The notion of an Euclidean domain is defined using the auxiliary machinery of a Euclidean function. But, I wonder, is that auxiliary machinery actually needed? More precisely, in the language $\{+,-,*,0,1\}$ of rings, is the class $K$ of those rings which are Euclidean domains, first-order axiomatizable, and if so, is it finitely first-order axiomatizable? Also, if it is not first-order axiomatizable, is the first-order theory of the class $K$ finitely axiomatizable?
2026-03-26 20:53:54.1774558434
Are Euclidean domains first-order axiomatizable in just the language of rings?
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No, the class of Euclidean domains is not elementary.
Let $\mathcal{Z}$ be any nonstandard model of the theory of the ring of integers (e.g. a nontrivial ultrapower). By overspill, there are instances of the Euclidean algorithm implemented internally to $\mathcal{Z}$ which are nonstandardly long; these prevent any genuine Euclidean function for $\mathcal{Z}$ for existing, since they would force any such function to assign an infinite value to any of the numbers occurring in this instance.