'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 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.

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 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)

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 this book, Professor Adkins undertakes an examination of certain key value-words in the period between Homer and the end of the fifth century. The behavior of these words both affected and was affected by the nature of the society in which their usage developed. The author shows how only with a complete understanding of the implications and significance of these value-words can the essence of the Greeks and their society be grasped.

This is an original and stimulating collection of articles by scholars trained in classics, moral philosophy, political science, literature, and intellectual history. Its principal objective is to convey to the modern reader a sophisticated understanding of Homeric and Classical Greek morality and how it differs from our own. Some of the articles focus primarily on Greek value concepts, especially the concept of arete. Others compare those concepts to modern notions of virtue and tolerance, as well as to the work of (...) contemporary literary figures and philosophers, including T.S. Eliot, Alasdair Macintyre, John Wallace, and Philippa Foot. Throughout, the juxtaposition of ancient and modern ideas and the worldviews they presuppose makes these readings both intellectually exciting and revealing. (shrink)

A publication by Professor Margenau is always of interest to persons concerned with philosophy of science. This is especially true, however, of his recently published book, The Nature of Physical Reality; for this book, dealing with basic epistemological problems arising from the development of modern quantum mechanics, is the most comprehensive and most systematic formulation of its author's philosophical position and is at the same time conceived as a “challenge” to “uncritical realism, unadorned operationalism, and radical empiricism”—to points of view, (...) that is, which Professor Margenau regards as “outmoded and in disharmony with the successful phases of contemporary physics”. (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 aim of this paper is to explore the conditions under which generalisation from single case studies, in the sense of making inferences concerning a wider class of phenomena beyond a case, is reasonable. Two sets of conditions, in particular, provide the basis for our consideration of this issue. The first is an exploration of the impressive amount of empirical knowledge that is contained within the theories that are used to make observations, to classify phenomena, and to understand and interpret (...) cases. The second is recognition of the role of a pattern of inference known as abduction, or inference to the best explanation, in drawing conclusions from case studies. It is argued that these two conditions, when applied to the procedures for conducting case studies, can be used to defend, as reasonable, the making of certain provisional generalisations from single cases. An epistemological procedure for testing and further justifying these generalisations, with a particular emphasis on dealing with the problem of confirmation bias, is then discussed and recommended. (shrink)

Philosophy of science and, more specifically, philosophy of quantum physics can be but special fields of a general philosophy of knowledge; and the problems arising in these fields can be evaluated properly only when they are seen under the perspective of the whole range of human knowledge. This paper deals with problems of quantum physics and, in particular, with the problem of scientific objects in quantum physics from the epistemological point of view previously defined in the author's books, A Philosophy (...) of Science and The Basis and Structure of Knowledge. The argument of the paper, however, has not been stated before in the direct and concise manner here attempted and may therefore contain elements of novelty even for those readers who are familiar with the books just mentioned. (shrink)

In a paper recently published in this quarterly I argued that modern quantum physics, as an integrated system of laws, supplements and completes in purely quantitative terms the fragmentary order of first-person experience, removing in a unique way ambiguities otherwise encountered at the level of common-sense things; and I contended that the choice of a different selective operator—purpose or value, let us say, rather than quantity—might entail an entirely different range and system of order. It is now my intention to (...) follow up this contention and to indicate at least in outline form some of the implications and possibilities of such a program; for it is possible that this undertaking may bring into new perspective some of the crucial problems which have vexed philosophers at all times when they have dealt with normative propositions and moral judgments. (shrink)

A short time ago, D. D. Lee wrote in Philosophy of Science: “Grammar contains in crystallized form the accumulated and accumulating experience, the Weltanschauung of a people.“ He thus called our attention once more to a theme which was much discussed during the 19th century but which has been in disrepute for some time in philosophical circles. It is a theme, however, which is not without merit. More than seven years of intensive study of primitive languages have convinced me that (...) the work started by Wilhelm von Humboldt deserves serious consideration even at present, and from the ethnological as well as from the philosophical point of view. (shrink)