It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure $\langle \omega; +, P\rangle$ , where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of $\langle\omega; S, P\rangle$ is decidable, where S is the successor function. The latter result is proved using a general (...) result of A. L. Semenov on decidability of monadic theories, and a proof of Semenov's result is presented. (shrink)

We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...) c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A\B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of "computable at density r" where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. (shrink)

We introduce the notion of "semi-r.e." for subsets of ω, a generalization of "semirecursive" and of "r.e.", and the notion of "weakly semirecursive", a generalization of "semi-r.e.". We show that A is weakly semirecursive iff, for any n numbers x 1 ,...,x n , knowing how many of these numbers belong to A is equivalent to knowing which of these numbers belong to A. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the (...) other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion "A is weakly semirecursive and recursive in K" in terms of recursive approximations to A. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised. (shrink)

We investigate operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K. We introduce notions of nontriviality for such operators, and use these to study which additional properties can be required of sets which can be completed to the jump by given operators of this kind.

It is shown that for every nonzero r.e. degree c there is a linear ordering of degree c which is not isomorphic to any recursive linear ordering. It follows that there is a linear ordering of low degree which is not isomorphic to any recursive linear ordering. It is shown further that there is a linear ordering L such that L is not isomorphic to any recursive linear ordering, and L together with its ‘infinitely far apart’ relation is of low (...) degree. Finally, an analogue of the recursion theorem for recursive linear orderings is refuted. (shrink)

It was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low₂ homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition $C\,\not \leqslant _T \,H$, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring (...) of pairs admits a pair of low₂ infinite homogeneous sets whose degrees form a minimal pair. (shrink)

We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω in (...) which the functions and relations have degree at most c. We show that every degree d ≥ 0 (ω) is the ωth jump degree of a Boolean algebra, but that for $n no Boolean algebra has nth-jump degree $\mathbf{d} > 0^{(n)}$ . The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties. (shrink)

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.

A study of the analogies between alchemy, Christian dogma, and psychological symbolism. Revised translation, with new bibliography and index.

This book examines the reactions of the friends and family of those who elect to die due to terminal illness. These surviving spouses, partners, relatives, and friends, in addition to coping with the death of a loved one, must also deal with the loved one's decision to die, thus severing the relationship. C. G. Prado examines how reactions to elective death are influenced by cultural influences and beliefs, particularly those related to life, death, and the possibility of an afterlife. Understanding (...) the role of these cultural influences on the grieving processes of survivors is a crucial step in allowing them to accept both intellectually and emotionally the finality of elective death and to deal with the decision of their loved one. (shrink)

In the 1950s and 60s, H. L. A. Hart's writings were responsible for the revitalizing of legal philosophy in the UK and worldwide. This new collection of essays by distinguished philosophers and legal theorists addresses the enduring significance of Hart's work, focusing upon some of its more neglected aspects, including his work on causation and the philosophy of action. Two essays discuss Hart and Honore;'s Causation in the Law, a book that has been undeservedly overlooked in many philosophical discussions of (...) causation. The majority of the essays consider Hart's views on the justification of punishment. They discuss Hart's dislike of retribution, his claim that harmless immorality should not be punished, and his search for a non-retributivist theory of mens rea and excuses. Other topics examined include Hart's definitions of key legal and philosophical concepts, including 'punishment', 'responsibility' and 'voluntariness', and his views on punishing negligent acts. (shrink)

This book is parapsychological study of the meaningful coincidence of events, extrasensory perception, and similar phenomena.

The Dushnik–Miller Theorem states that every infinite countable linear ordering has a nontrivial self-embedding. We examine computability-theoretical aspects of this classical theorem.

The concept of 'Archteypes' and the hypothesis of 'A Collective Unconscious' are two of Jung's better known and most exciting ideas. In this volume - taken from the Collected Works and appearing in paperback for the first time - Jung describes and elaborates the two concepts. Three essays establish the theoretical basis which are then followed by essays on specific archetypes. The relation of these to the process of individuation is examined in the last section. _The Archetypes and the Collective (...) Unconscious_ is one of Jung's central works. There are many illustrations in full colour. (shrink)

"From The collected works of C.G. Jung, volumes 4, 8, 12, 16"--P. [i].

Michel Foucault had a great influence upon a wide range of disciplines, and his work has been widely interpreted and is frequently referred to, but it is often difficult for beginners to find their way into the complexities of his thought. This is especially true for readers whose background is Anglo-American or "analytic" philosophy. C. G. Prado argues in this updated introduction that the time is overdue for Anglo-American philosophers to avail themselves of what Foucault offers. In this clear and (...) greatly-revised second edition, Prado focuses on Foucault's "middle" or genealogical work, particularly Discipline and Punish and Volume One of The History of Sexuality, in which Foucault most clearly comes to grips with the historicization of truth and knowledge and the formation of subjectivity. Understanding Foucault's thought on these difficult subjects requires working through much complexity and ambiguity, and Prado's direct and accessible introduction is the ideal place to start. (shrink)

_Psychological Types_ is one of Jung's most important and most famous works. First published by Routledge in the early 1920s it appeared after Jung's so-called fallow period, during which he published little, and it is perhaps the first significant book to appear after his own confrontation with the unconscious. It is the book that introduced the world to the terms 'extravert' and 'introvert'. Though very much associated with the unconscious, in _Psychological Types_ Jung shows himself to be a supreme theorist (...) of the conscious. In putting forward his system of psychological types Jung provides a means for understanding ourselves and the world around us: our different patterns of behaviour, our relationships, marriage, national and international conflict, organizational functioning. Appearing in paperback for the first time this central volume from Jung's Collected Works will be essential to anyone requiring a proper understanding of Jung's psychology. (shrink)

Reprint. Originally published: 1959; 1st Princeton/Bollingen pbk. ed. published: 1970.

For more than seven decades there has been a broad gap between how philosophy is conceived and practiced. Two ill-defined but well-recognized traditions have developed—the "analytic" and "Continental" schools of philosophy. The former traces its roots to philosophers like Frege, Russell, Moore, Wittgenstein, and the logical positivists. The latter has been heavily influenced by Nietzsche, Heidegger, Merleau-Ponty, Foucault, and Derrida, among others. The aim of this collection is to reconsider the often facile characterization of major thinkers as belonging to either (...) one or the other philosophical tradition. The contributors—philosophers from both sides of the divide working in different countries and contexts—all question the problematic conception that the two traditions are incommensurable. Each of their articles compares individual philosophers who have had a major influence on the analytic and Continental traditions with a view to clarifying their similarities and dissimilarities of approach. What this collection of thoughtful articles clearly demonstrates is that regardless of approach and precedents, analytic and Continental philosophers are all doing philosophy, and there are many important points of contact between them. The contributors are: Richard Rorty, whose thoughtful overview highlights the salient points in both traditions; Barry Allen ; Babette E. Babich ; David Cerbone ; Sharyn Clough ; Jonathan Kaplan ; Richard Matthews ; Carlos G. Prado ; Bjorn Torgrim Ramberg ; Mike Sandbothe ; Barry Stocker ; and Ed Witherspoon. (shrink)

Contents: Mandalas. I. A Study in the Process of Individuation. II. Concerning Mandala Symbolism Index Originally published in 1972. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in (...) the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Recently, Brukner and Zeilinger 3354) have claimed that the Shannon information is not well defined as a measure of information in quantum mechanics, adducing arguments that seek to show that it is inextricably tied to classical notions of measurement. It is shown here that these arguments do not succeed: the Shannon information does not have problematic ties to classical concepts. In a further argument, Brukner and Zeilinger compare the Shannon information unfavourably to their preferred information measure, I , with regard (...) to the definition of a notion of “total information content.” This argument is found unconvincing and the relationship between individual measures of information and notions of “total information content” investigated. We close by considering the prospects of Zeilinger's Foundational Principle as a foundational principle for quantum mechanics. (shrink)