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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. 

We characterize the completely determined Borel subsets of HYP as exactly the [Formula: see text] subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. 

We show that if F has a computable instance and the compositional product of F with itself is Weihrauch reducible to F, then F-diamond is Weihrauch reducible to F. The compositional product allows the use of two principles in sequence, while the diamond operator allows an arbitrary but finite number of uses of the given principle in sequence. This answers a question of Pauly.