Which elements are definable in $\Bbb N$ + $\Bbb Z$? Where an element, a, is definable if there exists a formula such that $\forall x(\phi(x) \rightarrow x = a) $.
I have that all elements of $\Bbb N$ are definable, since 0 is definable as the unique minimal element, and the next greatest is definable using the successor function.
I want to say that no element of $\Bbb Z$ is definable, but I'm not sure exactly why not?
Your argument that every element of the $\mathbb N$ part is definable looks valid.
In order to prove that no element of the $\mathbb Z$ part is definable, find an order automorphism that leaves no element of the $\mathbb Z$ part fixed. A definable element is necessarily fixed by every automorphism.