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$\mathbb C$ is an algebraic closed field with characteristic $0$, hence $Th(\mathbb C)$ is a recursive satisfiable complete theory, thus recursive axiomatizable. So if $\mathbb N$ is definable in $\mathbb C$, then $+$,$\cdot$ are definable, hence $0$,$1$,$<$,$S$(successor),$E$(power) are definable too.

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    You don't need $\pi$, it's enough to have the exponential function, because $\mathbb{Z}$ is precisely the set of elements in $\mathbb{C}$ which fixes the set of solutions to $exp(z)=1$ under multiplication.2013-08-03

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The theory of algebraically closed fields of characteristics zero is strongly minimal.

This means that every definable set [in every model] is finite or co-finite.

I took this fact from: A Guide to Classical and Modern Model Theory, p.77.

If you search for a reference about minimal and strongly minimal theories I am sure you can find more about this, including various proofs if you are interested in them.

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    @GEdgar: If you talk about "as we often do" then we can also use second-order formulas and then the integers can be defined...2012-11-29