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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)?

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up vote 14 down vote accepted

For the second example, let $M$ be the natural numbers and let $N$ be the integers greater than or equal to -1. Then $M$ is a substructure of $N$ but $M\models$ ``0 is the least element", while this is false in $N$. Thus the theory is not model complete.

The first example is similar, let $M$ be $[0,1]$ and let $N$ be $[-1,1]$. Again $M\models$ ``0 is the least element" but the extension $N$ does not.

One equivalent of model completeness is that every formula is equivalent to an existential formula. So theories like true arithmetic, the theory of the natural numbers in the language {$+,\cdot,0,1$}, are far from model complete.

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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.

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