Let M, N be L-structures. M is a substructure of N if M ⊆ N and for every constant symbol c, relation symbol R and function symbol f of L, we haveNote that since M is an L-structure, the condition on function symbols implies that M, as a subset of N , is closed under f N . We also say that N is an extension of M. The substructure is proper if its domain is a proper subset of the domain of the extension, in which case we write M < N . Note that M ≤ N ⇔ M < N ∨ M = N , whence this defines a partial order.If a subset S of N is closed under the interpretation of all constant and function symbols of N , then we have a substructure N ↾ S of N , called the restriction of N to S, on the domain S defined by c N ↾ S = c N , R N ↾ S = R N ∩ S arity(R) and f N ↾ S = f N ∩ S arity(f )+1 , for all constant, relation and function symbols, c, R, f , respectively.An embedding f : M → N from an L-structure M to an L-structure N is a function f : M → N , such that for each atomic L-formula φ( x) and for each m ∈ M, we have

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A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a "geometric technique" used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman's theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment KP P + P 1-Separation of ZF; and Gaifman's technique of iterated ultrapowers is employed to show that any countable model of GBC+"Ord is weakly compact" can be elementarily rankend-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of "strong rank-cut" is characterized (i) in terms of the theory GBC + "Ord is weakly compact", and (ii) in terms of fixed-point sets of self-embeddings.

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$ , expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$ , respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$ , and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins’ multiverse theory.