TITLE: What is a good example of a complete but not model-complete theory, and why?
QUESTION [10 upvotes]: The standard examples of complete but not model-complete theories seem to be:
- Dense linear orders with endpoints.
- The full theory $\mathrm{Th}(\mathcal{M})$ of $\mathcal{M}$, where $\mathcal{M} = (\mathbb{N}, >)$ is the structure of natural numbers equipped with the relation $>$ (and nothing else, i.e. no addition etc).
Can anyone explain or give a reference to show why any of these two theories are not model-complete, or give another example altogether of a complete but not model complete theory (with explanation)?

REPLY [3 votes]: A theorem of Chang asserts that any theory which is axiomatized by for-all there exists sentences and is categorical in some infinite power is model complete.
Lindstrom gave an example that completeness does not suffice.