})$-definable using a subset of $\omega_1$ as a parameter. This well-order is in $L(\mathcal{P}(\omega_1))$. Enhanced version of bounded forcing axioms are introduced that are strong enough to have the implications of the maximality principles mentioned above.