Hierarchies of forcing axioms II

Abstract

A $\Sigma^2_1$ truth for $\lambda$ is a pair $\langle Q,\psi\rangle$ so that $Q\subseteq H_\lambda, \psi$ is a first order formula with one free variable, and there exists $B\subseteq H_{\lambda+}$ such that $(H_{\lambda+} ; \in B) \models \psi[Q]$. A cardinal $\lambda$ is $\Sigma^2_1$ indescribable just in case that for every $\Sigma^2_1$ truth $\langle Q,\psi\rangle$ for $\lambda$, there exists $\overline{\lambda}$ < $\lambda$ so that $\overline{\lambda}$ is a cardinal and $\langle Q \cap H_{\overline{\lambda}}, \psi \rangle$ is a $\Sigma^2_1$ truth for $\overline{\lambda}$. More generally, an interval of cardinals $[\kappa, \lambda]$ with $\kappa \leq \lambda$ is $\Sigma^2_1$ indescribable if for every $\Sigma^2_1$ truth $\langle Q,\psi\rangle$ for $\lambda$, there exists $\overline{\kappa} \leq \overline{\lambda} < \kappa, \overline{Q} \subseteq H_{\overline{\lambda}}$, and $\pi: H_{\overline{\lambda}} \rightarrow H_{\lambda}$ so that $\overline{\lambda}$ is a cardinal, $\langle \overline{Q},\psi\rangle$ is a $\Sigma^2_1$ truth for $\overline{\lambda}$, and $\pi$ is elementary from $(H_{\overline{\lambda}} ; \in, \overline{\kappa}, \overline{Q})$ into $(H_{\lambda}; \in, \kappa, Q)$ with $\pi \upharpoonright \overline{\kappa} =$ id. We prove that the restriction of the proper forcing axiom to $\mathfrak{c}-linked posets requires a $\Sigma^2_1$ indescribable cardinal in L, and that the restriction of the proper forcing axiom to $\mathfrak{c}^+$-linked posets, in a proper forcing extension of a fine structural model, requires a $\Sigma^2_1$ indescribable 1-gap $[\kappa, \kappa^+]$. These results show that the respective forward directions obtained in Hierarchies of Forcing Axioms I by Neeman and Schimmerling are optimal.