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.

Big Ideas for Little Kids includes everything a teacher, a parent, or a college student needs to teach philosophy to elementary school children from picture books. Written in a clear and accessible style, the book explains why it is important to allow young children access to philosophy during primary-school education. Wartenberg also gives advice on how to construct a "learner-centered" classroom, in which children discuss philosophical issues with one another as they respond to open-ended questions by saying whether they agree (...) or disagree with what others have said. (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 show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, (...) and it was proved that Σ21-Kσ-regularity implies Σ21-u-regularity. Here we show that these properties are in fact equivalent. Since the proof of analytic u-regularity uses a game argument it was clear that determinacy implies u-regularity of all sets. Here we show that an inaccessible cardinal is enough to construct a model for projective u-regularity, namely it holds in Solovay's model. Finally we show that forcing with uniform trees is equivalent to Laver forcing. (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.

Assuming AD + DC, we characterize the self-dual boldface pointclasses which are strictly larger than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank [Formula: see text], and those of Wadge rank [Formula: see text] when ξ is limit.

v. 1. Editorial introduction -- v. 2. The English and Latin texts (i) -- v. 3. The English and Latin texts (ii).

This paper is an exposition and evaluation of the Agustín Rayo's views about the epistemology and metaphysics of mathematics, as they are presented in his book The Construction of Logical Space.

This volume is a study of the influence of Timaeus on the development of Western cosmology in three axial periods of European culture: Late Antiquity, Middle Ages and Renaissance.

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.

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)

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.

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)

Thomas Reid is now recognized as one of the towering figures of the Enlightenment. Best known for his published writings on epistemology and moral theory, he was also an accomplished mathematician and natural philosopher, as an earlier volume of his manuscripts edited by Paul Wood for the Edinburgh Reid Edition, Thomas Reid on the Animate Creation, has shown. The Correspondence of Thomas Reid collects all of the known letters to and from Reid in a fully annotated form. (...) Letters already published by Sir William Hamilton and others have been reedited, and roughly half of the letters included appear in print for the first time. Writing in 1802, Reid's disciple and biographer Dugald Stewart doubted that Reid's correspondence &"would be generally interesting.&" This collection proves otherwise, for the letters illuminate virtually every aspect of Reid's life and career and, in some instances, provide us with invaluable evidence about activities otherwise undocumented in his manuscripts or published works. Through his correspondence we can trace Reid's relations with contemporaries such as David Hume and his colleagues at both King's College, Aberdeen, and the University of Glasgow, as well as his engagement with the most controversial philosophical, scientific, and political issues of his day. If anything, the letters assembled here serve as the starting point for understanding Reid and his place in the Enlightenment. (shrink)

Thomas Hill, a leading figure in the recent development of Kantian moral philosophy, presents a set of essays exploring the implications of basic Kantian ideas for practical issues. The first part of the book provides background in central themes in Kant's ethics; the second part discusses questions regarding human welfare; the third focuses on moral worth-the nature and grounds of moral assessment of persons as deserving esteem or blame. Hill shows moral, political, and social philosophers just how valuable moral (...) theory can be in addressing practical matters. (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)

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)

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.

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.

The word “emotion” has named a psychological category and a subject for systematic enquiry only since the 19th century. Before then, relevant mental states were categorised variously as “appetites,” “passions,” “affections,” or “sentiments.” The word “emotion” has existed in English since the 17th century, originating as a translation of the French émotion, meaning a physical disturbance. It came into much wider use in 18th-century English, often to refer to mental experiences, becoming a fully fledged theoretical term in the following century, (...) especially through the influence of two Scottish philosopher-physicians, Thomas Brown and Charles Bell. This article relates this intellectual and semantic history to contemporary debates about the usefulness and meaning of “emotion” as a scientific term. (shrink)

Nietzsche is often regarded as a paradigmatically anti-theoretical philosopher. Bernard Williams has said that Nietzsche is so far from being a theorist that his text “is booby-trapped not only against recovering theory from it, but, in many cases, against any systematic exegesis that assimilates it to theory.” Many would apply this view especially to Nietzsche’s moral philosophy. They would say that even when he is making positive normative claims, as against just criticizing existing morality, his claims have neither the content (...) nor the organization characteristic of moral theory. (shrink)

Feeling good: four ways -- Finding that feeling -- The place of pleasure -- Knowing what's what -- Making things happen -- Being good -- Love and friendship -- Putting it together.

For early modern metaphysician Anne Conway, the world comprises creatures. In some sense, Conway is a monist about creatures: all creatures are one. Yet, as Jessica Gordon-Roth has astutely pointed out, that monism can be understood in very different ways. One might read Conway as an ‘existence pluralist’: creatures are all composed of the same type of substance, but many substances exist. Alternatively, one might read Conway as an ‘existence monist’: there is only one created substance. Gordon-Roth has done the (...) scholarship a great favor by illuminating these issues in Conway. However, this article takes issue with Gordon-Roth's further view that Conway ‘oscillates’ between the extremes of existence pluralism and monism. In its place, I argue we should read Conway as a priority monist: the whole of creation is ontologically prior to its parts. (shrink)

Respect, Pluralism, and Justice is a series of essays which sketches a broadly Kantian framework for moral deliberation, and then uses it to address important social and political issues. Hill shows how Kantian theory can be developed to deal with questions about cultural diversity, punishment, political violence, responsibility for the consequences of wrongdoing, and state coercion in a pluralistic society.

Serena Olsaretti brings together new essays by leading moral and political philosophers on the nature of desert and justice, their relations with each other and with other values.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (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.

§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)

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

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)