Absolute model companionship, forcibility, and the continuum problem

Abstract: Absolute model companionship (AMC) is a strict strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a "partial Morleyization" of the $\in$-theory $\mathsf{ZFC}+$"there are class many supercompact cardinals". We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}\tau$. Finally we characterize a strong form of Woodin's axiom $(*)$ as the assertion that the first order theory of $H{\aleph_2}$ as formalized in a certain natural signature is model complete.