'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)

We prove a strong boundedness theorem for dilators: if A ⊆ DIL is Σ 1 1 , then there is a recursive dilator D 0 such that ∀ D ∈ A.

We show how to get canonical models from in which the nonstationary ideal on ω1 is ω1 dense and there is no P-point.

We show how to get canonical models from in which the nonstationary ideal on ω1 is ω1 dense and there is no P-point.

Assume ZF + AD +V=L and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof = ω thenκis Rowbottom. We also establish some other partition properties.

We show the Borel Conjecture is consistent with the continuum large.

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)

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)

First Published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

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 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. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

For a serious book of philosophy, where better to begin to canvass various philosophical concepts and arguments than in relation to what is so familiar to every one of us –– the fact that we all have many and varied beliefs. The book is an introduction of philosophy, indeed intended as an introductory textbook. The author, as he wrote it, had both the teacher and the student in mind. He hopes it will prove a worthy contribution in the college, seminary (...) and university classroom, both interesting and serious. As well as thirteen clearly written chapters introducing the various topics, it is also provided with helpful summaries, tutorials, and work sheets. In considering belief we raise raises many of the central problems philosophers have discussed: knowledge, truth, justification, rationality, meaning, explanation, self deception, interpretation, reality, cause and effect, personal identity, theories, laws, hypotheses, the self, survival, God. Since belief is a universal phenomenon, it has unfortunately become common to understand the unqualified term ‘believer’ of the religious person. It seems strange to ask the question, ‘Are you a believer?’ outside the religious context. But we do when we are thinking of a particular theory or ideology or political attitude. We sometimes want to know whether she is ‘one of us!’ The author sometimes finds it convenient to illustrate his exposition by referring to religious beliefs. One does not have to be a religious believer to see that it is relevant and indeed interesting to do so. The history of philosophy provides many classical examples of such discussion. The book is of wide general interest. As well as doing service in the classroom, it will also prove its worth within other contexts. It will serve the aims of serious discussion groups, as well as providing a basis for regular and earnest individual study. We hope also that it will find a place with inquiring people of religious faith. (shrink)

Aesthetics, hermeneutics, criticism -- Social Werktreue and the subjectivization of aesthetics -- From musike to metaphysics -- Formalist aesthetics and musical hermeneutics -- Deconstructing the disciplinary divide -- The question of metaphor -- Mimesis and the hermeneutics of music -- Political critique and the politics of music criticism -- Toward a hermeneutics of music criticism.

This book presents the methodological framework of combining Multimodal Conversation Analysis (MCA) with Interpretative Phenomenological Analysis (IPA) to interpretively analyse translanguaging practices in educational contexts. Beginning with an overview of the three uses of translanguaging - translanguaging as a theory of language, as a pedagogical practice and as an analytical perspective - the book goes on to critically examine the different methodological approaches for analysing translanguaging practices in multilingual classroom interactions. It explains how MCA and IPA are useful methodologies for (...) understanding how and why translanguaging practices are constructed by participants in the classroom and discusses types of data collected and data collection procedures. The author, Kevin W. H. Tai, shows how combining these approaches enables researchers to study how translanguaging practices are constructed in multilingual classrooms and how teachers make sense of their own translanguaging practices at particular moments of classroom interaction. This detailed and concise guide is indispensable for students, practitioners, policymakers, and researchers from across the globe, particularly those working in the fields of applied linguistics and language education. (shrink)

Some years ago, in the course of a general critique of what has sometimes been referred to as the covering law theory of explanation, I made the claim that perfectly satisfactory explanations can often be provided by indicating only one or a few necessary conditions, where we remain ignorant of the sufficient conditions, of what we nevertheless claim to understand. What seemed to me one identifiable type of such explanations I called “explaining how-possibly,” because it was a type more naturally (...) given in response to the question how it could be that a certain thing happened than to the more familiar question why it did so. In the present paper I propose to review certain objections to this thesis which have come to my attention in the interval, and to make such concessions as these seem to require. In fact, the concessions will be minor, since the general thesis still seems to me quite defensible. (shrink)

First published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

First published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

Privacy is defined within a symbolic interaction framework in terms of identity definition and maintenance processes. It is argued that defining privacy within a symbolic interaction framework both generates a number of hypotheses involving the concept of privacy and allows the theorist to draw together several social psychological concepts within the one conceptual schema.

Periodic tables (PTs) are the ‘ultimate paper tools’ of general and inorganic chemistry. There are three fields of open questions concerning the relation between PTs and physics: (i) the relation between the chemical facts and the concept of a periodic system (PS) of chemical elements (CEs) as represented by PTs; (ii) the internal structure of the PS; (iii)␣The relation between the PS and atomistic quantum chemistry. The main open questions refer to (i). The fuzziness of the concepts of chemical properties (...) and of chemical similarities of the CE and their compounds guarantees the autonomy of chemistry. We distinguish between CEs, Elemental Stuffs and Elemental Atoms. We comment on the basic properties of the basic elements. Concerning (ii), two sharp physical numbers (nuclear charge and number of valence electrons) and two coarse fuzzy ranges (ranges of energies and of spatial extensions of the atomic orbitals, AOs) characterize the atoms of the CEs and determine the two-dimensional structure of the PS. Concerning (iii), some relevant ‘facts’ about and from quantum chemistry are reviewed and compared with common ‘textbook facts’. What counts in chemistry is the whole set of nondiffuse orbitals in low-energy average configurations of chemically bonded atoms. Decisive for the periodicity are the energy gaps between the core and valence shells. Diffuse Rydberg orbitals and minute spin–orbit splittings are important in spectroscopy and for philosophers, but less so in chemical science and for the PS. (shrink)