Subcomplete forcing principles and definable well-orders

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.