Constructing many atomic models in א 1 John Baldwin

As has been known since at least [?] and is carefully spelled out in Chapter 6 of [?], for every complete sentence ψ of Lω1,ω (in a countable vocabulary τ ) there is a complete, first order theory T (in a countable vocabulary extending τ ) such that the models of ψ are exactly the τ -reducts of the atomic models of T . This paper is written entirely in terms of the class AtT of atomic models of a complete first order theory T , but applies to Lω1,ω by this translation. Our main theorem, Theorem 2.8, asserts: Let T be any complete first-order theory in a countable language with an atomic model. If the pseudo-minimal types are not dense, then there are 2א1 pairwise non-isomorphic, full1 atomic models of T , each of size א1. The first section states some old observations about atomic models and develops a notion of ‘algebraicity’, dubbed pseudo-algebraicity for clarity, that is relevant in this context. We introduce the relevant analogue to strong minimality, pseudo-minimality, and state the pseudo-minimals dense/many models dichotomy. Section 3 expounds a transfer techinique, already used in [?] and [?] and applied here prove to Theorem 2.8. The gist of the method is to prove a model theoretic property is consistent with ZFC by forcing and then extend the model M of set theory witnessing this result to a model N , preserving the property and such that the property is absolute between V and N . Section 4 describes a forcing construction, which together with the results of Section 3, yields a proof of Theorem 2.8 in Section 5. The authors are grateful to Paul Larson and Martin Koerwien for many insightful conversations. 1An atomic model M is full if |φ(M,a)| = ||M || for every non-pseudo-algebraic formula φ(x, a) with a from M .