MARGINALIA ON A THEOREM OF WOODIN
Rasmus Blanck, Ali Enayat · Journal of Symbolic Logic · 2017
Abstract Let $\left\langle {{W_n}:n \in \omega } \right\rangle$ be a canonical enumeration of recursively enumerable sets, and supposeTis a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index $e \in \omega$ (that depends onT) with the property that if ${\cal M}$ is a countable model ofTand for some ${\cal M}$ -finite sets, ${\cal M}$ satisfies ${W_e} \subseteq s$ , then ${\cal M}$ has an end extension ${\cal N}$ that satisfiesT+We=s. Here we generalize Woodin’s theorem to all recursively enumerable extensionsTof the fragment ${{\rm{I}\rm{\Sigma }}_1}$ of PA, and remove the countability restriction on ${\cal M}$ whenTextends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.