# Mathematical Logic Quarterly

## Subcomplete forcing principles and definable well‐orders

### Abstract

It is shown that the boldface maximality principle for subcomplete forcing, ${\mathsf{MP}}_{\mathsf{SC}}\left({\mathbf{H}}_{{\omega }_{2}}\right)$, together with the assumption that the universe has only set many grounds, implies the existence of a well‐ordering of $\wp \left({\omega }_{1}\right)$ definable without parameters. The same conclusion follows from ${\mathsf{MP}}_{\mathsf{SC}}\left({\mathbf{H}}_{{\omega }_{2}}\right)$, 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 $x\subseteq \omega$, implies the existence of a well‐ordering of $\wp \left({\omega }_{1}\right)$ which is Δ1‐definable without parameters, and ${\mathrm{\Delta }}_{1}\left({\mathbf{H}}_{{\omega }_{2}}\right)$‐definable using a subset of ω1 as a parameter. This well‐order is in $\mathbf{L}\left(\wp \left({\omega }_{1}\right)\right)$. Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications of ${\mathsf{MP}}_{\mathsf{SC}}\left({\mathbf{H}}_{{\omega }_{2}}\right)$ mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed.

View all

View all