We can measure the complexity of a logical formula by counting the number of alternations between existential and universal quantifiers. Suppose that an elementary first-order formula $\varphi $ (in $\mathcal {L}_{\omega ,\omega }$) is equivalent to a formula of the infinitary language $\mathcal {L}_{\infty ,\omega }$ with n alternations of quantifiers. We prove that $\varphi $ is equivalent to a finitary formula with n alternations of quantifiers. Thus using infinitary logic does not allow us to express a finitary formula in a simpler way.

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism), which implies, in particular, that isometry of locally compact Polish metric spaces is Borel reducible to graph isomorphism. We show that potentially $\boldsymbol {\Pi }^{0}_{\alpha + 1}$ isomorphism relations are Borel reducible to equality on hereditarily countable sets of rank $\alpha $, $\alpha \geq 2$. We also study approximations of the Hjorth-isomorphism game, and formulate a condition ruling out classifiability by countable structures.

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete structures. As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable.

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by these relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a $\boldsymbol {\Sigma }^1_1$ complete equivalence relation. We then investigate the algorithmic properties of this reduction. We obtain that elementary bi-embeddability on the class of computable graphs is $\Sigma ^1_1$ complete with respect to computable reducibility and show that the elementary bi-embeddability and bi-embeddability spectra realized by graphs are related.

Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $ (or any regular $\kappa $ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the (basis for the) topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $ , when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms (geometric morphisms the inverse image of which preserves all $\kappa $ -small limits) into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory (with at most $\kappa $ many axioms), that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic.

We calculate the complexity of Scott sentences of scattered linear orders. Given a countable scattered linear order L of Hausdorff rank $\alpha $ we show that it has a ${d\text {-}\Sigma _{2\alpha +1}}$ Scott sentence. It follows from results of Ash [2] that for every countable $\alpha $ there is a linear order whose optimal Scott sentence has this complexity. Therefore, our bounds are tight. We furthermore show that every Hausdorff rank 1 linear order has an optimal ${\Pi ^{\mathrm {c}}_{3}}$ or ${d\text {-}\Sigma ^{\mathrm {c}}_{3}}$ Scott sentence and give a characterization of those linear orders of rank $1$ with ${\Pi ^{\mathrm {c}}_{3}}$ optimal Scott sentences. At last we show that for all countable $\alpha $ the class of Hausdorff rank $\alpha $ linear orders is $\boldsymbol {\Sigma }_{2\alpha +2}$ complete and obtain analogous results for index sets of computable linear orders.

We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal $\mathbb {L}_{\omega _1, \omega }$ sentence categorical on an end segment of cardinals below $\beth _\omega $ must be categorical also everywhere above $\beth _\omega $ . This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.

Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $-sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal:

Theorem.Assume${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $-sentence.

1. (1) If ψ is categorical in${\aleph _0}$and$1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $, then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals.

2. (2) If ψ is categorical in${\aleph _1}$, then ψ is categorical in all uncountable cardinals.

The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $-definable tame abstract elementary classes with primes.

We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show that the complexities are maintained in the sense that if the functor is ${\bf{\Delta }}_\alpha ^0$, then the interpretation that induces it is ${\rm{\Delta }}_\alpha ^{in}$ up to ${\bf{\Delta }}_\alpha ^0$ equivalence.

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:

• Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:

  1. (1) For all high-enough λ,Khas no long splitting chains.

  2. (2) For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.

  3. (3) For all high-enough λ,Khas a unique limit model of cardinality λ.

  4. (4) For all high-enough λ,Khas a superlimit model of cardinality λ.

  5. (5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.

  6. (6) There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.

This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Analogues of Scott’s isomorphism theorem, Karp’s theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An “interpolation theorem” (of a particular sort introduced by Barwise and van Benthem) for the infinitary quantificational boolean logic L∞ω holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definability property.

We introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint$\left( { \le \lambda ,k} \right)$-amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a complete sentence in ${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.

Let $L$ be a countable language. We say that a countable infinite $L$ -structure ${\mathcal{M}}$ admits an invariant measure when there is a probability measure on the space of $L$ -structures with the same underlying set as ${\mathcal{M}}$ that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of ${\mathcal{M}}$ . We show that ${\mathcal{M}}$ admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in $\text{Aut}({\mathcal{M}})$ of an arbitrary finite tuple of ${\mathcal{M}}$ fixes no additional points. When ${\mathcal{M}}$ is a Fraïssé limit in a relational language, this amounts to requiring that the age of ${\mathcal{M}}$ have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

Theorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.

The present article, which is a revised version of part of [Hu1], deals with various relations between models which might serve as exact formulations for the vague concept “similar” or “almost isomorphic”. One natural class of such formulations is equivalence in a given logic. Another way to express similarity is by potential isomorphism, i.e., isomorphism in some extension of the set-theoretic universe. The class of extensions may be restricted to give different notions of potential isomorphism. A third method is to study the winning strategies for an Ehrenfeucht-Fraïssé-game played between the two models, and the properties of the resulting equivalence and nonequivalence trees.

The basic question studied here is whether one such notion of similarity implies another. Some implications and counterexamples listed in this part are previously known or trivial, but all are mentioned for completeness' sake. Only models of cardinality ℵ1 are considered. Some results are therefore connected with the Continuum Hypothesis.