# Proving that unary predicates theory has no $T$-complete formulas

As a follow up to my previous question, as suggested in the comments, I would like to know a way how to prove that the theory of infinitely many unary predicates, is axiomatized by the sentences $$\sigma_{F, G}$$ (for $$F, G$$ disjoint finite subsets of $$\mathbb{N}$$) given by $$\exists x\left(\bigwedge_{j \in F} U_{j}(x) \wedge \wedge_{j \in G} \neg U_{j}(x)\right)$$ has no principal types. I know that the way to show it is through showing that no formula in $$\mathcal{L}$$ is $$T$$-complete.

To clarify, by theory of infinitely many unary predicates, I mean the language with unary predicate symbols $$\left(U_{n}: n \geq 1\right)$$. and $$T$$ a theory $$\operatorname{Th}(\mathcal{M})$$ where $$\mathcal{M}$$ is the following structure: take $$\mathcal{M}=\mathcal{P}(\mathbb{N})$$ and define the interpretation of each $$U_{n}$$ by taking $$U_{n}^{\mathcal{M}}(\alpha) \leftrightarrow n \in \alpha$$ for each $$\alpha \subseteq \mathbb{N}$$.

I am using defnitions as in thew original question:

Definitions

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 $$S_x(\Sigma)$$.

The source of this example is lecture notes by Henson.

• Don't erase your question. Commented Feb 2, 2021 at 2:55
• This should replace the broken link for Henson's notes: people.math.sc.edu/mcnulty/modeltheory/Henson.pdf
– user957499
Commented Feb 9, 2022 at 23:05

Think about reducts. For each finite set $$X\subseteq\mathbb{N}$$ (really, every finite sublanguage of $$L$$) and each $$a,b\in\mathcal{M}$$, write $$a\sim_Xb$$ iff $$\{n\in X: U_n(a)\}=\{n\in X: U_n(b)\}.$$ Now you can show that the following holds:
Suppose $$\varphi$$ is an $$L$$-formula only using symbols $$U_i$$ for $$i\in X$$, and $$a\sim_Xb$$. Then $$\mathcal{M}\models \varphi(a)\leftrightarrow\varphi(b)$$.
Now here's the key point about $$\mathcal{M}$$: for every finite $$X\subseteq\mathbb{N}$$ there are $$a,b\in\mathcal{M}$$ with $$tp(a)\not=tp(b)$$ but $$a\sim_Xb$$. For example, pick any $$a$$ whatsoever and let $$b=a\Delta\{\max(X)+1\}.$$ This lets us wrap things up as follows:
For each formula $$\varphi$$, let $$a,b$$ be elements of $$\mathcal{M}$$ with distinct types but which are $$\sim_{X_\varphi}$$-equivalent.