webcc/content/omitting-types.md

title: Omitting types date: 2016-03-15 00:00 tags: logic

Definition. A type p(x) for the theory T is a collection of formulas with a free variable x consitent with T.

-- see Michael Weiss' explanation of types.

Definition. A type p(x) is principal if there is a formula \alpha(x) such that p(x)=\\{ \psi(x) \mid \alpha \vdash \psi \\}

Definition. A type p(x) is full if for every formula \psi(x) either \psi \in p(x) or \neg \psi \in p(x).

A full type can be seen as a way of describing an a generic "element" of a theory. For any model M and an element m \in M we can consider the full type p(m) generated by m. It contains all the formlas \psi(m) that are true about m.

Omitting types theorem says that given a complete countable theory T in a countable language, there is a model M of T which omits countably-many non-principal full types.

Why is the non-principal condition needed?

If a theory T is complete, then any model of T realizes all the full principal types. For if p(x) is a complete principal isolated by \psi(x), then T \not\vdash \not \exists x.\psi(x), for otherwise the type p(x) would be inconsistent with T. Because T is complete it must be the case that T \vdash \exists x. \psi(x). Then such x realizes p(x).

What happens if the theory T is not complete?

Then it is not the case that all models of T realize all the full principle types. For instance take Peano arithmetic, and take T_1 = PA + Con(PA) and $T_2 = PA + \neg Con(PA). Both $T_1 and T_2 are consistent, thus they have models M_1 and M_2. Then denote by q(x) the type generated by the element 0 in M_1, and by p(x) the type generated by the element 0 in M_2. Both of those types are isolated by the formula $\phi(x) := \neg \exists y. y < x$, because \phi completely determines 0. Albeit those types are types for different theories, they are both types for a common theory PA. Furthemore, those types are complete and isolated. However, one of the types states the consistency of PA and the other one points out the inconsistency of PA. Thus both q and p cannot be realized in the same model.