A note on the Northcott property and undecidability

Abstract We uncover a natural relationship between the Northcott property for sets of algebraic numbers and the Julia Robinson number associated to sets of algebraic integers. This implies, in particular, that any subring of a ring of totally real integers having the Northcott property has undecidable first-order theory. Combining this theorem with previous results by the second author, we prove that the compositum of all totally real abelian extensions of $\mathbb {Q}$ of bounded degree $d$ has undecidable first-order theory.