A real x is -Kurtz random if it is in no closed null set . We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.

We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity. We also consider Low(M, Kurtz), the class of degrees a such that every element of M is a-Kurtz random. These are characterised when M is the class of Martin-Löf random, computably random, (...) or Schnorr random reals. We show that Low(ML, Kurtz) coincides with the non-DNR degrees, while both Low(CR, Kurtz) and Low(Schnorr, Kurtz) are exactly the non-high, non-DNR degrees. (shrink)

This essay is excerpted by Nathan Bupp from Paul Kurtz, The Turbulent Universe (Amherst, N.Y.: Prometheus Books, 2013). Copyright ©2013 by the Estate of Paul Kurtz. All rights reserved. Used by permission of the Estate and the publisher, www.prometheusbooks.com.

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.

The weakly random reals contain not only the Schnorr random reals as a subclass but also the weakly 1-generic reals and therefore the n -generic reals for every n . While the class of Schnorr random reals does not overlap with any of these classes of generic reals, their degrees may. In this paper, we describe the extent to which this is possible for the Turing, weak truth-table, and truth-table degrees and then extend our analysis to the Schnorr random and (...) hyperimmune reals. (shrink)

We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness (...) notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set. (shrink)

We show that a computable function $f:\mathbb R\rightarrow \mathbb R$ has Luzin’s property if and only if it reflects $\Pi ^1_1$ -randomness, if and only if it reflects $\Delta ^1_1$ -randomness, and if and only if it reflects ${\mathcal {O}}$ -Kurtz randomness, but reflecting Martin–Löf randomness or weak-2-randomness does not suffice. Here a function f is said to reflect a randomness notion R if whenever $f$ is R-random, then x is R-random as well. (...) If additionally f is known to have bounded variation, then we show f has Luzin’s if and only if it reflects weak-2-randomness, and if and only if it reflects $\emptyset '$ -Kurtz randomness. This links classical real analysis with algorithmic randomness. (shrink)

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ of distinct integers. Call such an $x$ UD random. Here it is shown that every Schnorr random real is UD random, but there are (...) Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random. (shrink)

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an (...) n such that n,2n,…,kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Π10 with computable measure, and Π10. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0,1}Nk. (shrink)

We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.

Polynomial clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone ${\fancyscript{C}}$ , a sequence ${B\in X^{\omega}}$ is ${\fancyscript{C}}$ -reducible to ${A \in {X}^{\omega}}$ if there is an algorithm that computes B from A using only effectively selected functions from ${\fancyscript{C}}$ . We show that if A is Kurtz random and (...) ${\fancyscript{C}_{1} \nsubseteq \fancyscript{C}_{2}}$ are polynomial clones, then there is a B that is ${\fancyscript{C}_{1}}$ -reducible to A but not ${\fancyscript{C}_{2}}$ -reducible to A. This implies a generalization of a result first proved by Lachlan (Z Math Logik Grundlagen Math 11:17–44, 1965) for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov–Loveland stochastic. (shrink)

There exist two main notions of typicality in computability theory, namely, Cohen genericity and randomness. In this article, we introduce a new notion of genericity, called partition genericity, which is at the intersection of these two notions of typicality, and show that many basis theorems apply to partition genericity. More precisely, we prove that every co-hyperimmune set and every Kurtz random is partition generic, and that every partition generic set admits weak infinite subsets, for various notions of weakness. (...) In particular, we answer a question of Kjos-Hanssen and Liu by showing that every Kurtz random admits an infinite subset which does not compute any set of positive effective Hausdorff dimension. Partition genericity is a partition regular notion, so these results imply many existing pigeonhole basis theorems. (shrink)

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences. (...) In what follows, we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general. (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.

In an opinion piece penned at the Great War’s onset yet apparently unpublished until now, the historian of religion Hermann Gunkel outlined the opportunities he saw for the German people in anticipation of their triumph. He believed this war could consummate what the Napoleonic Wars and the Unification of Germany had not. Gunkel hoped for true German unity, more liberal domestic politics, and spiritual restoration. Further still, he referred to a resurgence of piety on account of the conflict. On the (...) Great War’s centenary, this documentation speaks to the mobilizing, nationalist activities of Gunkel in particular and of academicians and theologians more generally. (shrink)

Paul Kurtz has been the dominant voice of secular humanism over the past thirty years. This compilation of his work reveals the scope of his thinking on the basic topics of our time and his many and varied contributions to the cause of free thought. It focuses on the central issues that have concerned Kurtz throughout his career: ethics, politics, education, religion, science, and pseudoscience. The chapters are linked by a common theme: the need for a new enlightenment, (...) one committed to the use of rationality and skepticism, but also devoted to realizing the highest values of humanist culture. Many writings included here were first published in magazines and journals long unavailable. Some of the essays have never before been published. They now appear as a coherent whole for the first time. Also included is an extensive bibliography of Kurtz's writings. Toward a New Enlightenment is essential for those who know and admire Paul Kurtz's work. It will also be an important resource for students of philosophy, political science, ethics, and religion. Among the chapters are: "Humanist Ethics: Eating the Forbidden Fruit"; "Relevance of Science to Ethics"; "Democracy without Theology"; "Misuses of Civil Disobedience"; "The Limits of Tolerance"; "Skepticism about the Paranormal: Legitimate and Illegitimate"; "Militant Atheism vs. Freedom of Conscience"; "Promethean Love: Unbound"; "The Case for Euthanasia"; and "The New Inquisition in the Schools.". (shrink)

Based on the author's ethnographic research in India, the book explores the psychology of Hinduism, and offers an innovative synthesis of psychoanylsis with modern anthropological theories of cultural difference. Stanley N. Kurtz offers a new interpretation of the multiple "mother goddesses" of Hinduism, and explores how this multiplicity is key to understanding early childhood experience in which a child is raised by many "mothers" in the Hindu joint family. Arguing that traditional psychoanalytic approaches to Indian culture have applied Western (...) models without regard to differing cultural circumstances, Kurtz suggests that immersion in a joint family plays a central role in the development of feelings and needs which are carried over into adult life. The book concludes with a briefr comparison of mothering in India and contemporary America. _All the Mothers Are One_ makes a significant contribution to the growing debate concerning the role of psychoanalysis in the interpretation of culture and the study of cross-cultural human development. (shrink)

First encounters : Jesuit logica in the late Ming and early Qing -- Haphazard overtures : logic in nineteenth-century Protestant writings -- Great expectations : Yan Fu and the discovery of European logic -- Spreading the word : logic in late Qing education and popular discourse -- Heritage unearthed : the discovery of Chinese logic -- Textbooks on logic adapted from Japanese, 1902-1911 -- Logical terms in early-twentieth-century textbooks.

Book reviewed in this article: Bioethics and Secular Humanism: The Search for a Common Morality. By H. Tristram Engelhardt, Jr.

Is life meaningful without religion? Can one be moral and not believe in God? While many Americans believe that God is necessary to secure moral order, Paul Kurtz argues that it is quite possible for rationalists and freethinkers to lead exemplary lives. Embracing the Power of Humanism is a collection of essays organized into five parts: "The Exuberant Life," "Independence," "Altruism," "Humanism," and "Ethical Truth" throughout which Kurtz provides nonbelievers with ethical guidelines and encourages all individuals to take (...) personal responsibility for moral excellence. (shrink)

Kurtz (philosophy, SUNY at Buffalo; editor, Free inquiry; and president, Prometheus Books) collects his essays, articles, and contributions to books written over the past thirty-five years. Annotation copyrighted by Book News, Inc., Portland, OR.

Kurtz argues that there are objective standards for judging truth claims in science, ethics, and philosophy. Of special interest is the application of the new skepticism to paranormal claims such as reincarnation and faith healing, and to religious beliefs, ethics and politics.