Published online by Cambridge University Press: 21 March 2017
Let $\left\langle {{W_n}:n \in \omega } \right\rangle$ be a canonical enumeration of recursively enumerable sets, and suppose T is 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 on T) with the property that if
${\cal M}$ is a countable model of T and for some
${\cal M}$-finite set s,
${\cal M}$ satisfies
${W_e} \subseteq s$, then
${\cal M}$ has an end extension
${\cal N}$ that satisfies T + We = s.
Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment ${{\rm{I}\rm{\Sigma }}_1}$ of PA, and remove the countability restriction on
${\cal M}$ when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.