We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group [Formula: see text] we introduce the [Formula: see text]-jump. We study the elementary properties of the [Formula: see text]-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups [Formula: see text], the [Formula: see text]-jump is proper in the sense that for any Borel equivalence relation [Formula: see text] the [Formula: see (...) text]-jump of [Formula: see text] is strictly higher than [Formula: see text] in the Borel reducibility hierarchy. On the other hand, there are examples of groups [Formula: see text] for which the [Formula: see text]-jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the full [Formula: see text]-tree, and relate these to iterates of the [Formula: see text]-jump. We also produce several new examples of equivalence relations that arise from applying the [Formula: see text]-jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank. (shrink)

More than any other thinker, Giorgio Agamben shows us that philosophy is also a matter of style and politics a matter of poetics. This book explores the unexpected and illuminating paths that his work traces across the territories of law and literature, linguistics, dance or cinema, in search of a new idea and practice of the community. It offers an irreplaceable introduction to one of the most fascinating thinkers of our time.'Jacques RanciereGathering some of the most important established and emerging (...) scholars to examine his body of work, this collection of essays seeks to explore Agamben's thought from these broader philosophical and literary concerns, underpinning its place within larger debates in continental philosophy. Including a contribution by Agamben himself, it is essential reading for anyone interested in his work.In the past five years, Giorgio Agamben has emerged as one of the most important continental philosophers. This burgeoning popularity of his work has largely been confined to a study of the homo sacer series. Yet these later 'political' works have their foundation in Agamben's earlier works on the philosophy of language, aesthetics and literature. From a philosophy of language and linguistics that leads to a broader theory of representation, Agamben develops a critical theory that attempts to explore the hiatuses and paradoxes that govern discursive practice across a broad range of disciplines. (shrink)

Love, hate, slavery, torture, addiction and death - as this book shows, only psychoanalysis can speak well of such matters. Psychoanalysis was the most important intellectual development of the 20th century, which left no practice from psychiatry to philosophy to politics untouched. Yet it was also in many ways an untouchable project, caught between science and poetry, medicine and hermeneutics. This unsettled, unsettling status has recently induced the philosopher Alain Badiou to characterise psychoanalysis as an 'antiphilosophy', that is, as a (...) practice that issues the strongest possible challenges to thought. Justin Clemens takes up the challenge of this denomination here, by re-examining a series of crucial psychoanalytic themes: addiction, fanaticism, love, slavery and torture. Drawing from the work of Freud, Lacan, Badiou, Agamben and others, Psychoanalysis is an Antiphilosophy offers a radical reconstruction of the operations and import of key psychoanalytic concepts and a renewed sense of the indispensable powers of psychoanalysis for today. (shrink)

§ 1. Introduction. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space.Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks : “The classification of Polish spaces up to (...) isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” Our Theorem 2.1 below quantifies precisely the enormity of this task.After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier.The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999], [2001]. (shrink)

In Logiques des Mondes, Paris, Seuil, 2006, Alain Badiou has produced a sequel to his magnum opus Being and Event. Whereas Being and Event primarily restricted itself to the relationship between ontology and the event, mathematics and poetry, the new book seriously extends and revises certain of its predecessor's. This article outlines some of the major doctrines, arguments, and motivations for the new work, as well as several points of possible difficulty.

We study the complexity of the classification problem for countable models of set theory (). We prove that the classification of arbitrary countable models of is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well‐founded models of.

In this article, I discuss how things go with the "Nothing" in the work of Alain Badiou, a topic which is evidently central to his thought, and which has received a great deal of attention in the commentary to date. As this problem is inaccessible outside of Badiou’s deployment of mathematics, I will suggest how accounts of Badiou’s work remain flawed insofar as they evade his mathematical demonstrations, and I attempt to clarify how mathematics operates in his system. I then (...) examine the consequences that such a use of mathematics has for Badiou’s doctrine of the nothing. I conclude with a discussion of some of the difficulties that the nothing continues to pose to Badiou, which have not (yet) been satisfactorily resolved. These difficulties devolve from the problematic of the incessantly doubling void. (shrink)

Following the publication of his magnum opus L’être et l’événement (Being and Event) in 1988, Alain Badiou has been acclaimed as one of France’s greatest living philosophers. Since then, he has released a dozen books, including Manifesto for Philosophy, Conditions, Metapolitics and Logiques des mondes (Logics of Worlds), many of which are now available in English translation. Badiou writes on an extraordinary array of topics, and his work has already had an impact upon studies in the history of philosophy, the (...) history and philosophy of science, political philosophy, aesthetics, psychoanalysis, and ontology. This volume takes up the challenge of explicating, extending and, in many places, criticising Badiou’s stunningly original theses. Above all, the essays collected here put Badiou’s concepts to the test in a confrontation with the four great headings that he himself has identified as essential to our humanity: science, love, art and politics. Many of the contributors have already been recognised as outstanding translators of and commentators on Badiou’s work; they appear here with fresh voices also destined to make a mark. (shrink)

We consider the complexity of the isomorphism relation on countable first-order structures with transitive automorphism groups. We use the theory of Borel reducibility of equivalence relations to show that the isomorphism problem for vertex-transitive graphs is as complicated as the isomorphism problem for arbitrary graphs and determine for which first-order languages the isomorphism problem for transitive countable structures is as complicated as it is for arbitrary countable structures. We then use these results to characterize the complexity of the isometry relation (...) for certain classes of homogeneous and ultrahomogeneous metric spaces. (shrink)

We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above \ in complexity.

Alain Badiou is one of the world's most influential living philosophers. Few contemporary thinkers display his breadth of argument and reference, or his ability to intervene in debates critical to both analytic and continental philosophy. Alain Badiou: Key Concepts presents an overview of and introduction to the full range of Badiou's thinking. Essays focus on the foundations of Badiou's thought, his "key concepts" - truth, being, ontology, the subject, and conditions - and on his engagement with a range of thinkers (...) central to his philosophy, including Plato, Descartes, Spinoza, Heidegger and Deleuze. (shrink)

The Heideggerian rupture in the history of philosophy in the name of a phenomenological and poetic ontology has provided an opening which many of the key figures in twentieth century continental thought have exploited. However, this opening was marked by Heidegger himself as an ambiguous one, insofar as metaphysics was perhaps integrally ‘onto-theology,’ that is, ultimately continuous with the world-historical capture of the thought of being. This piece argues that the philosophy of Alain Badiou, which departs from the recognition that (...) Heidegger is the ‘last universally recognised philosopher’, provides the means for a radical reconsideration of the philosophy-theology relationship in its specifically Heideggerian form, involving as it does further questions of science and technology, the status of the poem, and the nature of ontological thought as such. We argue that, through the deployment of mathematics as ontology, the Gordian knot of onto-theology and its legion of consequences can be cut, and a new assemblage of many of the key Heideggerian motifs can be put into play: the poem, history, and philosophy itself. (shrink)

This book collects the work of leading scholars on Alain Badiou and G.W.F. Hegel, creating a dialogue between, and a critical appraisal of, these two central figures in European philosophy.

This is a unique collection presenting work by Alain Badiou and commentaries on his philosophical theories. It includes three lectures by Badiou, on contemporary politics, the infinite, cinema and theatre and two extensive interviews with Badiou – one concerning the state of the contemporary situation and one wide ranging interview on all facets of his work and engagements. It also includes six interventions on aspects of Badiou's work by established scholars in the field, addressing his concept of history, Lacan, Cinema, (...) poetry, and feminism; and four original essays by young and established scholars in Australia and New Zealand addressing the key concerns of Badiou's 2015 visit to the Antipodal region and the work he presented there. With new material by Badiou previously unpublished in English this volume is a valuable overview of his recent thinking. Critical responses by distinguished and gifted Badiou scholars writing outside of the European context make this text essential reading for anyone interested in the development and contemporary reception of Badiou's thought. (shrink)

The theoretical writings of Jacques Lacan, Gilles Deleuze and Alain Badiou stand at the heart of contemporary European thought. While the combined corpus of these three figures contains a significant number of references to each otherOCOs work, such references are often simply critical, obscure - or both. Lacan Deleuze Badiou guides us through the crucial, under-remarked interrelations between these three thinkers, identifying the conceptual passages, connections and disjunctions that underlie the often superficial statements of critique, indifference or agreement. Working through (...) the rubrics of the contemporary, time, the event and truth, Bartlett, Clemens and Roffe present a new, lucid account of where these three thinkers stand in relation to one another and why their nexus remains unsurpassed as a point of reference for contemporary thought itself."e. (shrink)

A reconsideration of the philosophical destiny of education What is education? This volume thinks through this question from a range of perspectives unique perspectives. Revealing the contentions and possibilities of a new engagement with the question of education, it will give you new insights into education: what it is, what it is not, and what is to be done about it. At a time when education is so important as to be considered an essential human right, yet is under attack (...) from funding cuts, this book will open the thinking on education onto new and important territory. (shrink)

The controversies unleashed by psychoanalysis never seem to stop repeating themselves. If what psychoanalysis has to say is true, then, by its own lights, it has to be controversial. Controversies are thus a privileged place to see this truth and this resistance in violent and lurid action. Take infant experience and bastardry. Every kid is a bit of a bastard, and the establishment of this infantile bastardry conditions subsequent repetitions of the organism: that breast is persecuting me, these are not (...) my real parents, I did not borrow your kettle. Just how much of a bastard is this baby? The answers psychoanalysis comes up with depend on how it formulates the vicissitudes of differential repetitions, formations of the unconscious. Yet there remains something puzzling about repetition: if eros is constantly getting itself into nasty situations as a matter of course, are there still other factors (perhaps even more sinister) at work? Because of his refusal to dismiss his own puzzlement, Jacques Lacan persistently returned to the relation between desire and drive, reformulating his own theory as he went. At one moment, as we shall see, he comes to discriminate between a surprising number of (at least 3!) kinds of death. (shrink)

“Almost in the same historical moment when Galileo directed all modern physics to the reading of that book which Nature was supposed to have written herself in geometric or, subsequently, algebraic signs, the modern novel and modern theatre stepped in as evidence that modern readers and spectators enjoy the effects of those fictions most of all when they are altogether free of science.” Friedrich Kittler, “Man as a drunken town musician”.

As a philologist, Nietzsche had to be a materialist – a materialist of letters. If letters are not life, however, they are the indices of its limits. You can’t live except at the limit; to get to a limit, you have to reconstruct a genealogy for yourself; once you know where you are, you have the opportunity to lose yourself again, this time effectively. Life is whatever will have greeted you in that loss, the disappearance at the limit.

We define the notion of a weakly pointed tree, and characterize the amount of genericity necessary to prevent a uniformly branching tree being weakly pointed. We use these ideas to show there is no topological analogue of a measure-theoretic selection theorem of Graf and Mauldin.

The interdisciplinary relevance of Spinoza today.