'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 the Borel Conjecture is consistent with the continuum large.

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 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 1938. The aim of this book is to expound Kant’s _Critique of Judgement _by interpreting all the details in the light of what Kant himself declares to be his fundamental problem. _A Commentary on Kant’s Critique of Judgement _provides an excellent introduction to Kant’s third critique, and will be of interest to students of philosophy.

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

'Uncertainty and chance' is a subject with a broad span, in that there is no academic discipline or walk of life that is not beset by uncertainty and chance. In this book a range of approaches is represented by authors from varied disciplines: natural sciences, mathematics, social sciences and medical sciences. At one extreme, this volume is concerned with the foundations of probability. At the other extreme, we have scholars who acknowledge the concept of chance and uncertainty but do not (...) cope with it by means of systematic measurement or quantative analysis. (shrink)

"First published by Oxford University Press, 1916."--Title page verso.

The significance of the dating of Enmann's Kaisergeschichte in the controversy which has long surrounded the Historia Augusta is common knowledge to all scholars who have more than a nodding acquaintance with the period. Enmann himself concluded that the KG ended with or shortly after Diocletian's accession. This was a necessary hypothesis for Enmann in 1884 because the H.A. had clearly used the KG and the self-proclaimed authorship and dating of the former were generally accepted. None the less in Enmann's (...) opinion the resemblances between Victor and Eutropius continued into the sole reign of Constantius II down to the Battle of Strasbourg in 357 and Enmann was compelled to ascribe this to a ‘continuation’ of the KG. (shrink)

In assessing the value of any historical work it is necessary for the investigator to undertake the often frustrating and tedious task of Quellenforschung. In the case of the De Caesaribus the first substantial attempts began in Germany in 1873 and 1874 with the appearance of two important studies by A. Enmann and A. Cohn. Enmann sought to explain the mass of verbal similarities, numerous errors and shared idiosyncrasies to be found in Victor's De Caesaribus, Eutropius' Breviarium 7–10, and parts (...) of the Historia Augusta by postulating a common source which has subsequently come to be known as the Kaisergeschichte. He was so convinced by his findings that he stated , ‘Victor und Eutrop es beide als Haupt – wenn nicht als einzige Quelle benutzt haben’. Working quite independently Cohn's examination of the first eleven chapters led him to conclude that from the beginning of Augustus' sole reign to the death of Domitian Victor, Eutropius and the Epitomator followed a common source which he entitled Suetonius auctus, an account excerpted from Suetonius with some new material added. Enmann, however, had already described the Kaisergeschichte's early chapters in this way and the two were readily identified. One major problem remained. The traditional date of the completion of the HA was still believed to be 305/6; consequently if the K.G. had been used by its authors then the K.G. must have been written by that year. Nevertheless notable similarities between Victor and Eutropius continued apparently till 357 and Enmann therefore suggested a continuation of the K.G. until that date. (shrink)