'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 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.

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.

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)

In this paper I attempt to trace certain aspects of the evolution of theories of space-time and mechanics as revealed by a brief comparative study of Newtonian theory, Robb's theory, general relativity, and Milne's kinematical relativity. The first object is to emphasise how each theory leaves us in a position in which the succeeding one appears as a perfectly natural next step in the development of ideas. The second object is to show how, in spite of superficial differences in character, (...) the theories all necessarily possess the same general structure constituted by the presence of hypotheses, from which certain general mathematical relations are deduced, which in their turn are used to predict relations between observable quantities. (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)

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)

Anime is exploding on the worldwide stage! Anime has been a staple in Japan for decades, strongly connected to manga. So why has anime become a worldwide sensation? A cursory explanation is the explosion of online streaming services specializing in anime, like Funimation and Crunchyroll. Even more general streaming services like Netflix and Amazon have gotten in on the game. Anime is exotic to Western eyes and culture. That is one of the reasons anime has gained worldwide popularity. This strange (...) aesthetic draws the audience in only to find it is deeper and more sophisticated than its surface appearance. Japan is an honor and shame culture. Anime provides a platform to discuss “universal” problems facing human beings. It does so in an amazing variety of ways and subgenres, and often with a sense of humor. The themes, characters, stories, plotlines, and development are often complex. This makes anime a deep well of philosophical, metaphysical, and religious ideas for analysis. International scholars are represented in this book. There is a diversity of perspectives on a diversity of anime, themes, content, and analysis. It hopes to delve deeper into the complex world of anime and demonstrate why it deserves the respect of scholars and the public alike. (shrink)

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