We show that every ℵα can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1 but whose automorphism group fails the topological Vaught conjecture on analytic sets. We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.

Using the theory of Borel equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism.

This article explores the contribution of the virtue of integrity to the concept of mature care. The virtue of integrity is understood as both a personal and a social virtue. The argument is that the virtue of integrity is a necessary condition for providing mature care. An example from a psychiatric acute ward illustrates that a nurse acting with the virtue of integrity displays clear self‐boundaries and self‐respect as well as respect towards the inherent integrity of the patient. The article (...) also addresses some problems related to lack of the virtue of integrity by discussing the problems connected to detachment and rigidity. (shrink)

If there is a good $\Delta^1_3$ wellordering of the reals, then there is a $\Pi^1_1$ wellfounded relation for which the comparison relation is not projective.

Open peer commentary on the article “Elementary Students’ Construction of Geometric Transformation Reasoning in a Dynamic Animation Environment” by Alan Maloney. Upshot: Parnorkou and Maloney’s article presents an interesting, well-structured and clearly described study of children’s reasoning about mental rotations. Specifically, Parnorkou and Maloney deploy the microworld Graphs ’n Glyphs, and use it as a “window on thinking-in-change” as they observe and interview children who use it. Reading the article raised a few questions for me about the role of body (...) syntonicity in learning about rotation of geometric shapes, and I wonder where the authors feel these foundational concepts fit in with their research. (shrink)

Open peer commentary on the article “Circularity and the Micro-Macro-Difference” by Manfred Füllsack. Upshot: Füllsack provides a convincing argument for viewing circularity through a systems sciences perspective and for seeing micro-level emergence as an explanatory lens for phenomena that are circular at the macro-level. However, as an educator focusing on reasoning about circular macro-level phenomena through explanations at the micro-level, I see a series of issues relating designing appropriate learning experiences and fundamentally defining what this kind of thinking looks like.

There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group — and hence, there are 2α0 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility.

The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.

This paper shows that while there seems to be more or less a general acceptance for plurality as a condition of world politics and at least a vague commitment to a pluralist ideal, the challenge remains to formulate a fruitful account of international pluralism. While dominating approaches to international theory present international pluralism as essentially a by-product and instrumental, this paper suggest an alternative way to conceive of international pluralism when defending the ancient concept variety as a better guide to (...) approach both the understanding of plurality as the human condition and the notion of international pluralism. The paper concludes that it is preferable to accept a variety of pluralist conceptions rather than go on searching for a theoretical conception standing above the controversy; accepting pluralism in a sense involves rejecting just one version of pluralism. (shrink)

We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π03 iff the invariants are countable sets of reals, it is potentially Π04 iff the invariants are countable sets of countable sets of reals, and so on. The proofs (...) use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models. (shrink)

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may (...) at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL are measurable. John Steel's analysis also settled a number of structural questions regarding HODL, such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals. (shrink)

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin Σ 1 2 equivalence relation there is a Δ 1 3 real, N, over which every equivalence class is generic--and hence there is a good Δ 1 2 (N ♯ ) wellordering of the equivalence classes. Analogous results are obtained for Π 1 2 and Δ 1 2 quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is shown to imply that (...) every Π 1 2 prewellorder has rank less than $\underset{\sim}{\delta}^1_2$. (shrink)

In the presence of large cardinals, or sufficient determinacy, every equivalence relation in L(R) either admits a wellordered separating family or continuously reduces E 0.

As social, locative, and mobile media render the intimate public and the public intimate, this volume interrogates how this phenomenon impacts art practice and politics. Contributors bring together the worlds of art and media culture to rethink their intersections in light of participatory social media. By focusing upon the Asia-Pacific region, they seek to examine how regionalism and locality affect global circuits of culture. The book also offers a set of theoretical frameworks and methodological paradigms for thinking about contemporary art (...) practice more generally. (shrink)

There exists a model in a countable language having a unique element which is not definable in $\scr{L}_{\omega _{1},\omega}$.

Assuming $\underset{\sim}{\Pi}^1_2$ determinacy, the model L[ T 2 ] does not depend on the choice of T 2.

Unquestionably, the zeitgeist of Web 2.0 is symbolized by the dominance of social networking sites (SNS) and user-created content (UCC). MySpace, Facebook, and Cyworld mini-hompy are but a few examples of SNS that are becoming increasingly part of urban everyday life and interwoven into the historicity of the Internet. Web 2.0 has promised much about new forms of participation, creation, collaboration, and authorship, and yet within each location, we can find examples of both empowerment and exploitation. This is particularly the (...) case in the divergent region of the Asia-Pacific. Rather than the region being a sum of “imagined communities” (Anderson 1983), this paper argues that the distributed social networks of Web 2.0 UCC is formed, informed, and maintained through the perpetual process of “imaging communities.” These imaging communities can be seen in the visual, textual, and aural modes of UCC and can be seen to reflect the region’s new technocultural cartographies. (shrink)

If some infinitary sentence provides a counterexample to Vaught's Conjecture, then there is an infinitary sentence which also provides a counterexample but has no model of cardinality bigger than ℵ₁.

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to be (...) “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, (...) such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that theBorel cardinalitiesof the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.One of the first things to be addressed concerns the complexity of these equivalence relations. This question forappears in [1]. (shrink)

Let E be a Σ1 1 equivalence relation for which there does not exist a perfect set of inequivalent reals. If 0# exists or if V is a forcing extension of L, then there is a good ▵1 2 well-ordering of the equivalence classes.

A treeable ergodic equivalence relation of integer cost is generated by a free action of the free group on the corresponding number of generators. Every countable treeable ergodic equivalence relation is induced by the free action of some countable group.

Assuming every real has a sharp, we prove that for any $\mathop \prod \limits_\~ _1^1 $ equivalence relation either Borel reduces E₀ or in a $\mathop \Delta \limits_\~ _3^1 $ manner allows the assignment of bounded subsets of ω₁ as complete invariants.

It is possible to completely characterize which countable models generated by 0# exist inL. This in turn has applications in the study of analytic equivalence relations; for instance, ifE is∑ 1 1 and every invariant∑ 1 1 (0#) set isΔ 1 1 , thenE has at most ℵ0 many equivalence classes.

Suppose that any two Π12 sets are comparable in the sense of Wadge degrees. Then every real has a dagger. This argument proceeds by using the Dodd-Jensen core model theory to show that x ε ωω along with, say, “0† implies the existence of a Π12 norm of length u2. As a result of more recent work by John Steel, the same argument will extend to show that the Wadge comparability of all Π12 sets implies Π12 determinacy.

Michael Oakeshott’s distinction between ‘civil association’ and ‘enterprise association’ has inspired international society theorists to conceive of international society as not just a ‘purposive association’ constructed by states to satisfy their interests but also as a ‘practical association’ providing formal and pragmatic rules that are not instrumental to particular goals of state policy. While this article is supportive of the Oakeshottian turn in international society theory, it suggests that somewhat different conclusions can be drawn from it. The article sketches out (...) an alternative conception of international ‘civil association’, one that transcends the boundaries of communities. It is argued that such a notion of civil association is both possible and at the same time anchored in the experiences of the modern state. It is suggested that this notion of international civil association, when sustained by an adequate legal conception, promotes the enforcement of moral and political responsibility acr... (shrink)

There is an ergodic, measure preserving, countable Borel equivalence relation E on a standard Borel probability space (X, µ) such that E\c is not essentially free on any conull C ⊂ X.

We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.

We show that there is a complete, consistent Borel theory which has no "Borel model" in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel. We also investigate Borel isomorphisms between Borel structures.

The existence of a countable complete theory with exactly ℵ 1 many minimal models is independent of ZFC + ¬CH.

This Handbook presents key ideas of philosophers and social theorists whose ideas inform process approaches to organization studies. Each chapter addresses the background and context of this thinker, their work (with a focus on the processual elements), and the potential contribution to organization and management research.

Regulation of the subcellular localization of certain proteins is a mechanism for the regulation of their biological activities. FGF‐2 can be produced as distinct isoforms by alternative initiation of translation on a single mRNA and the isoforms are differently sorted in cells. High molecular weight FGF‐2 isoforms are not secreted from the cell, but are transported to the nucleus where they regulate cell growth or behavior in an intracrine fashion. 18 kDa FGF‐2 can be secreted to the extracellular medium where (...) it acts as a conventional growth factor by binding to and activation of cell‐surface receptors. Furthermore, following receptor‐mediated endocytosis, the exogenous FGF‐2 can be transported to the nuclei of target cells, and this is of importance for the transmittance of a mitogenic signal. The growth factor is able to interact with several intracellular proteins. Here, the mode of action and biological role of intracellular FGF‐2 are discussed. BioEssays 28: 504–514, 2006. © 2006 Wiley Periodicals, Inc. (shrink)