This is a nice application of the omitting types theorem that characterises -categorical theories in countable languages .
The main notion is that of a type. If is a set of -sentences a type over is a set of -formulas in some fixed finite tuple of free variables such that is maximally consistent with , i.e. for each finite subset of there is a model of , and is maximal such that this happens.
For the rest of this worksheet, fix a countable first-order language and a complete -theory which has infinite models. (Hence has no finite models - why is this?)
Suppose that has a type which is not isolated. I.e. suppose has a type which has no support. Show that is not -categorical.
Suppose that every type over is isolated. Let . Show that there are finitely many types over in the variables . Hint: Enumerate all formulas in as , and for consider
as the infinite paths through a certain tree. How many distinct consistent paths are there?
A type in free variables is called an -type.
Suppose has finitely many -types for each . Show that every type is isolated. (Hint: consider the tree in the previous exercise.)
Suppose that every type over is isolated and given countable models show that . (Use back-and-forth.)
For a complete theory with infinite models in a countable language , the following are equivalent.
A variation of this result is also useful.
Suppose is a complete -theory with infinite models and has countably many -types for each . Show that there is a countable model such that
Such a model is called a prime models. There is up to isomorphism at most one countable prim model of any complete theory . However, there is no converse to the result in the last exercise: find such a theory with a prime model and uncountably many -types for some .