We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.

Let T be a complete, countable, first-order theory with a finite number of countable models. Assuming that dcl(∅) is infinite we show that T has the strict order property.

Let $\mathcal M=$ be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders. Motivated by Rubin’s work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the “geometric” simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest (...) condition characterizes, up to definitional equivalence, theories of colored orders expanded by equivalence relations with convex classes. (shrink)

Let T be a complete, countable, first-order theory having infinite models. We introduce types directed by constants, and prove that their presence in a model of T guaranties the maximal number of non-isomorphic countable models : I=2.

We prove a dichotomy theorem for minimal structures and use it to prove that the number of non-isomorphic countable elementary extensions of an arbitrary countable, infinite first-order structure is infinite.

We prove that a countable, complete, first-order theory with infinite dcl( $ \theta $ ) and precisely three non-isomorphic countable models interprets a variant of Ehrenfeucht’s or Peretyatkin’s example.

We introduce the notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the former is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that (...) they are equivalence relations and prove that invariants of non-orthogonal types are closely related. The techniques developed are applied to prove that in the case of a binary, stationarily ordered theory with fewer than countable models, the isomorphism type of a countable model is determined by a certain sequence of invariants of the model. In particular, we confirm Vaught's conjecture for binary, stationarily ordered theories. (shrink)

We prove that a Kueker theory with infinite dcl(Ø) does not have the strict order property and that strongly minimal types are dense: any non-algebraic formula is contained in a strongly minimal type.

We prove that the existence of a nonisolated type having a finite domain and which is orthogonal to øin a 1-based theory implies that it has a continuum nonisomorphic countable models.

We consider semi-isolation on the locus of a strongly non-isolated, RK-minimal type in a small theory, and we prove that its asymmetry (as a binary relation) is caused by a specific form of the strict order property: the partial definability of semi-isolation.

We introduce notions of strong and eventual strong non-isolation for types in countable, stable theories. For T superstable or small stable we prove a dichotomy theorem: a regular type over a finite domain is either eventually strongly non-isolated or is non-orthogonal to a NENI type . As an application we obtain the upper bound for Lascar’s rank of a superstable theory which is one-based or trivial, and has fewer than 20 non-isomorphic countable models.

We prove that every type of finite Cantor-Bendixson rank over a model of a first-order theory without the strict order property is definable and has a unique nonforking extension to a global type. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Let T be a complete, superstable theory with fewer than ${2^{\aleph_{0}}}$ countable models. Assuming that generic types of infinite, simple groups definable in T eq are sufficiently non-isolated we prove that ω ω is the strict upper bound for the Lascar rank of T.

We prove: Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding has 2 ℵ 0 nonisomorphic countable models. Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

We answer a question of Cassidy and Kolchin about the universality of the constrained closure of a differential field by working in a larger category of models.

We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also prove that weak quasi-o-minimality of a theory with respect to one definable linear order implies weak quasi-o-minimality with respect to any other such order.

We study semi-isolation as a binary relation on the locus of a complete type and prove that—under some additional assumptions—it induces the strict order property.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 63.3 (2002) 539-545 [Access article in PDF] Lewis White Beck on Reasons and Causes Paul Guyer Essays by Lewis White Beck: Five Decades as a Philosopher. Edited by Predag Cicovacki. Rochester: University of Rochester Press, 1998. Pp. xxxii, 244. This volume reissues twelve previously uncollected pieces by the late Lewis White Beck (1913-1997) and also includes a reminiscence by a former colleague, (...) an introduction by the editor, and a bibliography of Beck's work. Beck was of course the most distinguished American Kant scholar of his time, who made enduring contributions to scholarship with his translations of Kant's writings in moral philosophy, originally published in Critique of Practical Reason and Other Writings in Moral Philosophy (Chicago, 1949); his trenchant articles on the analytic-synthetic distinction from the early 1950s, collected in his Studies in the Philosophy of Kant (Indianapolis, 1965); his unsurpassed A Commentary on Kant's Critique of Practical Reason (Chicago, 1960); his monumental Early German Philosophy: Kant and His Predecessors (Cambridge, Mass., 1969); his various articles on Kant's treatment of causation, collected in Essays on Kant and Hume (New Haven, 1978); and in his independent philosophical work, The Actor and the Spectator (New Haven, 1975). The present volume, edited by a student from Beck's last years at the University of Rochester, where he spent most of his career, includes articles on a variety of subjects published between 1939 and 1984, and it might seem to be little more than an act of piety toward a now departed but still revered teacher. But a core group of these articles are intimately connected with Beck's central scholarly and philosophical concerns and are well worth republication and renewed study.The articles included in the volume are three early pieces on several central issues in the philosophical debates of the time, "The Synoptic Method" (1939), "The Formal Properties of Ethical Wholes" (1941), and "The Principle of Parsimony in Empirical Science" (1943); an early piece on aesthetics, "Judgments [End Page 539] of Meaning in Art" (1944); two pieces on the tensions between causal explanations and rational evaluations of human behavior, "Psychology and the Norms of Knowledge" (1954) and "Conscious and Unconscious Motives" (1966); Beck's 1971 Presidential Address to the Eastern Division of the American Philosophical Association, a skeptical discussion of "Extraterrestrial Intelligent Life"; an overview of philosophies of time, "World Enough, and Time" (1979); a piece on styles of philosophical writing called "Philosophy As Literature" (1980); and then three pieces on Kant, published after the second of Beck's two earlier anthologies of his articles on Kant: "Kant on the Uniformity of Nature" (1981), "Five Concepts of Freedom in Kant" (1987), and "What Have We Learned from Kant?" (1984). With the last paper, the editor departs from the strictly chronological order he has previously employed, presumably in order to end the volume on a suitably general and retrospective note.I will focus my comments on the two systematic papers on psychological explanation and rational evaluation as well as on the first two Kant papers, all of which touch upon the issues that were central to Beck in his philosophically most important works, the 1960 Commentary on Kant's Critique of Practical Reason and the 1975 The Actor and the Spectator, and which point toward what was Beck's leading idea in both his philosophical and interpretative work: it concerns the idea that the causal explanation of human behavior as a subset of natural events and the rational explanation and justification of human actions are two entirely separate perspectives on the same subject-matter that can be rendered compatible only by the thought that each, causal explanation of behavior on the one hand and rational assessment of actions on the other, is only a regulative ideal for our conduct of inquiry and thus that neither represents a metaphysically privileged point of view. This was an approach that Beck claimed to have been foreshadowed in Kant's resolution of the "Antinomy of Teleological Judgment" in his last great critique, the Critique of the Power... (shrink)