Why is the noetherian ring property not definable in first-order logic?

Question

I am reading this paper on the connection between model theory and algebraic geometry.

https://math.uchicago.edu/~may/REU2015/REUPapers/Zhang,Victor.pdf

On page 9, I have trouble understanding Example 19: it says:

This shows the property of being a noetherian ring is not expressible as a set of sentences in first-order logic, otherwise $\text{Th}(\mathbb Z)$ would model such a set and our example model would be a noetherian ring.

I do not know why "otherwise $\text{Th}(\mathbb Z)$ would model such a set and our example model would be a noetherian ring". May I please ask for a more expanded explanation? Thank you for any help!

Answer 1

If being a noetherian ring can be expressed as a set of sentences in first-order logic of ring theory, that means, there is a set of first-order sentences $\mathcal S$ (in the language of ring theory), such that a ring $R$ is noetherian iff $\mathcal S\subset \text{Th}(R)$.

If so, then since $\mathbb Z$ is noetherian, $\mathcal S\subset \text{Th}(\mathbb Z)$. As $\text{Th}(\mathbb Z)\subset \text{Th}(M)$ (where $M$ is the special example given in Example 19, which is shown to be non-noetherian), we have $\mathcal S\subset\text{Th}(M)$, so $M$ is noetherian, a contradiction.