Set-theoretic inevitability

Abstract

After defining the concept of inevitability for statements with the modal logic of forcing, we raise the question of which statements are inevitable under $\mathsf{ZFC}$ and under large cardinal hypotheses. We show that $\mathsf{SOA}$ (defined as ${\mathsf{ZFC}}^{-}$ + $\mathsf{V}=\mathsf{HC}$) plus the ${\mathbf{\Sigma }}_2^1$-perfect set property is $\mathsf{ZFC}$-inevitable, and that Projective Determinacy is inevitable in the theory of $\mathsf{ZFC}$ with a proper class of Woodin cardinals. We then discuss the possible conceptual significance of these results for the foundations of set theory.

arXiv comment20 pages