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Abstract For any subset$$Z \subseteq {\mathbb {Q}}$$, consider the set$$S_Z$$of subfields$$L\subseteq {\overline {\mathbb {Q}}}$$which contain a co-infinite subset$$C \subseteq L$$that is universally definable inLsuch that$$C \cap {\mathbb {Q}}=Z$$. Placing a natural topology on the set$${\operatorname {Sub}({\overline {\mathbb {Q}}})}$$of subfields of$${\overline {\mathbb {Q}}}$$, we show that ifZis not thin in$${\mathbb {Q}}$$, then$$S_Z$$is meager in$${\operatorname {Sub}({\overline {\mathbb {Q}}})}$$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$$\mathcal {O}_L$$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$$\exists $$-definable subset of an algebraic extension of$${\mathbb Q}$$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.