Constructing many atomic models in $\aleph_1$

TL;DR

This paper introduces pseudo-algebraicity to analyze atomic models of first-order theories, showing that under certain conditions, there are many non-isomorphic atomic models of size .

Contribution

It establishes a new approach using pseudo-algebraicity to construct numerous atomic models in  size for theories with non-dense pseudo-minimal types.

Findings

If pseudo-minimal types are not dense, then there are 2^{} non-isomorphic atomic models of size .

The notion of pseudo-algebraicity helps in understanding the diversity of atomic models.

The results apply to any complete first-order theory with an atomic model in a countable language.

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 $\aleph_1$.