The Ryll-Nardzewski Theorem

This is a nice application of the omitting types theorem that characterises 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 0 -categorical.

Exercise.

Suppose that every type over T is isolated. Let 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 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 . 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.

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 . 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 .