We prove that the classes of UHF algebras and AF algebras, while not axiomatizable, can be characterized as those C*-algebras that omit certain types in the logic of metric structures.

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In the target article, “Revive and Refuse: Capacity, Autonomy, and Refusal of Care After Opioid Overdose,” Kenneth D. Marshall and collaborators (2024) highlight important complexities in the care...

We realize the Jiang-Su algebra, all UHF algebras, and the hyperfinite II1factor as Fraïssé limits of suitable classes of structures. Moreover by means of Fraïssé theory we provide new examples of AF algebras with strong homogeneity properties. As a consequence of our analysis we deduce Ramsey-theoretic results about the class of full-matrix algebras.

The Connes Embedding Problem asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the statement that every type II1 tracial von Neumann algebra has a computable universal theory.

Shelah, S., C. Laflamme and B. Hart, Models with second order properties V: A general principle, Annals of Pure and Applied Logic 64 169–194. We present a general framework for carrying out the construction in [2-10] and others of the same type. The unifying factor is a combinatorial principle which we present in terms of a game in which the first player challenges the second player to carry out constructions which would be much easier in a generic extension of the (...) universe, and the second player cheats with the aid of ♦. Section 1 contains an axiomatic framework suitable for the description of a number of related constructions, and the statement of the main theorem 1.9 in terms of this framework. In Section 2 we illustrate the use of our combinatorial principle. The proof of the main result is then carried out in Sections 3–5. (shrink)

Historians of the eugenics movement have long been ambivalent in their examination of the links between British hereditary researchers and Nazi Germany. While there is now a clear consensus that American eugenics provided significant material and ideological support for the Germans, the evidence remains less clear in the British case where comparatively few figures openly supported the Nazi regime and the left-wing critique of eugenics remained particularly strong. After the Second World War British eugenicists had to push back against the (...) accusation that their science was intrinsically dictatorial or totalitarian and, as a result, many of their early perceptions of the Nazis were ignored or rationalised away. Further, historians in recent years have focused more directly on the social reformist elements of eugenics, discussing the links between hereditary science and the birth control and feminist movements in addition to others. While undoubtedly making valuable contributions to the scholarly understanding of the eugenic milieu in the interwar years, these studies have neglected to recontextualize the sentiments of British eugenicists who did indeed view the Nazi government positively in the early years of the 1930s. This article argues that there was a significant, though not numerically sizable, faction in the British eugenics movement, though mostly outside the Eugenics Society itself, in the early 1930s that viewed the Nazi Germany as an admirable state for its implementation of eugenic principles. One of these figures was later interned by his own government for being too closely aligned with the German regime, though he argued that this affinity was driven by the quest for scientific truth rather than politics. Eugenics in Britain thus contained a greater diversity of views toward Germany than scholars have previously assumed, warranting more research into the individuals and organizations harbouring these views. (shrink)

A countable unidimensional theory without the omitting types order property (OTOP) has prime models over pairs and is hence classifiable. We show that this is not true for uncountable unidimensional theories.

Let T be simple, work in Ceq over a boundedly closed set. Let p ∈ S(θ) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose (p, Q) is ACL-embedded in Q (see definitions below). Then Aut(p/Q) with its action on pC is type-definable in Ceq over θ. In particular, if p ∈ S(θ) is internal in a stably-embedded type-definable set Q, and pC υ Q is stably-embedded, then Aut(p/Q) is type-definable with its action (...) on pC. (shrink)

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 prove the Main Gap for the class of a -models (sufficiently saturated models) of an arbitrary stable 1-based theory T . We (i) prove a strong structure theorem for a -models, assuming NDOP, and (ii) roughly compute the number of a -models of T in any given cardinality. The analysis uses heavily group existence theorems in 1-based theories.

We present a structure theorem for superstable quasi-varieties without DOP. We show that every algebra in such a quasi-variety weakly decomposes as the product of an affine algebra and a combinational algebra, that is, it is bi-interpretable with a two sorted structure where one sort is an affine algebra, the other sort is a combinatorial algebra and the only non-trivial polynomials between the two sorts are certain actions of the affine sort on the combinatorial sort.

We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form.

Historians of the eugenics movement have long been ambivalent in their examination of the links between British hereditary researchers and Nazi Germany. While there is now a clear consensus that American eugenics provided significant material and ideological support for the Germans, the evidence remains less clear in the British case where comparatively few figures openly supported the Nazi regime and the left-wing critique of eugenics remained particularly strong. After the Second World War British eugenicists had to push back against the (...) accusation that their science was intrinsically dictatorial or totalitarian and, as a result, many of their early perceptions of the Nazis were ignored or rationalised away. Further, historians in recent years have focused more directly on the social reformist elements of eugenics, discussing the links between hereditary science and the birth control and feminist movements in addition to others. While undoubtedly making valuable contributions to the scholarly understanding of the eugenic milieu in the interwar years, these studies have neglected to recontextualize the sentiments of British eugenicists who did indeed view the Nazi government positively in the early years of the 1930s. This article argues that there was a significant, though not numerically sizable, faction in the British eugenics movement, though mostly outside the Eugenics Society itself, in the early 1930s that viewed the Nazi Germany as an admirable state for its implementation of eugenic principles. One of these figures was later interned by his own government for being too closely aligned with the German regime, though he argued that this affinity was driven by the quest for scientific truth rather than politics. Eugenics in Britain thus contained a greater diversity of views toward Germany than scholars have previously assumed, warranting more research into the individuals and organizations harbouring these views. (shrink)