It has been widely discussed whether the existence of other universes than the known universe is a purely metaphysical consideration. This paper argues to the contrary that the existence of other universes is a meaningful hypothesis, which can be tested based on observable data. The paper applies the perspective of observer bias as an approach for assigning probabilities to the mutually excluding hypotheses of universe vs. multiverse, i.e., whether the known universe is the only universe, or just one universe in (...) an ensemble of universes, a so-called multiverse. The basic idea of the paper relies on the following consideration. In a multiverse, the vast majority of observers would live in universes that are more life friendly. Thus, conditional on there being a multiverse, we should expect to find ourselves in a universe with values of the fundamental parameters that provide particularly fertile grounds for life. In contrast, if there is only a single universe, it may well be the case that there is still a few observers even if the parameters are far from ideal for life, and in that case, those observers will find values of the parameters that are not ideal for intelligent life. It may well be the case that, among all parameter configurations that allow life at least somewhere, those that are not ideal for life far outnumber those that are ideal for life. Based on this elementary consideration, the paper proceeds to propose a quantitative framework to determine probabilities for either hypothesis. In particular, it is described how a future ab initio determination of some of the factors in the Drake equation may be used to infer which one of the two hypotheses is the more likely. A quantitative approach to this end is proposed. Expressing these factors of the Drake equation in the context of the two hypotheses, a general likelihood approach is first described. Then, to support intuition, example computations are provided, illustrating how an actual hypothesis test would work in practice. (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)

We reformulate, in the context of continuous logic, an oscillation theorem proved by G. Hjorth and give a proof of the theorem in that setting which is similar to, but simpler than, Hjorth's original one. The point of view presented here clarifies the relation between Hjorth's theorem and first-order logic.

S. Adams, W. Ambrose, A. Andretta, H. Becker, R. Camerlo, C. Champetier, J.P.R. Christensen, D.E. Cohen, A. Connes. C. Dellacherie, R. Dougherty, R.H. Farrell, F. Feldman, A. Furman, D. Gaboriau, S. Gao, V. Ya. Golodets, P. Hahn, P. de la Harpe, G. Hjorth, S. Jackson, S. Kahane, A.S. Kechris, A. Louveau,, R. Lyons, P.-A. Meyer, C.C. Moore, M.G. Nadkarni, C. Nebbia, A.L.T. Patterson, U. Krengel, A.J. Kuntz, J.-P. Serre, S.D. Sinel'shchikov, T. Slaman, Solecki, R. Spatzier, J. Steel, D. Sullivan, (...) S. Thomas, A. Valette, V.S. Varadarajan, B. Velickovic, B. Weiss, J.D.M. Wright, R.J. Zimmer. (shrink)

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When [Formula: see text] for countable abelian [Formula: see text], Solecki [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343], Ding and Gao showed that for such G, the (...) orbit equivalence relation must in fact be potentially [Formula: see text], while conjecturing that the optimal bound could be [Formula: see text]. We show that the optimal bound is [Formula: see text] by constructing an action of such a group G which is not potentially [Formula: see text], and show how to modify the analysis of [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343] to get this slightly better upper bound. It follows, using the results of Hjorth et al. [Borel equivalence relations induced by actions of the symmetric group, Ann. Pure Appl. Logic 92 (1998) 63–112], that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets. (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)

We study a natural subclass of continuous actions of Polish groups on Polish spaces which we call eventually open actions. We prove that this property characterizes the actions endowed with a complete system of hereditarily countable invariant structures introduced by Hjorth as a generalization of Scott sentences.

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)

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.

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.

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)

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.

§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)

What is to be learned from the chaotic downfall of the Weimar Republic and the erosion of European liberal statehood in the interwar period vis-a-vis the ongoing European crisis? This book analyses and explains the recurrent emergence of crises in European societies. It asks how previous crises can inform our understanding of the present crisis. The particular perspective advanced is that these crises not only are economic and social crises, but must also be understood as crises of public power, order (...) and authority. In other words, it argues that substantial challenges to the functional and normative setup of democracy and the rule of law were central to the emergence and the unfolding of these crises. The book draws on and adds to the rich ’crises literature’ developed within the critical theory tradition to outline a conceptual framework for understanding what societal crises are. The central idea is that societal crises represent a discrepancy between the unfolding of social processes and the institutional frameworks that have been established to normatively stabilize such processes. The crises at issue emerged in periods characterized by strong social, economic and technological transformations as well as situations of political upheaval. As such, the crises represented moments where the existing functional and normative grid of society, as embodied in notions of public order and authority, were severely challenged and in many instances undermined. Seen in this perspective, the book reconstructs how crises unfolded, how they were experienced, and what kind of responses the specific crises in question provoked. -/- Table of Contents -/- Introduction: European Crises of Public Power: From Weimar until Today, Poul F. Kjaer & Niklas Olsen / Part I: Semantics, Notions and Narratives of Societal Crisis / 1. What Time Frame Makes Sense for Thinking About Crises?, David Runciman / 2. The Stakes of Crises, Janet Roitman / Part II: Weimar and the Interwar Period: Ideologies of Anti-Modernism and Liberalism / 3. The Crisis of Modernity – Modernity as Crisis: Towards a Typology of Crisis Discourses in Interwar East Central Europe and Beyond, Balázs Trencsényi / 4. European Legitimacy Crisis – Weimar and Today: Rational and Theocratic Authority in the Schmitt-Strauss Exchange, John P. McCormick / 5. Crisis and the Consumer: Reconstructions of Liberalism in Twentieth Century Political Thought , Niklas Olsen / Part III: The Causes of Crises: From Corporatism to Governance / 6. The Constitutionalization of Labour Law and the Crisis of National Democracy , Chris Thornhill / 7. The Crisis in Labour Law: From Weimar to Austerity Ruth Dukes / 8. From the Crisis of Corporatism to the Crisis of Governance, Poul F. Kjaer / Part IV: The Euro and the Crisis of Law and Democracy / 9. What is left of the European Economic Constitution II? From Pyrrhic Victory to Cannae Defeat Christian Joerges / 10. Reflections on Europe’s “Rule of Law Crisis”, Jan-Werner Müller. 11. Democracy under Siege: The Decay of Constitutionalisation and the Crisis of Public Law and Public Opinion, Hauke Brunkhorst/ Part V: The Consequences of Crises and the Future of Europe / 12. Crises and Extra-Legality: From Above and From Below, William E. Scheuermann / 13. “We could all go Down the Road of Lebanon” – Crisis Thinking on the Anti-Muslim Far Right, Mikkel Thorup / 14. Conclusions and Perspectives: The Re-Constitution of Europe, Poul F. Kjaer & Niklas Olsen Index . (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.

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)

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.

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)

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.

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.

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.

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.

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

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.

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.