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Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete.

I think there is some sentimental value in working with a theory that is simultaneously consistent and complete but the vast majority of pure mathematics nowadays seems to be done in ZFC. Of course, if one cannot interpret Robinson arithmetic a huge chunk of interesting mathematics becomes immediately unaccessible (e.g. Fermat's last theorem can not be formulated) so there may be good reasons for this.

So I would like to learn about complete and consistent theories that are not too boring. Tarski's Euclidean geometry is an example (which is also culturally significant in that many students are exposed to it in one way or another).

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There are many structures which have decidable theories, and these theories are necessarily consistent and complete. For instance:

  1. The theory of any given finite structure
  2. Primitive recursive arithmetic
  3. The theory of $(\mathbb R, +, \cdot, <, x \mapsto e^x)$
  4. The theory of algebraically-closed commutative fields of characteristic zero
  5. The theory of the random graph
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    $\begingroup$ Decidable theories need not be complete. For instance, 4 is not complete unless you also specify the characteristic of the field. $\endgroup$
    – Wojowu
    Commented Jun 11, 2021 at 22:48
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    $\begingroup$ Strictly speaking a decidable theory need not be consistent either, according to most definitions. :P $\endgroup$
    – Noah Schweber
    Commented Jun 11, 2021 at 22:52
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    $\begingroup$ I would say that decidable theories need not be consistent either, since it's pretty easy to decide whether a formula is proven by the inconsistent theory (answer: it is). $\endgroup$
    – Jonas Frey
    Commented Jun 11, 2021 at 22:53
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    $\begingroup$ Yes; thank you all for pointing this out. I was thinking of the theories of structures with decidable theories. Edited to correct. $\endgroup$
    – Luke Serafin
    Commented Jun 11, 2021 at 23:54
  • $\begingroup$ The theory of $(\mathbb R, +, \cdot, <, x \mapsto e^x)$ is not known to be decidable -- that is still open: en.wikipedia.org/wiki/Tarski%27s_exponential_function_problem $\endgroup$
    – user44143
    Commented Aug 18, 2021 at 22:31

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