The Atomic Model Theorem

We investigate the complexity of several classical model theoretic theorems about prime and atomic models and omitting types. Some are provable in RCA0, others are equivalent to ACA0. One, that every atomic theory has an atomic model, is not provable in RCA0 but is incomparable with WKL0, more than Π1 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Π1 conservative over RCA0. A priority argument with Shore blocking shows that it is also Π1-conservative over BΣ2. We also provide a theorem provable by a finite injury priority argument that is conservative over IΣ1 but implies IΣ2 over BΣ2, and a type omitting theorem that is equivalent to the principle that for every X there is a set that is hyperimmune relative to X. Finally, we give a version of the atomic model theorem that is equivalent to the principle that for every X there is a set that is not recursive in X, and is thus in a sense the weakest possible natural principle not true in the ω-model consisting of the recursive sets. Hirschfeldt’s research was partially supported by NSF Grants DMS-0200465 and DMS-0500590. Shore’s research was partially supported by NSF Grants DMS-0100035 and DMS-0554855. Slaman’s research was partially supported by NSF Grants DMS-9988644 and DMS-0501167. All the authors thank the Institute for Mathematical Sciences of the National University of Singapore for its generous support during the period July 18 – August 15, 2005, when part of this research was done.