'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)

Moral sensitivity and moral reasoning are essential competencies biomedical researchers have to develop to make ethical decisions in their daily practices. Previous research has shown that these competencies can be developed through ethics education. However, it is unclear which underlying mechanisms best support the development of these competencies. In this article we argue that the development of moral sensitivity and moral reasoning can be fostered through teaching strategies that tap into students’ moral imagination. We describe how moral imagination can stimulate (...) the development of these competencies through three different merits of moral imagination. Moral imagination can help students to 1) transfer and apply abstract moral concepts to concrete situations and contexts, 2) explore the perspective of others, 3) explore and foresee the moral consequences of different decisions and actions. We explain these three merits of moral imagination in the context of biomedical research and present a theoretical model for how these merits can be used to stimulate the development of moral sensitivity and moral reasoning. Furthermore, we describe multiple teaching strategies for biomedical curricula that tap into the three merits of moral imagination. These teaching strategies can inspire teachers to design ethics education that activates students’ moral imagination for the development of moral sensitivity and moral reasoning. (shrink)

The ongoing debate on the ethics of using artificial intelligence (AI) in military contexts has been negatively impacted by the predominant focus on the use of lethal autonomous weapon systems (LAWS) in war. However, AI technologies have a considerably broader scope and present opportunities for decision support optimization across the entire spectrum of the military decision-making process (MDMP). These opportunities cannot be ignored. Instead of mainly focusing on the risks of the use of AI in target engagement, the debate about (...) responsible AI should (i) concern each step in the MDMP, and (ii) take ethical considerations and enhanced performance in military operations into account. A characterization of the debate on responsible AI in the military, considering both machine and human weaknesses and strengths, is provided in this paper. We present inroads into the improvement of the MDMP, and thus military operations, through the use of AI for decision support, taking each quadrant of this characterization into account. (shrink)

This article examines, experimentally, whether inequality affects the social capital of trust in non-market and market settings. We consider three experimental treatments, one with equality, one with inequality but no knowledge of the income of other agents, and one with inequality and knowledge. Inequality, particularly when it is known, has a corrosive effect on trusting behaviours in this experiment. Agents appear to be less sensitive to known relative income differentials in markets than they are in the non-market settings, but trust (...) in markets appears generally more vulnerable to the introduction of inequality than in the non-market setting. (shrink)

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

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.