Logic Workshop

There is a natural way to formulate fragments of Todorcevic’s strong reflection principle (SRP) which are associated to forcing classes more restrictive than the class of all stationary set preserving forcing notions. The fragment associated to the subcomplete forcings (SC-SRP), while retaining many crucial consequences of SRP, is compatible with CH, and even Jensen's Diamond Principle. In particular, the saturation of the nonstationary ideal, a celebrated consequence of SRP, does not follow from its subcomplete fragment. In fact, adding CH to SC-SRP results in a principle which outright contradicts the saturation of the nonstationary ideal. A specific form of diagonal reflection of stationary sets of ordinal was used by Paul Larson to separate SRP from Martin's Maximum: that form of diagonal reflection follows from MM, but not from SRP. The surprising initial observation is that it does follow from SC-SRP + CH. The key reason for this is that SC-SRP + CH implies the nonsaturation of the nonstationary ideal. Thus, an apparent weakness of SC-SRP + CH turns out to be a strength in this context.

I will introduce the concepts involved and present some further results along these lines. The picture that emerges is that in the context of SC-SRP, saturation and diagonal reflection work against each other.

This is joint work with Hiroshi Sakai.

Tennenbaum's theorem states that no non-standard model of PA is computable. Hence, no unsound extension of PA has computable models. Pakhomov recently showed that this consequence of Tennenbaum's theorem is fragile; it depends on the signature in which PA is presented. In particular, there is a theory T such that (i) T is definitionally equivalent to PA (this is a strong form of bi-interpretability) and (ii) every consistent r.e. extension of T has a computable model. Pakhomov's techniques yield analogous results for ZF and other canonical systems. He asked whether there is a consistent, r.e. theory T such that no theory which is definitionally equivalent to T has a computable model. We answer this question with an ad hoc construction. This is joint work with Patrick Lutz.