The Ryll-Nardzewski Theorem

This is a nice application of the omitting types theorem that characterises $ℵ_{0}$ -categorical theories in countable languages $L$ .

The main notion is that of a type. If $T$ is a set of $L$ -sentences a type over $T$ $p (\overline{x})$ is a set of $L$ -formulas in some fixed finite tuple of free variables $\overline{x}$ such that $p (\overline{x})$ is maximally consistent with $T$ , i.e. for each finite subset $\{ϕ_{1} (\overline{x}) ϕ_{1} (\overline{x}) \dots ϕ_{1} (\overline{x})\}$ of $p (\overline{x})$ there is a model of $T \cup \{\exists \overline{x} (ϕ_{1} (\overline{x}) \land ϕ_{1} (\overline{x}) \land \dots \land ϕ_{1} (\overline{x}))\}$ , and $p (\overline{x})$ is maximal such that this happens.

For the rest of this worksheet, fix a countable first-order language $L$ and a complete $L$ -theory $T$ which has infinite models. (Hence $T$ has no finite models - why is this?)

Exercise.

Suppose that $T$ has a type which is not isolated. I.e. suppose $T$ has a type which has no support. Show that $T$ is not $ℵ_{0}$ -categorical.

Exercise.

Suppose that every type over $T$ is isolated. Let $n \in ℕ$ . Show that there are finitely many types over $T$ in the variables $x_{1}, \dots, x_{n}$ . Hint: Enumerate all formulas in $x_{1}, \dots, x_{n}$ as $σ_{0} (\overline{x}), σ_{1} (\overline{x}), \dots$ , and for $A \subseteq ℕ$ consider

p_{A} (\overline{x}) = \{σ_{i} (\overline{x}): i \in A\} \cup \{\neg σ_{i} (\overline{x}): i \notin A\}

as the infinite paths through a certain tree. How many distinct consistent paths are there?

Definition.

A type in $n$ free variables $x_{1}, \dots, x_{n}$ is called an $n$ -type.

Exercise.

Suppose $T$ has finitely many $n$ -types for each $n \in ℕ$ . Show that every type is isolated. (Hint: consider the tree in the previous exercise.)

Exercise.

Suppose that every type over $T$ is isolated and given countable models $M, N ⊨ T$ show that $M N$ . (Use back-and-forth.)

Hence deduce

Theorem.

For a complete theory $T$ with infinite models in a countable language $L$ , the following are equivalent.

$T$ is $ℵ_{0}$ -categorical.
$T$ has finitely many $n$ -types for each $n \in ℕ$ .
every type over $T$ is isolated.

A variation of this result is also useful.

Exercise.

Suppose $T$ is a complete $L$ -theory with infinite models and $T$ has countably many $n$ -types for each $n \in ℕ$ . Show that there is a countable model $M ⊨ T$ such that

Every type realised in $M$ is isolated, and
for any $N ⊨ T$ there is an elementary embedding $M \to N$ .

Such a model is called a prime models. There is up to isomorphism at most one countable prim model of any complete theory $T$ . However, there is no converse to the result in the last exercise: find such a theory $T$ with a prime model and uncountably many $n$ -types for some $n$ .