By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size $\beth_n$ with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting types over T, H(T), is at least $\beth_\omega$ . (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly $\beth_\omega$ ). If T ≠ true arithmetic, then $H(T) = \beth_{\omega1}$ . If $\delta \not\rightarrow (\rho)^{ , then any completion of (...) Peano Arithmetic has a model of size δ with no set of indiscernibles of size ρ. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L. (shrink)

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let ε denote the structure of the computably enumerable sets under inclusion, $\varepsilon = (\{W_e\}_{e\in \omega}, \subseteq)$ . We previously exhibited a first order ε-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has (...) a certain "slowness" property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A ∈ ε there exists B in the orbit of A such that X ≤ T B under relative Turing computability (≤ T ). We produce B using the Δ 0 3 -automorphism method we introduced earlier. (shrink)

We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let [Formula: see text] is the Turing degree of a [Formula: see text] set J ≥T0″}. Let [Formula: see text] such that [Formula: see text] is upward closed in [Formula: see text]. Then there is an ℒ property [Formula: see text] such that [Formula: see text] if and only if there is an A where A ≡T F and [Formula: see text]. (...) A corollary of this is that, for all n ≥ 2, the high n computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin's Invariance Conjecture. (shrink)

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, (...) ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable. We use A* to denote the equivalent class of A under the ideal of finite sets. (shrink)

Post in 1944 began studying properties of a computably enumerable set A such as simple, h-simple, and hh-simple, with the intent of finding a property guaranteeing incompleteness of A . From the observations of Post and Myhill , attention focused by the 1950s on properties definable in the inclusion ordering of c.e. subsets of ω, namely E = . In the 1950s and 1960s Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating ∄-definable properties (...) of A , like maximal, hh-simple, atomless, to the information content of A . Harrington and Soare gave an answer to Post's program for definable properties by producing an ∄-definable property Q which guarantees that A is incomplete and noncomputable, but developed a new Δ 3 0 -automorphism method to prove certain other properties are not ∄-definable. In this paper we introduce new ∄-definable properties relating the ∄-structure of A to deg, which answer some open questions. In contrast to Q we exhibit here an ∄-definable property T which allows such a rapid flow of elements into A that A must be complete even though A may possess many other properties such as being promptly simple. We also present a related property NL which has a slower flow but fast enough to guarantee that A is not low, even though A may possess virtually all other related lowness properties and A may simultaneously be promptly simple. (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)

The luminaries of late thirteenth-century Europe took great interest in the mysterious fifth-century author known as Dionysius the Areopagite. They typically read Dionysius not in the original Greek, but in a Latin edition prepared sometime in the middle of the thirteenth century. This edition, which appeared first in Paris and later circulated all over Western Europe, was no mere translation. In addition to the famous translation made by Eriugena in the ninth century, it contained translations of scholia on the Dionysian (...) texts made by Anastasius the Librarian, alternative readings provided by Anastasius and other Latin readers, as well as excerpts from Eriugena's own theological masterwork, the Periphyseon. University scholars such as Albert the Great and Thomas Aquinas thus learned Dionysian mystical theology not only from his text, but from the seven-hundred year interpretive tradition that literally surrounded it on the page. (shrink)

We show that if A and $\widehat{A}$ are automorphic via Φ then the structures $S_{R}(A)$ and $S_{R}(\widehat{A})$ are $\Delta_{3}^{0}-isomorphic$ via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

The medieval fascination with the mysterious language of Dionysius the Areopagite is nowhere more evident than in the thirteenth-century textbook edition of his treatise on liturgical rites. Dionysius employed unfamiliar Greek to describe people, actions, and texts that would have been perfectly familiar to his readers. The Latin translation used in the thirteenth-century textbook strives to preserve this unfamiliarity, but commentaries are introduced between its lines and paragraphs, disrupting its ability to bewilder and surprise. These commentaries make the Dionysian text (...) less mysterious, while also slightly altering its meaning. In the hands of the commentators, Dionysius becomes less interested in the aesthetic mystery of the liturgy, and more interested in credal orthodoxy. To read text and commentary together is to confront seven hundred years of competing voices speaking on the nature and purpose of the Christian church. (shrink)

The Dionysian corpus makes virtually no statement about the authority of kings or the structure of nations, but it has nevertheless repeatedly been the subjectof political analysis. Several scholars have recently sketched out a Dionysian politics by drawing analogies between the Dionysian church and the city, and between the Dionysian bishop and the emperor. These analogies are of limited usefulness. They show that Dionysius does employ Platonic political language to describe the ecclesiastical hierarchy, but they risk overlooking or downplaying the (...) hierarchy’s non-temporal, and therefore non-political, activity. A more fruitful ground for developing a Dionysian politics may be found in his brief discussion of the legal hierarchy, which provides practical instruction for action in the temporal realm without direct reference to the contemplative activity of the church. (shrink)

From the late nineteenth century onwards, Asian Buddhist monks have been associated in American thought with science, rationality and anti-colonialism. Though the narrative of nineteenth century ‘Buddhist Modernism’ is routinely invoked to explain this, a more illuminating genealogy of this ‘modernist monasticism’ identifies deeper roots in anti-Catholicism. This paper explores these roots through a genealogy of the Buddhist Modernist Monk. Beginning with the seventeenth century travel journals of Jesuit missionaries, it winds its way through varied British rhetorics to nineteenth century (...) Sri Lanka, and ends in Chicago, at the World’s Parliament of Religion of 1893. There, these intertwined discourses coalesced in the form of the Buddhist Modernist Monk: a figure now familiar and beloved in American culture as an embodiment of compassion and rationality, yet with a history of prejudice and politics that has yet to be meaningfully explored. As we acknowledge anti-Catholicism’s centrality to the history of the Modernist Monk, we are necessarily reminded of the moral ambivalence of the ‘science-religion’ dichotomy that fuels his mystique. At minimum, future analyses must critique the presumption of such supra-historical binaries, and deploy an open framework attentive to the contradictions and relations of reciprocal determination that characterize his genealogy. (shrink)

The twentieth century discovered the concept of sacred place largely through the work of Martin Heidegger and Mircea Eliade. Their writings on sacred place respond to the modern manipulation of nature and secularization of space, and so may seem distinctively postmodern, but their work has an important and unacknowledged precedent in the Neoplatonism of Late Antiquity and the early Middle Ages. Sacred Place in Early Medieval Neoplatonism traces the appearance and development of sacred place in the writings of Neoplatonists from (...) the third to ninth centuries, and sets them in the context of present-day debates over place and the sacred. (shrink)

Apparently alone among medieval Christians, Eriugena argues that all life is immortal. He relies on Plato’s Timaeus as his primary source for this claim, but he modifies the argument of the Timaeus considerably. He turns Plato’s cosmic soul into the genus of life, thereby taking a treatise that originally dealt with cosmology and using it to explore the ontological significance of definition. All species that fall under the genus of life must be immortal, because a mortal species would contradict the (...) genus. No later medieval author would take up Eriugena’s arguments explicitly, although Aquinas comes close. The two thirteenth-century thinkers to address universal immortality seriously—Aquinas and Bonaventure—argue against it, but they are more faithful than Eriugena himself to a literal reading of the Timaeus. (shrink)

This book is a translation of a key commentary on the Book of Changes, or Yijing, perhaps the most broadly influential text of classical China. The Yijing first appeared as a divination text in Zhou-dynasty China and later became a work of cosmology, philosophy, and political theory as commentators supplied it with new meanings. While many English translations of the Yijing itself exist, none are paired with a historical commentary as thorough and methodical as that written by the Confucian scholar (...) Cheng Yi, who turned the original text into a coherent work of political theory. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Clement of AlexandriaL. Michael HarringtonEric Osborn. Clement of Alexandria. Cambridge-New York: Cambridge University Press, 2005. Pp. xviii + 324. Cloth, $85.00.With Clement of Alexandria, Eric Osborn returns to the subject of his 1957 book, The Philosophy of Clement of Alexandria, but its style and themes more closely resemble his more recent studies of second-century Christian thinkers: Tertullian, First Theologian of the West (Cambridge, 1997) and Irenaeus of Lyons (...) (Cambridge, 2001). Osborn's Clement, Tertullian, and Irenaeus are bound together by more than the century in which they happened to live. They share an optimism—a confidence that the world is finally comprehensible—and an argumentative theology, developed in response to the Gnostics, who, Osborn says in his book on Irenaeus, "were strong on picture and myth but weak on argument" (xiii).Osborn's approach to Clement is, as he describes it, analytic or problematic, meaning that "we simply ask what problems forced Clement to write and where he found Christian teaching in need of elucidation" (xii-xiii). Three problems concern Clement: (1) "how can the narrative of the kerygma (what God does) be translated into a metaphysic (who God is)?"; (2) "how two distinct beings, father and son, constitute one God"; and (3) "an answer had to be found for those who divided faith and knowledge." Each problem is addressed by means of a theme—divine economy for the first, reciprocity for the second, and salvation by faith for the third—and each of the book's three major parts devotes itself to one of these themes.Having laid out this systematic division, Osborn generally follows the logic of Clement's own thought rather than an imposed structure, developing his argument by careful collation and paraphrase. As a result, by the end of the book, the reader will feel thoroughly and effortlessly initiated in Clement's major works: the Protrepticus, the Paedagogus, and the Stromateis. The book thus serves as a thorough introduction to Clement, without aiming so low that it is of no interest to the scholar.Osborn's central argument unfolds from his observation that "Clement, more than any other early Christian writer, knew and enjoyed Greek philosophy and literature" (2). Clement defends the rationality shared between Christians and Greeks against the irrationalists of his time: the theosophers, or Gnostics (215). Osborn, in turn, defends Clement against the irrationalists of our own time, represented in his book by Wolfhart Pannenberg. The "divine irrational freedom" praised by Pannenberg is replaced in Osborn's Irenaeus and Clement by "universal wisdom" (47). The universality of wisdom allows great and dispassionate natures to reach the truth—whether they are Christians or Greeks—though each is like one limb of the truth, severed from the rest. Convinced that Christ will reunite these disiecta membra, "Clement converses with them and joins their broken lights" (104).In footnotes and parenthetical references throughout the book, Osborn argues against the approaches to Clement in doxography and Discourse Power theory, precisely because they are unable to treat him as a philosopher (79). The doxographer can show Clement's dependence on earlier philosophers, but is unable to see how he differs from them (xii). The Discourse Power theorist can see that Clement's system differs from the Gnostic system, but is able to explain the difference only as a competition between two social structures, without seeing that only one of them aspires to logical coherence (215).This dismissal of doxography and Discourse Power theory may strike the reader as too thoroughgoing, especially if we distinguish, as Osborn himself does in his book on Tertullian, philosophy as "a matter of system and speculation" from philosophy as "a matter of argument" (198). Clement's optimism depends on his confidence in system and speculation, in so far as they can be defended by argument. But if argument can defend two different systems equally well, then we may wonder why Clement chooses the system he does. Here the doxographer, who can show the provenance of his thought, becomes essential, as does the Discourse Power theorist, who can set Clement's choice of system in the context of the contemporary social fabric that inevitably affects even the... (shrink)

This article asserts the need for the ethical analysis of regulatory policy. The article explores the conventional wisdom surrounding the proper role of government, the function of law, the role of lawmakers, the nature of business, and the relationship between business and government. It is the traditional thinking regarding these fundamental aspects of our social life which creates barriers to the ethical analysis of regulatory policy. It is argued that, in spite of the persistence of agency theories of the firm, (...) a stakeholder theory of the firm best approximates a true descriptive and normative view of business organizations. If the role of government is to maximize the full range of public — private relationships for any given series of inputs, and the role of the firms is to maximize the balance of diverse stakeholders' interests, then a stakeholders' interests paradigm becomes the natural foundation for the ethical analysis of policies which regulate business. (shrink)