This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable (...) within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies. The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:uippraisals from the 'Past THE CLEAREST INTELLECT OF OUR AGEl H UGH MACLENNAN 19°7-199° R cently I have been rereading Bertrand Russell, and in so doing I suddenly realized that lowe to this man a good deal of such happiness as I enjoy. Over the years I had forgotten how great my debt was, but when I reread one of his books which I first read as a student, (...) I understood the extent to which he had liberated me from the psychological sadism of Calvin and Knox, to whom I had been unduly exposed when young. It is probably impossible for anyone entirely to escape once these two have got their hands around his throat, but Russell enabled me to escape a litde. 1 [This essay was recently discovered during the cataloguing for BRACERS of an obscure class of correspondence. Mr. Joseph w. Devine had sent Russell a clipping of the essay's appearance in' the Philadelphia Sunday But/etin, of 15 July 1962. I had long thought, from reading Elspeth Cameron's Hugh MacLennan: a Writers Life (1981), that he showed signs not only of having been influenced by Russell but of being the sOrt of person who is attracted to Russell's way of thinking. A promising tide in the same scholar's bibliography ofMacLennan turned out to be a fuller version of the clipping. "The Clearest Intellect of Our Age" is reprinted from The Montreal Star, 14 July 1962, "Entertainments" sec., p. 5. Russell did influence MacLennan. Neither the Russell Archives nor the inventory of the Maclennan papers at the Universiry of Calgary reveals personal contact berween the rwo. I wish I had asked MacLennan about contacts and influence seventeen years ago, when I sent him an early issue of Russell He commented: ''1 found your journal fascinating and never before had read that beautiful thing Russell wrote in his early days with Whitehead:" He is referring to the essay on the status of women in No.6, now in Papers 12.-Ed.] russell: the Journal ofthe Benrand Russell Archives McMaster Unive..ity Libraty Press noS. 11 (summer 1991): 83-5 ISSN 00,6-01631 84 HUGH MACLENNAN I suppose he really is the greatest man alive, in the sense that his teaching grows and his teaching is good, but it is pointless to argue the point between him and Churchill. It is wonderful, however, that many think he is our greatest man and that he has become an international hero to millions of the thoughtful youth. He must be the first pure logician who ever was that. Until he took his stand against nuclear war, he never appealed to the passions. He lacked the aggressions which youth admires. He never said: "Come live with me and I will be your master"; or, "Follow me and I will purge you of your sins." He had no use for self-sacrifice as such. He simply said, in effect: "Happiness is good. If you desire it, you must be reasonable." As he was intelligent enough to know that anyone who teaches that happiness is good, and teaches how to achieve it, is sute to be considered subversive to a Christian-commercial society, he bore no personal resentment when society persecuted him. The young know, of course, that Russell is now assumed by the experts to be the most important philosopher since Hume, but that is not why they love him. They love him for the sublime civilization of his courage, for his refusal to let an intensely kind nature blunt his respect for truth. They love him because he loves life even in the twentieth century; and they love him most of all because he is engage. And of course they also love the wit which refuses to be made tongue-tied by those who insist that whenever one speaks of sex or religion one must be solemn. Russell's wit is worth the study of any writer. It is based on mathematics, and Russell would agree, I think, with the remark Camillien Houde once made to me when I asked the famous mayor of Montreal how he was always able to make people laugh. Houde replied... (shrink)

We give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version ofLand then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.

We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the central (...) issue for the Inner Model Program. (shrink)

We investigate large cardinal axioms beyond the level of ω-huge in context of the universality of the suitable extender models of [Suitable Extender Models I, J. Math. Log.10 101–339]. We show that there is an analog of ADℝ at the level of ω-huge, more precisely the construction of the minimum model of ADℝ generalizes to the level of Vλ+1. This allows us to formulate the indicated generalization of ADℝ and then to prove that if the axiom holds in V at (...) a proper class of λ then in every suitable extender model, the axiom holds at a proper class of λ. (shrink)

Journal of Mathematical Logic, Ahead of Print. Asperó and Schindler have completely solved the Axiom [math] vs. [math] problem. They have proved that if [math] holds then Axiom [math] holds, with no additional assumptions. The key question now concerns the relationship between [math] and Axiom [math]. This is because the foundational issues raised by the problem of Axiom [math] vs. [math] arguably persist in the problem of Axiom [math] vs. [math]. The first of our two main theorems is that Axiom (...) [math] is equivalent to Axiom [math], and as a corollary we show that Axiom [math] fails in all the known models of [math]. This suggests that [math] actually refutes Axiom [math]. Our second main theorem is that the [math] Conjecture holds assuming [math]. This is the strongest partial result known on this conjecture which is one of the central open problems of [math]-theory and [math]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [math]-expressible and which implies [math], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [math]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [math] for which some property involving only quantification over [math] holds. (shrink)

Asperó and Schindler have completely solved the Axiom [Formula: see text] vs. [Formula: see text] problem. They have proved that if [Formula: see text] holds then Axiom [Formula: see text] holds, with no additional assumptions. The key question now concerns the relationship between [Formula: see text] and Axiom [Formula: see text]. This is because the foundational issues raised by the problem of Axiom [Formula: see text] vs. [Formula: see text] arguably persist in the problem of Axiom [Formula: see text] vs. (...) [Formula: see text]. The first of our two main theorems is that Axiom [Formula: see text] is equivalent to Axiom [Formula: see text], and as a corollary we show that Axiom [Formula: see text] fails in all the known models of [Formula: see text]. This suggests that [Formula: see text] actually refutes Axiom [Formula: see text]. Our second main theorem is that the [Formula: see text] Conjecture holds assuming [Formula: see text]. This is the strongest partial result known on this conjecture which is one of the central open problems of [Formula: see text]-theory and [Formula: see text]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [Formula: see text]-expressible and which implies [Formula: see text], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [Formula: see text]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [Formula: see text] for which some property involving only quantification over [Formula: see text] holds. (shrink)

The main theorem is that the Ultrafilter Axiom of Woodin :115–37, 2011) must fail at all cardinals where the Axiom I0 holds, in all non-strategic extender models subject only to fairly general requirements on the non-strategic extender model.

We prove that if $V = L [s]$ where $s$ is an $\omega$ -sequence of ordinals then the GCH holds.

The results of this paper concern the effective cardinal structure of the subsets of [ω1]<ω1, the set of all countable subsets of ω1. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.

We prove that every countable relation on the enumeration degrees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}, is uniformly definable from parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}. Consequently, the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document} is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order (...) theory of the enumeration degrees of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Sigma^0_2$\end{document} sets is not decidable. (shrink)

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set (...) and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry. (shrink)

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...) with this feature in the sense that any other $\Sigma _1^2 $ -statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of $V_\lambda $ . If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies¬CH. (shrink)

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...) third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails. (shrink)

J. Łoś raised the following question: Under what conditions can a countable partially ordered set be extended to a dense linear order merely by adding instances of comparability ? We show that having such an extension is a Σ 1 l -complete property and so there is no Borel answer to Łoś's question. Additionally, we show that there is a natural Π 1 l -norm on the partial orders which cannot be so extended and calculate some natural ranks in that (...) norm. (shrink)

In this study, we investigated whether differences in the experience of regret may be a potential explanation for damaging behaviours associated with psychopathy and criminal offending. Participants were incarcerated offenders (n = 60) and non-incarcerated controls (n = 20). Psychopathic traits were characterised with the Psychopathic Checklist: Screening Version. Regret was assessed by responses to outcomes on a simulated gambling task. Incarcerated offenders experienced a reduced sense of regret as compared to non-incarcerated controls. We obtained some evidence that specific psychopathic (...) factors and facets could differentially relate to the experience and use of emotions. Our data provide initial evidence of important associations between negative emotions and decision behaviour in the context of criminal offending. (shrink)

One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.

We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...) the Determinacy Transfer Theorem for arbitrary [Formula: see text], namely the statement that [Formula: see text] determinacy implies [Formula: see text] determinacy. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:27 Franciscan Studies 63 (2005) FRANCISCAN INSTITUTE MEDAL HONOREES The Franciscan Institute Medal for 2005 was awarded to two scholars of international renown in the world of Franciscan studies: David Flood, OFM and David Burr. Both honorees are known throughout the world particularly for their scholarly work on Peter John Olivi – a friar of the late 13th century whose critical importance for Franciscan history has now come to (...) be appreciated largely thanks to the work of these two scholars. The Medals were conferred at an Academic Convocation in connection with the Feast of St. Bonaventure in July 2005. David Flood, OFM is the 17th recipient of the Franciscan Institute Medal. David Burr is the 18th recipient of the Franciscan Institute Medal. MICHAEL BLASTIC 28 DAVID E. FLOOD, OFM 17th Recipient of the Franciscan Institute Medal Official Citation It is both an honor and a joy for me to celebrate and recount some of the significant contributions of Brother David Flood, OFM, to Franciscan studies and life. Brother David has been a member of the St. Joseph Province of the Friars Minor in Montreal, Canada, since 1950. He received a B.A. in philosophy from Laval University, Quebec City, studied theology at the Friars Convent in Montreal, and was ordained in May, 1958. He also received a M.A. in English and Literature from the University of Montreal in 1959. From 1958 until 1961 he taught English Literature at St. Francis College in Biddeford, Maine, and while teaching he published studies on English topics including Keats and Beckett. Between 1961 and 1965 while residing in Cologne, Germany, he studied with Kajetan Esser, OFM, at Mönchengladbach, while reading philosophy and history at the University of Cologne. He received his doctorate from that university in 1965 with the dissertation Die Regula non Bullata der Minderbrüder (published in 1967 by Dietrich Coelde Verlag). Returning to Canada, he taught church history in Montreal from 1965-1966. In 1967-1968 he held a fellowship at the Institute for European History in Mainz, working on a critical edition of Peter Olivi’s Rule Commentary (published by Franz Steiner Verlag, 1972). During the 1970’s and 1980’s, David worked with various Franciscan groups, especially the Franciscan Federation, in North America while residing in Chicago, Illinois. From time to time during that period he returned to Mönchengladbach for Franciscan research and writing. After returning to Montreal in 1991, he began working with Gedeon Gál, OFM, in 1994, and visited the Franciscan Institute regularly as they worked together on the Chronicle of Nicholas Minorita (The Franciscan Institute, 1996). He also worked on the exegetical works of Peter Olivi with Gedeon, publish- FRANCISCAN MEDAL 29 ing a critical edition of Olivi’s Principles of Sacred Scripture, and Postilla ’s on Isaiah and 1 Corinthians (The Franciscan Institute, 1997). He became a permanent member of the Franciscan Institute in January, 1998, and continued working on editions of Olivi’s exegetical works, completing the Commentary on the Acts of the Apostles (The Franciscan Institute, 2001). David is presently completing work on the edition of Olivi’s Commentary on the Book of Genesis which he hopes to complete within a year. In addition to his work on textual editions and scholarly papers, David has lectured widely in North America and Europe, as well as in Africa. He just recently returned from a semester in Rome where he taught a course on the Regula non Bullata at the Scuola Superiore di Studi Medievali at the Antonianum. The bibliography of David’s publications number more than fifty entries and includes a significant number of critical text editions, each of which is recognized as a quality example of medieval text editing. In addition to his edition of the Early Rule and Olivi’s Rule Commentary, David also has published editions of the Rule commentaries of Hugh of Digne (1980), David of Augsburg (1993), and John of Wales (2002). In addition to the exegetical texts and the Rule Commentary of Olivi, David edited Olivi’s Quaestio de Mendicitate, and together with David Burr, his fellow honoree this evening, he published an edition of Olivi’s Sixteenth Question on Evangelical Perfection dealing with poverty... (shrink)

We extend work of H. Friedman, L. Harrington and P. Welch to the third level of the projective hierarchy. Our main theorems say that (under appropriate background assumptions) the possibility to select definable elements of non-empty sets of reals at the third level of the projective hierarchy is equivalent to the disjunction of determinacy of games at the second level of the projective hierarchy and the existence of a core model (corresponding to this fragment of determinacy) which must then contain (...) all real numbers. The proofs use Sacks forcing with perfect trees and core model techniques. (shrink)

'The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.' David Hilbert (1862-1943). This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen (...) world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: Can infinity be found in the real universe? Finally, the book returns to questions of philosophical and theological aspects of infinity"--Provided by publisher. (shrink)

Introduction: occupation -- Impossibilities -- Irreconcilability -- Kaddish -- Authoritarianism: a case study -- Rosenzweig's patient -- Beyond.

Population-level biomedical research offers new opportunities to improve population health, but also raises new challenges to traditional systems of research governance and ethical oversight. Partly in response to these challenges, various models of public involvement in research are being introduced. Yet, the ways in which public involvement should meet governance challenges are not well understood. We conducted a qualitative study with 36 experts and stakeholders using the World Café method to identify key governance challenges and explore how public involvement can (...) meet these challenges. This brief report discusses four cross-cutting themes from the study: the need to move beyond individual consent; issues in benefit and data sharing; the challenge of delineating and understanding publics; and the goal of clarifying justifications for public involvement. The report aims to provide a starting point for making sense of the relationship between public involvement and the governance of population-level biomedical research, showing connections, potential solutions and issues arising at their intersection. We suggest that, in population-level biomedical research, there is a pressing need for a shift away from conventional governance frameworks focused on the individual and towards a focus on collectives, as well as to foreground ethical issues around social justice and develop ways to address cultural diversity, value pluralism and competing stakeholder interests. There are many unresolved questions around how this shift could be realised, but these unresolved questions should form the basis for developing justificatory accounts and frameworks for suitable collective models of public involvement in population-level biomedical research governance. (shrink)

MacNeilage's target article develops a theory for the evolution of human speech articulation along the lines of “slot- filler” structure. His content/frame schema commits him to the tenets of embodiment, muscle sense, and a memory for speech. My commentary ties these aspects together in their historical and current perspective.

Controversies over how to define the word religion have persisted for decades. It is a term of art and of academic study, but also one of governance, technologies, and of networks; it is a concept whose diversity is often its own worst enemy. Religion is as much a fuzzy set of conceptualizations and generalizations about a range of human activities as it is an authorizing system of persons, ideas, and practices. What is Religion?: Debating the Academic Study of Religion invites (...) readers to eavesdrop on scholarly debates over the limits of, and uses for, a word commonly used but infrequently defined in a precise manner. This volume takes the temperature of the modern field of Religious Studies by inviting a diverse group of scholars to offer their own substantive contribution that builds on the shared opening prompt, Religion is.... Their essays document the current state of the field and its various sub-fields, assess the progress that has been made over the past generation, and propose new directions for future work. Seventeen of the international field's leading scholars show how they work with each other's definition, or, sometimes, the lack of a definition. Of interest to students, scholars, and general readers alike, What is Religion? will provoke debate and provide insights into the state of the field. (shrink)

This article uses Kalman P. Bland’s The Artless Jew as a way to think about the recent history of the study of Judaism. The discipline’s preoccupation with disembodied texts has led to a way to conceptualize and situate Jews and Judaism that leaves certain blind spots and lacunae within our dominant narratives. To illumine some of these, the article focuses on ritual and what we can learn about the study of ritual in Judaism – and the study of Judaism more (...) generally – by connecting it not to the particularities of Jewishness, but to the ostensible universalism of larger fields of study, such as the academic study of religion. (shrink)

Although performance pressure has desirable consequences, there is evidence that it can produce unintended outcomes as employees tend to engage in dysfunctional and unethical behaviors to meet performance goals. Thus, the process through which employees think and behave unethically under performance pressure deserves more research attention. This study goes beyond the stress-appraisal perspective and investigates whether and when performance pressure influences individual work mindsets and behaviors from a moral reasoning perspective. Specifically, we contend that performance pressure is related to employee (...) expediency through moral decoupling. We further hypothesize dialectical thinking and moral identity to be the boundary conditions of the proposed relationships. Analyses of data from a field study in three waves provide support for most of the hypotheses. In particular, we find that moral decoupling accounts for additional variance after we control for the stress-appraisal effect of performance pressure on employee expediency. The study offers several contributions to the literature on performance pressure and unethical behavior. (shrink)