We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property is first order.

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness.

We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if, for each prime p, the p-primary part of A is either finite or it coincides with the Prüfer p-group. We also provide a model-theoretic description of the model companions we obtain.

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.

L. van den Dries proved that the theory of n-valued rings has a model companion. We show here that this result is still true when the valuation rings are required to satisfy given inclusion relations (we restrict ourselves to the case of residual characteristic zero).

In an earlier paper we proved that a universal Horn class generated by finitely many finite structures has a model companion. If the language has only finitely many fundamental operations then the theory of the model companion admits a primitive recursive elimination of quantifiers and is primitive recursive. The theory of the model companion is ℵ0-categorical iff it is complete iff the universal Horn class has the joint embedding property iff the universal Horn class is generated by a single finite structure. In the last section we look at structure theorems for the model companions of universal Horn classes generated by functionally complete algebras, in particular for the cases of rings and groups.