This article is a contribution to the study of extensions of arbitrary models of $\mathsf {ZF}$ (Zermelo–Fraenkel set theory), with no regard to countability or well-foundedness of the models involved. Our main results include the theorems below; in Theorems A and B, ${\mathcal {N}}$ is said to be a conservative elementary extension of $\mathcal {M}$ if $\mathcal { N}$ elementarily extends $\mathcal {M}$, and the intersection of every $ {\mathcal {N}}$-definable set with the universe of $\mathcal {M}$ is $\mathcal {M} $-definable (parameters allowed). In Theorem B, $\mathsf {ZFC}$ is the result of augmenting $\mathsf {ZF}$ with the axiom of choice.

Theorem A. Every model $\mathcal {M}$ of $\mathsf {ZF}+\exists p\left ( \mathrm {V}=\mathrm {HOD}(p)\right ) $ has a conservative elementary extension ${\mathcal {N}}$ that contains an ordinal above all of the ordinals of $\mathcal {M}$.

Theorem B. If ${\mathcal {N}}$ is a conservative elementary extension of a model $\mathcal {M}$ of $ \mathsf {ZFC}$, and ${\mathcal {N}}$ has the same natural numbers as $\mathcal {M}$, then $\mathcal {M}$ is cofinal in ${\mathcal {N}}$.

Theorem C. Every consistent extension of $ \mathrm {ZF}$ has a model $\mathcal {M}$ of power $\aleph _{1}$ such that $\mathcal {M}$ has no proper end extension to a model of $\mathsf {ZF}$.

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic.

That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.

We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate (⊨*) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension Θ of ZF there is a finitely axiomatizable extension *Θ′ of GB that is a conservative extension of Θ. We also prove a conservative extension result that justifies the use of ⊨* to characterize ground models for forcing constructions.