Souslin quasi-orders and bi-embeddability of uncountable structures Alessandro Andretta Luca Motto Ros Author address: Universita` di Torino, Dipartimento di Matematica, Via Carlo Alberto, 10, 10123 Torino, Italy E-mail address: [email protected] Universita` di Torino, Dipartimento di Matematica, Via Carlo Alberto, 10, 10123 Torino, Italy E-mail address: [email protected] arXiv:1609.09292v2 [math.LO] 18 Mar 2019 Contents 1. Introduction 1 2. Preliminaries and notation 14 3. The generalized Cantor space 22 4. Generalized Borel sets 30 5. Generalized Borel functions 37 6. The generalized Baire space and Baire category 41 7. Standard Borel κ-spaces, κ-analyticquasi-orders,andspacesofcodes 47 8. Infinitary logics and models 55 9. κ-Souslin sets 65 10. The main construction 76 11. Completeness 85 12. Invariant universality 91 13. An alternative approach 106 14. Definable cardinality and reducibility 115 15. Some applications 126 16. Further completeness results 132 Indexes 147 Concepts 147 Symbols 148 Bibliography 151 iii Abstract We provide analogues of the results from [FMR11, CMMR13] (which correspond to the case κ = ω) for arbitrary κ-Souslin quasi-orders on any Polish space, for κ an infinite cardinal smaller than the cardinality of R. These generalizations yield a variety of results concerning the complexity of the embeddability relation between graphs or lattices of size κ, the isometric embeddability relation between complete metric spaces of density character κ, and the linear isometric embeddability relation between (real or complex) Banach spaces of density κ. Received by the editor March 19, 2019. 2010 Mathematics Subject Classification. 03E15, 03E60, 03E45, 03E10, 03E47. Key words and phrases. Generalized descriptive set theory, infinitary logics, κ-Souslin sets, determinacy, (bi-)embeddability, uncountable structures, non-separable metric spaces, non-separable Banach spaces. The authors would like to thank the anonymous referee for carefully reading the manuscript, pointing out inaccuracies, and suggesting changes that substantially improved the paper. All remaining errors are responsibility of the authors alone. v 1. INTRODUCTION 1 1. Introduction 1.1. What we knew. 1.1.1. Equivalence relations and classification problems. The analysis of definable equiva- lence relations on Polish spaces (or, more generally, on standard Borel spaces) has been one of the most active areas in descriptive set theory for the last two decades — see [FS89, Kec99, Kan08, Gao09] for excellent surveys of the subject. The main goal of this research area is to classify equivalence relations by means of reductions: if E and F are equivalence relations on Polish or standard Borel spaces X and Y , then f : X → Y reduces E to F if and only if (1.1) x E x′ ⇔ f(x) F f(x′), for every x, x′ ∈ X. In order to obtain nontrivial results one usually imposes definability assump- tions on E and F and/or on the reduction f. For example one may assume that • E and F are Borel or analytic1 and f is continuous or Borel (as it is customary when studying actions of Polish groups on standard Borel spaces [BK96, Hjo00, KM04]), or that • E and F are projective and f is Borel [HS79], or that • E, F , and f belong to some inner model of determinacy, such as L(R)[Hjo95, Hjo00]. When the reducing function f is Borel we say that E is Borel reducible to F (in symbols E ≤B F ) and that f is a Borel reduction of E to F . If E ≤B F ≤B E, then E and F are said to be Borel bi-reducible, in symbols E ∼B F . Borel reducibility is usually interpreted both as a topological version of the ubiquitous notion of classification of mathematical objects, and as a tool for computing cardinalities of quotient spaces in an effective way. If E and F are equivalence relations, then it can be argued that the statement E ≤B F is a precise mathematical formulation of the following informal assertions: • the problem of classifying the elements of X up to E-equivalence is no more complex than the problem of classifying the elements of Y up to F -equivalence; • the elements of X can be classified (in a definable way) up to E-equivalence using the F - equivalence classes as invariants; • the quotient space X/E has cardinality less or equal than the cardinality of X/F , and this inequality can be witnessed in a concrete and definable way. The theory of Borel reducibility has been used to gauge the complexity of many natural problems. One striking example is given by the classification of countable structures up to isomorphism briefly described below, whose systematic study was initiated by H. Friedman and Stanley in [FS89]. In a pioneering work in the mid 70s [Vau75] Vaught observed that by ω identifying the universe of a countable structure with ω, the collection ModL of countable L- structures can be construed as a Polish space, and the isomorphism relation =∼ on this space is an analytic equivalence relation. In order to exploit this identification most effectively, first- order logic must be replaced by its infinitary version Lω1ω, where countable conjunctions and disjunctions are allowed [Kei71]: by a theorem of Lopez-Escobar the (Lω1ω-)elementary classes, ω that is the sets Modσ of all countable models of an Lω1ω-sentence σ, are exactly the Borel ω subsets of ModL which are invariant under isomorphism. It follows that each elementary class ω is a standard Borel subspace of ModL, and that the restriction of the isomorphism relation to ω ∼ ω ∼ω Modσ, denoted in this paper either by =↾ Modσ or by =σ, is an analytic equivalence relation, whose complexity can then be analyzed in terms of Borel reducibility. Other classification problems whose complexity with respect to ≤B has been widely studied in the literature include e.g. the classification of Polish metric spaces up to isometry [CGK01, GK03, Cle12, CMMR18b] and the classification of separable Banach spaces up to linear isometry [Mel07] or up to isomorphism [FLR09]. 1 1 As customary in descriptive set theory, analytic sets are also called Σ1. 2 CONTENTS 1.1.2. Quasi-orders and embeddability. The concept of reduction from (1.1) can be applied to arbitrary binary relations, such as partial orders and quasi-orders (also known as preorders) [HMS88]. Quasi-orders are reflexive and transitive relations, and their symmetrization gives rise to an equiv- ⊏ alence relation. For example, the embeddability relation ∼ between structures (graphs, combi- natorial trees, lattices, quasi-orders, partial orders, . ) is a quasi-order, and its symmetrization ⊏ ω is the relation ≈ of bi-embeddability. The restriction of ∼ to the Borel set Modσ, denoted by ⊏ ω ⊏ω ∼↾ Modσ or by ∼σ, is an analytic quasi-order. Similarly, the relation of bi-embeddability on ω ω ω Modσ, denoted by ≈↾ Modσ or ≈σ, is an analytic equivalence relation. In [LR05, Theorem 3.1] Louveau and Rosendal proved that it is not possible to classify (in a reasonable way) all countable structures up to bi-embeddability, as the embeddability relation ⊏ ∼ is as complex as possible with respect to ≤B (whence also ≈ is as complex as possible). ⊏ω Theorem 1.1 (Louveau-Rosendal). The embeddability relation ∼CT on countable combina- torial trees (i.e. connected acyclic graphs) is a ≤B-complete analytic quasi-order, that is: ⊏ω (a) ∼CT is an analytic quasi-order on CTω, the Polish space of countable combinatorial trees, and ⊏ω (b) every analytic quasi-order is Borel reducible to ∼CT. ω Thus also the bi-embeddability relation ≈CT on CTω is a ≤B-complete analytic equivalence rela- tion. Remark 1.2. In an abstract setting, it is not hard to find ≤B-complete quasi-orders and equivalence relations. In fact it is easy to show that for every pointclass Γ closed under Borel preimages, countable unions, and projections, the collection of quasi-orders (respectively: equiv- alence relations) in Γ admits a ≤B-complete element, i.e. there is a quasi-order (respectively, an equivalence relation) UΓ ∈ Γ such that R ≤B UΓ, for any quasi-order (respectively: equiv- alence relation) R ∈ Γ — see [LR05, Proposition 1.3] and the ensuing remark. Examples of 1 Γ as above are the projective classes Σn’s and S(κ), the collection of all κ-Souslin sets — see Definition 9.1. However such ≤B-complete UΓ’s are usually obtained by ad hoc constructions. In contrast, Theorem 1.1 provided the first concrete, natural example of a ≤B-complete analytic ω 2 equivalence relation: the relation ≈CT of bi-embeddability on combinatorial trees. A ≤B-complete analytic equivalence relation must contain a non-Borel equivalence class, since it reduces the equivalence relation ω ω (1.2) EA := {(x, y) ∈ 2 × 2 | x = y ∨ x, y ∈ A} , where A ⊆ ω2 is a proper analytic set. On the other hand, the equivalence classes of an equiva- lence relation EG induced by a continuous (or Borel) action of a Polish group G are Borel [Kec95, 1 Theorem 15.14], even when EG is Σ1 and not Borel. Therefore there is a striking difference be- tween the isomorphism relation =∼ (which is induced by a continuous action of the group Sym(ω) ⊏ of all permutations on the natural numbers) and the embeddability relation ∼: even a very sim- ∼ 1 ple relation like the EA above is not Borel reducible to =, while any Σ1 equivalence relation (in 1 ⊏ fact: any Σ1 quasi-order) is Borel reducible to ∼. Hjorth isolated a topological property, called turbulence, that characterizes when an equivalence relation induced by a continuous Polish group action is Borel reducible to =∼ [Hjo00]. ⊏ω The next example shows that ∼CT is also complete for partial orders of size ℵ1, in the sense ⊏ω that each such partial order embeds into the quotient order of ∼CT.