Can we deploy creative practices to critically address the fatal interlocking of global surveillance technologies, neo-colonial expansionism, environmental degradation and the lethal threat of drone warfare? Throughout the following conversation, Shona Illingworth and Anthony Downey examine these and other questions in relation to the recent publication of Topologies of Air (Sternberg Press and The Power Plant, 2022). Edited by Downey, the book includes discussion and documentation of two major bodies of work by Illingworth, including Topologies of Air (2021) (...) and Lesions in the Landscape (2015), alongside an extended series of essays that analyse the psychological and environmental impact of military, industrial and corporate transformations of airspace and outer space. Employing interdisciplinary research and collaborative processes, Illingworth’s practice, as detailed in the discussion below, uses creative methodologies to visualize and interrogate this proliferating exploitation of airspace. The conversation between Illingworth and Downey also outlines the work of the Airspace Tribunal, an ongoing series of public hearings that brings together diverse disciplines, methodologies, knowledge and lived experiences to propose a new human right that will counter the colonization of the sky and, in time, protect individuals, communities and ecologies from ever-increasing threats from above. (shrink)

We characterize the class of c.e. degrees that bound a critical triple as those degrees that compute a function that has no ω-c.e. approximation.

It is still commonly supposed that Descartes based his argument for the mind-body distinction on the law of the indiscernibility of identicals. I argue that this interpretation is very unlikely to be correct. I explain three contemporary versions of this interpreta- tion and say why I reject it. Basically, use of this law for Descartes’s conclusion would require reference to human bodies or else the supposition, for the purpose of the argument, of reference to human bodies. But at the time (...) Descartes formulated his argument, he would not have allowed either. Instead, I amend a different interpretation which others have found, and briefly defend it against one famous objection. (shrink)

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new (...) characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members. (shrink)

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

Robert I. Soare, Automorphisms of the Lattice of Recursively Enumerable Sets. Part I: Maximal Sets.Manuel Lerman, Robert I. Soare, $d$-Simple Sets, Small Sets, and Degree Classes.Peter Cholak, Automorphisms of the Lattice of Recursively Enumerable Sets.Leo Harrington, Robert I. Soare, The $\Delta^0_3$-Automorphism Method and Noninvariant Classes of Degrees.

Patrick Downey finds comedy at the heart of the Western philosophical and theological tradition. In Serious Comedy Downey tracks tragedy and comedy from the beginning of Western thought to the twentieth century, beginning with an in-depth examination of Aristotle and three Platonic dialogues: the Republic, the Phaedrus, and the Symposium. In the book's second section Downey argues that the Bible is at heart a comedic narrative and analyzes the philosophical and theological implications of this comedy. In the (...) third section Downey traces comedy and tragedy in classic and modern philosophers such as Machiavelli, Dante, Hegel, Nietzsche, and Kierkegaard. Readers with an interest in classical philosophy and theology, political philosophy, and narrative theory will be particularly interested in this insightful and thoughtful perspective on the Western tradition. (shrink)

This book presents new results in computability theory, a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field's connections with disparate areas of mathematical logic and mathematics more generally have grown deeper, and now have a variety of applications in topology, group theory, and other subfields. This monograph establishes new directions in the field, blending classic results with modern research areas such as algorithmic randomness. The significance of the book lies not only (...) in the depth of the results contained therein, but also in the fact that the notions the authors introduce allow them to unify results from several subfields of computability theory. (shrink)

Informal construction -- Formal construction -- Limiting results.

We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K (...) for random α. Following work of Downey, Hirschfeldt and LaForte, we say that αK β iff there is a constant such that for all n, . We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that lim supn K−K=∞ where Ω is Chaitin's random real. One is based upon work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that μ=0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n ≠ m then the initial segment complexities of the natural n- and m-random sets and Ω︀) are different. The techniques introduced in this paper have already found a number of other applications. (shrink)

Slavoj Zizek is operating from a position of certainty, a position discovered by Jacques Lacan in Seminar XI. In this essay, I examine this position of certainty ("Gewissheit") and the ways this position is distinct from both existential phenomenology and post-structuralism, ultimately arguing that for structuralist psychoanalysis to function requires an intentional forgetting of being.

In response to the many pressures facing public higher education, public universities are experimenting with business-oriented practices that seem likely to alter their nature and purposes. In this paper, we examine several hybrid experiments—new organizational strategies intended deliberately, sometimes explicitly, to hybridize the traditional norms and practices associated with academia and business at one emblematic public university. These cases illustrate how each hybrid experiment is a tacit response to existing norms and strategies that govern the university–business boundary, initiated as a (...) hedge against the challenging fiscal and political climate. Taken together, they do not lead to a unitary and/or linear spread of business codes and practices. Instead, what some have referred to as “business logic” appears multifaceted, having many elements that are deployed, institutionalized, and perceived differently in different contexts, even within a single university. (shrink)

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is ${\Sigma _1^1 }$ -complete. This result and proof have a number of nice corollaries: the Scott rank of ε is $\omega _1^{{\rm{CK}}}$ + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly $\Delta (...) _\alpha ^0 $ orbit (from the proof). (shrink)

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that (...) no set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ⋃ c) ⋂ (b ⋃ c), a ⋃ c | b ⋃ c, and c < a ⋃ b.

However, the relation of the novels to ancient philosophy remains under-studied. This volume is intended to open up some of the issues involved.