Here reprinted from the 1902 Macmillan edition.

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of (...) model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

Hrushovski originated the study of “flat” stable structures in constructing a new strongly minimal set and a stable 0-categorical pseudoplane. We exhibit a set of axioms which for collections of finite structure with dimension function δ give rise to stable generic models. In addition to the Hrushovski examples, this formalization includes Baldwin's almost strongly minimal non-Desarguesian projective plane and several others. We develop the new case where finite sets may have infinite closures with respect to the dimension function δ. In (...) particular, the generic structure need not be ω-saturated and so the argument for stability is significantly more complicated. We further show that these structures are “flat” and do not interpret a group. (shrink)

Grossberg and VanDieren have started a program to develop a stability theory for tame classes. We name some variants of tameness and prove the following. Let K be an AEC with Löwenheim-Skolem number ≤κ. Assume that K satisfies the amalgamation property and is κ-weakly tame and Galois-stable in κ. Then K is Galois-stable in κ⁺ⁿ for all n<ω. With one further hypothesis we get a very strong conclusion in the countable case. Let K be an AEC satisfying the amalgamation property (...) and with Löwenheim-Skolem number ℵ₀ that is ω-local and ℵ₀-tame. If K is ℵ₀-Galois-stable then K is Galois-stable in all cardinalities. (shrink)

This article reports the findings from a study that investigates the relationship between ethical climates and police whistle-blowing on five forms of misconduct in the State of Georgia. The results indicate that a friendship or team climate generally explains willingness to blow the whistle, but not the actual frequency of blowing the whistle. Instead, supervisory status, a control variable investigated in previous studies, is the most consistent predictor of both willingness to blow the whistle and frequency of blowing the whistle. (...) Contrary to popular belief, the results also generally indicate that police are more inclined than civilian employees to blow the whistle in Georgia - in other words, they are less inclined to maintain a code of silence. (shrink)

We use κ-free but not Whitehead Abelian groups to constructElementary Classes (AEC) which satisfy the amalgamation property but fail various conditions on the locality of Galois-types. We introduce the notion that an AEC admits intersections. We conclude that for AEC which admit intersections, the amalgamation property can have no positive effect on locality: there is a transformation of AEC's which preserves non-locality but takes any AEC which admits intersections to one with amalgamation. More specifically we have: Theorem 5.3. There is (...) an AEC with amalgamation which is not (N₀, N₁)-tame but is 2(N0, ∞)-tame; Theorem 3.3. It is consistent with ZFC that there is an AEC with amalgamation which is not (≤ N₂, ≤ N₂)-compact. (shrink)

CHAPTER 1 Combinatorial Geometries and Infinitary Logics In this chapter we introduce two of the key concepts that are used throughout the text. ...

We give a general account of the goals of axiomatization, introducing a variant on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and (...) thus are an immodest complete descriptive axiomatization of that subject. (shrink)

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a vocabulary $\tau $ with a single ternary relation R. We prove that for every integer k there exist $2^{\aleph _0}$ -many integer valued functions $\mu $ such that (...) each $\mu $ determines a distinct strongly minimal Steiner k-system $\mathcal {G}_\mu $, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture. (shrink)

We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...) dependence of certain versions on (highly independent) cardinal arithmetic. (shrink)

In this paper we study abstract elementary classes of modules. We give several characterizations of when the class of modules A with is abstract elementary class with respect to the notion that M1 is a strong submodel M2 if the quotient remains in the given class.

In [5], examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper, we give examples of complete ‐sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A For every natural number k, there is a class $K_k $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{k - 1} $ , but $K_k $ has the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_{k - 3} $ and has models of cardinality $\mathfrak{N}_{k (...) - 1} $ . More strongly, we can have disjoint amalgamation up to $\mathfrak{N}_\alpha $ for α > ω, but have a bound on size of models. Theorem B For every countable ordinal a, there is a class $K_\alpha $ defined by a sentence in $L_{\omega 1.\omega } $ that has no models of cardinality greater than $ \supset _{\omega 1} $ , but K does have the disjoint amalgamation property on models of cardinality less than or equal to $\mathfrak{N}_\alpha $ . Finally we show that we can extend the $\mathfrak{N}_\alpha $ to $ \supset _\alpha $ in the second theorem consistently with ZFC and while having $\mathfrak{N}_i \ll \supset _i $ for 0 > i < α. Similar results hold for arbitrary ordinals α with |α|= K and $L_{k + ,\omega } $. (shrink)

This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference and a detailed proof that in a suitable extension of ZFC, there is a complete sentence of \ that has maximal models in cardinals cofinal in the first measurable cardinal and, of course, never again.

IMITATION is a matter of such familiarity to us all that it goes usually unattended to: so much so that professed psychologists have left it largely undiscussed. Whether it be one of the more ultimate facts or not, suppose we assume it to be so; let us then see what we can explain by it, and where we may be able to trace its influence in the developed mind.

We prove two results on the stability spectrum for Lω1,ω. Here [Formula: see text] denotes an appropriate notion of Stone space of m-types over M. Theorem for unstable case: Suppose that for some positive integer m and for every α μ, K is not i-stable in μ. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness (...) in the theory under study. In this paper, we expound the construction of tree indiscernibles for sentences of Lω1,ω. Further we provide some context for a number of variants on the Ehrenfeucht–Mostowski construction. (shrink)

In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis.

Djordjević [Dj 1] proved that under natural technical assumptions, if a complete L n -theory is stable and has amalgamation over sets, then it has arbitrarily large finite models. We extend his study and prove the existence of arbitrarily large finite models for classes of models of L n -theories (maybe omitting types) under weaker amalgamation properties. In particular our analysis covers the case of vector spaces.

We provide a general framework for studying the expansion of strongly minimal sets by adding additional relations in the style of Hrushovski. We introduce a notion of separation of quantifiers which is a condition on the class of expansions of finitely generated models for the expanded theory to have a countable ω-saturated model. We apply these results to construct for each sufficiently fast growing finite-to-one function μ from 'primitive extensions' to the natural numbers a theory T μ of an expansion (...) of an algebraically closed field which has Morley rank 2. Finally, we show that if μ is not finite-to-one the theory may not be ω-stable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume's Knave and the Interests of JusticeJason Baldwin, doctoral student in philosophyHume's account of the artificial virtues of justice and promise-keeping developed in Book III, Part ii of the Treatise is among the most provocative elements of his ethics. His goal there is to tell a naturalistic story of the origin and moral standing of these virtues, a story that makes no appeal to any irreducibly moral motives or (...) properties but instead grounds even our most basic and apparently binding obligations in the same thoroughly human passions and sentiments with which Hume explains all our actions. Several modern critics, including David Gauthier and Stephen Darwall, have claimed that Hume's story of the artificial virtues is inconsistent with his account of natural human motivations, and have further claimed that Hume himself became aware of this problem in the second Enquiry, where he discussed it in the famous "sensible knave" passage in the last paragraphs of the "Conclusion." They therefore claim that Hume abandoned in the Enquiry what is usually taken to be his official Treatise account of justice and promise-keeping, and may also have intended a different account in the Treatise itself.This paper argues that the sensible knave passage neither signals nor requires any significant change in Hume's official Treatise story of the artificial virtues. Part 1 summarizes this official account, Part 2 explains the challenge Gauthier and Darwall believe the sensible knave poses to this account, and Part 3 defends the standard Humean account from this challenge. Whatever other problems Hume's account of justice and promise-keeping may have, the free rider problem, as developed by Gauthier and Darwall through the character of the sensible knave, is not among them.1. The Interest Account of Artificial Virtues1.1 Hume's Predecessor TraditionsHume makes few explicit references to the English-speaking predecessors whose work informs his thinking about the artificial virtues, and I shall follow his example. [End Page 277] But without pretending to survey that rich and varied history, we may usefully think of Hume's approach to justice and its allied virtues as an attempt to preserve the strengths of two opposed predecessor traditions.1 The first tradition, represented most prominently by Hobbes but also by Mandeville, explains the origin of virtues generally, including Humean artificial virtues, in purely naturalistic terms, without any appeal to divinely ordained ends or observer-independent moral facts. Hume shares this humanistic aspiration, but he disavows the cynical skepticism and crude egoism which are usually taken to accompany it. He wants to tell a story about justice which will allow us to admire its genuine moral worth even as we acknowledge its human pedigree.The second tradition, a reaction to the first, is the moral sense school represented by Shaftesbury, Butler, and especially Hutcheson. This tradition insists, against the cynical egoists, on a moral reality to which our sentiments or consciences give us immediate access, an access secured by God's design. Also characteristic of the moral sense theorists is the claim that human nature contains benevolent motives which are the objects of the moral sense's approval and which are just as basic to our constitution as are selfish motives. Hume avails himself of both moral sense strands, albeit in attenuated form. He agrees that justice and injustice are known by the sentiments, and he also believes that benevolent first-order motives are a part of human nature. But Hume rejects the moral sense tradition's Christian teleology, and he is also less optimistic about the potential of benevolence to explain such complex moral behaviors as are involved in our institutions of justice and promise-keeping.Hume's standard Treatise story of the origin of justice and promise-keeping, which we may call the "interest account," has two stages. In the first stage, Hume explains how the practices which constitute justice and promise-keeping arise through the working of our natural interests; in this stage Hume might be seen as working in the broadly naturalistic-egoist tradition. In the second stage, Hume explains how these practices and the motives which underlie them become the objects of moral evaluation, i.e., why... (shrink)

Let be a class of models with a notion of ‘strong’ submodel and of canonically prime model over an increasing chain. We show under appropriate set-theoretic hypotheses that if K is not smooth , then K has many models in certain cardinalities. On the other hand, if K is smooth, we show that in reasonable cardinalities K has a unique homogeneous-universal model. In this situation we introduce the notion of type and prove the equivalence of saturated with homogeneous-universal.

We introduce the notion T does not omit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property . For any theory T, form a new theory $T_{\rm Aut}$ by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If T is a stable theory, T does not admit obstructions if and only if $T_{\rm Aut}$ has a model companion. (...) The proof involves some interesting new consequences of the nfcp. (shrink)

The projective plane of Baldwin 695) is model complete in a language with additional constant symbols. The infinite rank bicolored field of Poizat 1339) is not model complete. The finite rank bicolored fields of Baldwin and Holland 371; Notre Dame J. Formal Logic , to appear) are model complete. More generally, the finite rank expansions of a strongly minimal set obtained by adding a ‘random’ unary predicate are almost strongly minimal and model complete provided the strongly minimal set is ‘well-behaved’ (...) and admits ‘exactly rank k formulas’. The last notion is a geometric condition on strongly minimal sets formalized in this paper. (shrink)