Constructing many atomic models in $\aleph_1$

Abstract

We introduce the notion of pseudo-algebraicity to study atomic models of first order theories (equivalently models of a complete sentence of $L_{\omega_1,\omega}$. Theorem: 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 $2^{\aleph_1}$ pairwise non-isomorphic atomic models of $T$, each of size $\aleph_1$.