# Unary predicates theory with no principal types

Definition

Take a satisfiable set of $$L$$-sentences $$\Sigma$$ and variables $$x = x_1, \ldots, x_n$$. Denote by $$S_x(\Sigma)$$ the set of all $$\Sigma$$-realizable $$x$$-types in $$L$$.

A type $$p(x) \in S_x(\Sigma)$$ is principal if it contains a $$\Sigma$$-complete formula. Equivalently, the singleton $$\{p(x)\}$$ is an open set in $$S_x(\Sigma)$$, or principal $$x$$-types are exactly the isolated points of $$S_x(\Sigma)$$.

Problem

I want to find a theory with no principal types realizable in $$T$$. I was suggested theory of infinitely many independent unary predicates is such a theory.

Attempt

I realize I'd have to prove that no formula in $$\mathcal{L}$$ is $$T$$-complete. To that end, I think I would need to show that if and only if $$\phi(x)$$ is a single-variable formula not containing a unary predicate symbol $$U_n$$ then both $$\phi(x) \wedge U_n(x)$$ and $$\phi(x) ∧ \neg U_n(x)$$ are realized in $$T$$. It seems intuitive, yet I do not know how to prove it.

How could I go about proving it?

You have the right idea, so I will just work out your argument. Basically $$T$$ just says nothing. So any $$\mathcal{L}$$-structure is a model. So suppose for a contradiction that a type $$p(x)$$ is isolated by $$\phi(x)$$ and let $$a \in M$$ realise $$p(x)$$. Let $$U_n$$ be a predicate that does not occur in $$\phi(x)$$. Now let $$\mathcal{L}'$$ be the language $$\mathcal{L}$$ without the predicate $$U_n$$. Let $$M'$$ be the $$\mathcal{L}'$$-reduct of $$M$$. Note that still $$M' \models \phi(a)$$. We define an $$\mathcal{L}$$-structure $$M_1$$ by taking $$M'$$ and interpreting $$U_n$$ as just $$\{a\}$$. We define another $$\mathcal{L}$$-structure $$M_2$$ by taking $$M'$$ and interpreting $$U_n$$ as the empty set. Now $$M_1 \models \phi(a) \wedge U_n(a)$$ and $$M_2 \models \phi(a) \wedge \neg U_n(a)$$, contradicting that $$\phi(a)$$ isolates $$p(x)$$.
• @hesse Ah, that is a different example. The reason I took the empty theory (so $T = \emptyset$) is because then the models are very easy: they are just the $\mathcal{L}$-structures. There is still a good reason to try this more complicated example: the theory there is complete, so we see that even in a complete theory we can have that no type is principal. If you still have questions about that particular construction I suggest you open a new question, as a follow-up. Commented Jan 29, 2021 at 23:26