We work in the context of ω-stable theories. We obtain a natural, algebraic equivalent of ENI-NDOP and discuss recent joint proofs with Shelah that if an ω-stable theory has either ENI-DOP or is ENI-NDOP and is ENI-deep, then the set of models of T with universe ω is Borel complete.

We show that if a countable structure M in a finite relational language is not cellular, then there is an age-preserving $N \supseteq M$ such that $2^{\aleph _0}$ many structures are bi-embeddable with N. The proof proceeds by a case division based on mutual algebraicity.

Fix a weakly minimal (i.e. superstable U-rank 1) structure M. Let M∗ be an expansion by constants for an elementary substructure, and let A be an arbitrary subset of the universe M. We show that all formulas in the expansion (M∗,A) are equivalent to bounded formulas, and so (M,A) is stable (or NIP) if and only if the M-induced structure AM on A is stable (or NIP). We then restrict to the case that M is a pure abelian group with (...) a weakly minimal theory, and AM is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of (Z,+). Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form (M,A). Most notably, we show that if (G,+) is a weakly minimal additive subgroup of the algebraic numbers, A⊆G is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of A is a root of unity, then (G,+,B) is superstable for any B⊆A. (shrink)

We introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory $T$ is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model $M$ of $T$ has an expansion $(M,A)$ by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct, and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic (...) structure. (shrink)

Given a sequence {a n } in (0,1) converging to a rational, we examine the model theoretic properties of structures obtained as limits of Shelah-Spencer graphs G(m, m -αn ). We show that in most cases the model theory is either extremely well-behaved or extremely wild, and characterize when each occurs.

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure M is cellular if and only if M is \-categorical and mutually algebraic. Second, if a countable structure M in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually (...) algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen. (shrink)

We give an example of a countable theory $T$ such that for every cardinal $\lambda \geq \aleph_2$ there is a fully indiscernible set $A$ of power $\lambda$ such that the principal types are dense over $A$, yet there is no atomic model of $T$ over $A$. In particular, $T$ is a theory of size $\lambda$ where the principal types are dense, yet $T$ has no atomic model.

We prove that if M is any model of a trivial, weakly minimal theory, then the elementary diagram T(M) eliminates quantifiers down to Boolean combinations of certain existential formulas.

A class K of structures is controlled if for all cardinals λ, the relation of L∞,λ-equivalence partitions K into a set of equivalence classes . We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled 69–88).

We describe techniques for constructing models of size continuum inωsteps by simultaneously building a perfect set of enmeshed countable Henkin sets. Such models have perfect, asymptotically similar subsets. We survey applications involving Borel models, atomic models, two-cardinal transfers and models respecting various closure relations.

We sharpen Hájek’s Completeness Theorem for theories extending predicate product logic, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi\forall}$$\end{document}. By relating provability in this system to embedding properties of ordered abelian groups we construct a universal BL-chain L in the sense that a sentence is provable from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi\forall}$$\end{document} if and only if it is an L-tautology. As well we characterize the class of lexicographic sums that have this universality property.

We study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers R. We then show that a definable function in an o-minimal expansion of R enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of R. (...) Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group. (shrink)

We define a new topology on the space of strong types of a given theory and use it to state an omitting types theorem for countably saturated models of the theory. As an application we show that if T is a small, stable theory of finite weight such that every elementary extension of the countably saturated model is ω-saturated then every weakly saturated model is ω-saturated.

A countable, atomically stable structure U in a finite, relational language has fewer than 2 ω non-isomorphic substructures if and only if U is cellular. An example shows that the finiteness of the language is necessary.

A class K of structures is controlled if, for all cardinals λ, the relation of L ∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the ω-independence property is not controlled.

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, (...) show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density. (shrink)

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion Q such that, in any Q-generic extension of the universe, there are non-isomorphic models M 1 and M 2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if `c.c.c.' is replaced by other cardinal-preserving adjectives. We also give (...) an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings. (shrink)

We describe the Ax-Kochen definable subsets of the value group of a Hensel field and apply our results to a problem on identifying invariant factors in Hecke algebras.

We study classes of atomic models \ of a countable, complete first-order theory T. We prove that if \ is not \-small, i.e., there is an atomic model N that realizes uncountably many types over \\) for some finite \ from N, then there are \ non-isomorphic atomic models of T, each of size \.

We characterize when the elementary diagram of a mutually algebraic structure has a model complete theory, and give an explicit description of a set of existential formulas to which every formula is equivalent. This characterization yields a new, more constructive proof that the elementary diagram of any model of a strongly minimal, trivial theory is model complete.

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete one-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory T is not $\omega $ -stable, then the elementary diagram of some countable model of T has a Borel complete reduct.

We explore the existence and the size of infinite models of categorical theories having cardinality less than the size of the associated Tarski-Lindenbaum algebra. Restricting to totally transcendental, categorical theories we show that “Every tiny model is countable” is independent of ZFC. IfT is trivial there is at most one tiny model, which must be the algebraic closure of the empty set. We give a new proof that there are no tiny models ifT is not totally transcendental and is non-trivial.

We consider the question of when, given a subset A of M, the setwise stabilizer of the group of automorphisms induces a closed subgroup on Sym(A). We define s-homogeneity to be the analogue of homogeneity relative to strong embeddings and show that any subset of a countable, s-homogeneous, ω-stable structure induces a closed subgroup and contrast this with a number of negative results. We also show that for ω-stable structures s-homogeneity is preserved under naming countably many constants, but under slightly (...) weaker conditions it can be lost by naming a single point. (shrink)

On one way of talking about a traditional metaethical topic, realists accept that some items appear on the list of what exists in the moral or more broadly normative domain of inquiry. They then divide over whether those items are like what science and experience suggest that all other items on the list of what exists across all domains are like – naturalistic and secular. Reductive naturalists answer this further question affirmatively. Why don’t nonnaturalists? I explore the answer that it’s (...) because normative entities are “just too different” in the sense that they are countably different things. However, I argue that this answer rests on the subtle presupposition that the normative domain doesn’t also contain uncountable entities of the sort that analytic metaphysicians call “stuff”. Taking intuitions of difference seriously points the way to a novel form of metaethical reductive naturalism. The key is to identify the stuff that matters. (shrink)

Throughout his career, Derek Parfit made the bold suggestion, at various times under the heading of the "Normativity Objection," that anyone in possession of normative concepts is in a position to know, on the basis of their competence with such concepts alone, that reductive realism in ethics is not even possible. Despite the prominent role that the Normativity Objection plays in Parfit's non-reductive account of the nature of normativity, when the objection hasn't been ignored, it's been criticized and even derided. (...) We argue that the exclusively negative attention that the objection has received has been a mistake. On our reading, Parfit's Normativity Objection poses a serious threat to reductivism, as it exposes the uneasy relationship between our a priori knowledge of a range of distinctly normative truths and the typical package of semantic commitments that reductivists have embraced since the Kripkean revolution. (shrink)

Some rules seem more important than others. The moral rule to keep promises seems more important than the aesthetic rule not to wear brown with black or the pool rule not to scratch on the eight ball. A worrying number of metaethicists are increasingly tempted to explain this difference by appealing to something they call “authoritative normativity” – it’s because moral rules are “authoritatively normatively” that they are especially important. The authors of this chapter argue for three claims concerning “authoritative (...) normativity”: (1) that motivation for it originates in a parochial conception of “normative flavours”; (2) that arguments against alternative ways of explaining e.g., morality’s importance are overblown; and (3) that there are strong reasons against theorizing in terms of “authoritative normativity”. The overarching aim is not to show that “authoritative normativity” is incoherent or entirely unmotivated. Rather, it is to discuss some of “authoritative normativity’s” deeper commitments to correct the impression that is it largely benign or that it is entailed by familiar and popular “metanormative” positions. Authoritative normativity, it seems to us, is a solution in search of a problem. (shrink)

Are words like ‘woman’ or ‘man’ sex terms that we use to talk about biological features of individuals? Are they gender terms that we use to talk about non-biological features e.g. social roles? Contextualists answer both questions affirmatively, arguing that these terms concern biological or non-biological features depending on context. I argue that a recent version of contextualism from Jennifer Saul that Esa Diaz-Leon develops doesn't exhibit the right kind of flexibility to capture our theoretical intuitions or moral and political (...) practices concerning our uses of these words. I then float the view that terms like 'woman' or 'man' are polysemous, arguing that it makes better sense of the significance of some forms of criticisms of mainstream gender ideology. (shrink)

It is often said that normative properties are “just too different” to reduce to other kinds of properties. This suggests that many philosophers find it difficult to believe reductive theses in ethics. I argue that the distinctiveness of the normative concepts we use in thinking about reductive theses offers a more promising explanation of this psychological phenomenon than the falsity of Reductive Realism. To identify the distinctiveness of normative concepts, I use resources from familiar Hybrid views of normative language and (...) thought to develop a Hybrid view of normative concepts. In addition to using this new Hybrid view to explain why reductive theses are difficult to believe, I show how to preserve several patterns of inference involving normative concepts that, intuitively, it is possible to make, and hence answer an important recent challenge to Hybrid views from Mark Schroeder. (shrink)

Upon doing something generous for someone with whom you are close, some kind of reciprocity may be appropriate. But it often seems wrong to actually request reciprocity. This chapter explores the wrongness in making these requests, and why they can nevertheless appear appropriate. After considering several explanations for the wrongness at issue (involving, e.g. distinguishing oughts from obligation, the suberogatory, imperfect duties, and gift-giving norms), a novel proposal is advanced. The requests are disrespectful; they express that their agent insufficiently trusts (...) the hearer to recognize their own reasons to reciprocate, and the close relationship between them morally requires this kind of trust. This proposal is articulated and situated in the recent discussion of the normativity of requests. The chapter concludes by explaining how these requests may appear appropriate. Agents in these relationships often have standing to make requests, though the standing is lacking in these particular cases. (shrink)

A critical survey of various positions on the nature, use, possession, and analysis of normative concepts. We frame our treatment around G.E. Moore’s Open Question Argument, and the ways metaethicists have responded by departing from a Classical Theory of concepts. In addition to the Classical Theory, we discuss synthetic naturalism, noncognitivism (expressivist and inferentialist), prototype theory, network theory, and empirical linguistic approaches. Although written for a general philosophical audience, we attempt to provide a new perspective and highlight some underappreciated problems (...) about normative concepts. (shrink)

Many prominent ethicists, including Shelly Kagan, John Rawls, and Thomas Scanlon, accept a kind of epistemic modesty thesis concerning our capacity to carry out the project of ethical theorizing. But it is a thesis that has received surprisingly little explicit and focused attention, despite its widespread acceptance. After explaining why the thesis is true, I argue that it has several implications in metaethics, including, especially, implications that should lead us to rethink our understanding of Reductive Realism. In particular, the thesis (...) of epistemic modesty in ethics implies a kind of epistemic modesty about the metaphysical nature of ethics, if Reductive Realism about the metaphysical nature of ethics is true, and it implies that normative concepts are indispensable to practical deliberation in a way that answers an influential objection to Reductive Realism from Jonathan Dancy, David Enoch, William FitzPatrick, and Derek Parfit. (shrink)

Ethicists struggle to take reductive views seriously. They also have trouble conceiving of some supervenience failures. Understanding why provides further evidence for a kind of hybrid view of normative concept use.