Non existence of Prime models

Question

Let $L$ be a countable language. Let $T$ be a complete $L$ theory. We know that if $T$ is small, then there is a prime model of the theory. But $\text{Th}(\mathbb{N},+,\times,0,1)$ is not small but it has a prime model.

As far as I'm aware there is no necessary and sufficient condition for the existence of prime models. But I would like to see how to show certain theories do not have prime models. For example, how would you show that any consistent extension of ZFC doesn't have a prime model? (I strongly suspect that is the case).

Edit 1: To clarify: I'm asking if there is a "standard" argument that you can try when you suspect a theory doesn't have a prime model.

Answer 1

There is a simple necessary and sufficient condition for the existence of a prime model: "isolated types are dense". If $T$ is a complete theory in a countable language, then

1. If $M\models T$, then $M$ is prime if and only if $M$ is countable and atomic (meaning that $M$ realizes only isolated types over the empty set).
2. $T$ has a countable atomic model if and only if isolated types are dense (meaning that for every formula $\varphi(\overline{x})$, there is an isolated type $p(\overline{x})$ in $S^n(\varnothing)$ such that $\varphi\in p$).

These facts should be proven in every model theory textbook - they're Theorems 4.2.8 and 4.2.10 in Marker, for example. The situation is more subtle if $L$ is uncountable (largely because it's harder to omit types), and I suspect there's not a satisfying general criterion for the existence of prime models in that context.

One more comment: To prove that $T$ does not have a prime model, you need to come up with a formula $\varphi(\overline{x})$ which is not contained in any isolated type. This means $[\varphi(\overline{x})]$ is a perfect set in the Stone space, so there must be continuum-many types containing $\varphi(\overline{x})$. This is why small theories have prime models.

Answer 2

I address the part of your question which asks "how would you show that any consistent extension of ZFC doesn't have a prime model?" In fact (assuming of course that ZF is consistent), some consistent extensions of ZFC do have prime models.

Let $M$ be a model of ZFC in which each element is definable. (Such "pointwise definable" models of set theory were the subject of this question.) Then $T=\operatorname{Th}(M)$ is a consistent extension of ZFC, and $M$ is a prime model of $T.$

P.S. For the existence of pointwise definable models of ZFC, see the answers to this question, or consult the paper of J. D. Hamkins, D. Linetsky, and J. Reitz, Pointwise definable models of set theory, Journal of Symbolic Logic 78(1), pp. 139-156, 2013.