Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment $\mathcal {V}$. In this context, we construct atomic formulas and define the regular fragment of our enriched logic by taking conjunctions and existential quantification of those. We then characterize $\mathcal {V}$-categories of models of regular theories as enriched injectivity classes in the $\mathcal {V}$-category of structures. These notions rely on the choice of an orthogonal factorization system $(\mathcal {E},\mathcal {M})$ on $\mathcal {V}$ which will be used, in particular, to interpret relation symbols and existential quantification.

We explore the possibilities for elementary embeddings $j : M \to N$ , where M and N are models of ZFC with the same ordinals, $M \subseteq N$ , and N has access to large pieces of j. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line ${\mathbb R}$ .

Let P be a forcing notion and $G\subseteq P$ its generic subset. Suppose that we have in $V[G]$ a $\kappa{-}$ complete ultrafilter1 , 2 W over $\kappa $ . Set $U=W\cap V$ .

We study the notion of non-trivial elementary embeddings under the assumption that V satisfies ZFC without Power Set but with the Collection Scheme. We show that no such embedding can exist under the additional assumption that it is cofinal and either is a set or that the scheme of Dependent Choices of arbitrary length holds. We then study failures of instances of Collection in symmetric submodels of class forcings.

We give a sharper version of a theorem of Rosický, Trnková and Adámek [13], and a new proof of a theorem of Rosický [12], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)-extendible cardinals.

Working in the theory ”ZF + There is a nontrivial elementary embedding j : V → V“, we show that a final segment of cardinals satisfies certain square bracket finite and infinite exponent partition relations. As a corollary to this, we show that this final segment is composed of Jonsson cardinals. We then show how to force and bring this situation down to small alephs. A prototypical result is the construction of a model for ZF in which every cardinal μ ≥ ℵ2 satisfies the square bracket infinite exponent partition relation . We conclude with a discussion of some consistency questions concerning different versions of the axiom asserting the existence of a nontrivial elementary embedding j: V → V. By virtue of Kunen's celebrated inconsistency result, we use only a restricted amount of the Axiom of Choice.