# 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 $T$ $p ( x _ )$ is a set of $L$-formulas in some fixed finite tuple of free variables $x _$ such that $p ( x _ )$ is maximally consistent with $T$, i.e. for each finite subset $ϕ 1 ( x _ ) ϕ 1 ( x _ ) … ϕ 1 ( x _ )$ of $p ( x _ )$ there is a model of $T ∪ ∃ x _ ϕ 1 ( x _ ) ∧ ϕ 1 ( x _ ) ∧ … ∧ ϕ 1 ( x _ )$, and $p ( x _ )$ 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 $n ∈ ℕ$. Show that there are finitely many types over $T$ in the variables $x 1 , … , x n$. Hint: Enumerate all formulas in $x 1 , … , x n$ as $σ 0 ( x _ ) , σ 1 ( x _ ) , …$, and for $A ⊆ ℕ$ consider

$p A ( x _ ) = σ i ( x _ ) i ∈ A ∪ ¬ σ i ( x _ ) i ∉ A$

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

Definition.

A type in $n$ free variables $x 1 , … , x n$ is called an $n$-type.

Exercise.

Suppose $T$ has finitely many $n$-types for each $n ∈ ℕ$. 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 $n ∈ ℕ$.
• 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 $n ∈ ℕ$. 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 $M → N$.

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 $n$.