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Suppose $T$ is a complete first order theory with a finite axiomatization. Must $T$ be $\aleph_0$-categorical? If not are there any simple examples of finitely axiomatized complete first order theories which are not $\aleph_0$-categorical?

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From a quick look at David Marker's model theory textbook (GTM 217), the answer is no. In exercise 2.5.16 he gives a finitely axiomatizable theory that is $\aleph_1$-categorical and without finite models (thus complete), but not $\aleph_0$-categorical. The theory looks to be anything but simple, however. I hope that someone can chime in with a nicer example. – Garrett Ervin Sep 28 '13 at 4:08
up vote 15 down vote accepted

The answer is no: for a simple example, take $Th(\mathbb Z,<)$. The axioms are:

  • $<$ defines a linear order;

  • every element has a immediate successor and an immediate predecessor.

Getting examples which are lower in the classification (e.g. stable) is much harder, and this is the point of the exercise in Marker's book which Garrett mentioned in the comments above.

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