The Atomic Model Theorem and Type Omitting

by Denis R. Hirschfeldt, Richard A. Shore, and Theodore A. Slaman



Status: published in the Transactions of the American Mathematical Society 361 (2009) 5805 - 5837.

Availability: PostScript, DVI, and PDF

Abstract. 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 Pi11 conservative over RCA0 and strictly weaker than all the combinatorial principles of Hirschfeldt and Shore [2007] that are not Pi11 conservative over RCA0. A priority argument with Shore blocking shows that it is also Pi11-conservative over BSigma2. We also provide a theorem provable by a finite injury priority argument that is conservative over ISigma1 but implies ISigma2 over BSigma2, 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 omega-model consisting of the recursive sets.



drh@math.uchicago.edu