Set-theoretic inevitability

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