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 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)

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 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)

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.

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)

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.

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)

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.

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 523 Aristotle's Theology: A Commentary on Book A of the Metaphysics. By Leo Elders. (Assen, The Netherlands: Royal VanGorcum Ltd., 1972) In 1961 Leo Elders published a book under the title Aristotle's Theory o] the One with the subtitle "A Commentary on Book X of the Metaphysics." Five years later he published Aristotle's Cosmology, subtitled "A Commentary on the De Caelo." Continuing his "commentary " approach to (...) Aristotle's works, he returns in the present work to the Metaphysics, restricting himself in this case to an examination of Book Lambda. The author is convinced that the "central doctrine of the Book is the conception of an unmoved Mover, who is unchangeable and eternal; his being is subsistent thinking; he is the final cause of all movement and the Good desired by the cosmos" (p. 1). In addition to a detailed commentary on each of the ten chapters of Book Lambda, the book contains seven introductory chapters which discuss the relation of Aristotle's theology to that of his Greek predecessors, particularly Plato, and attempts to solve such problems as the date and structure of Metaphysics Lambda and its place in the Aristotelian corpus. The author believes Metaphysics Lambda to be a collection of short studies which belong to mature Aristotelianism but do not form a coherent treatise. Because the eighth chapter asserts a multiplicity of Unmoved Movers, which runs counter to the position taken in the rest of the book, Elders takes the position that the eighth chapter is a short essay by one of Aristotle's collaborators or students which was inserted into the book by a later editor. One of the significant difficulties with the book is that the author reads a number of anachronistic ideas back into Greek philosophy. He assumes that we all know, especially the author himself, what "theology" means to the Greeks and that whatever its meaning it is not only continuous with the Christian conception of theology but can best be understood with the aid of the Christian approach to theology. Is it not best to regard the Unmoved Mover simply as the Perfect Scientist or Perfect Knower, the final cause of those who desire knowledge? Of course, it is true that Aristotle refers to the Unmoved Mover as divine (theos)----what,after all, deserves the designation "divine" except that which is perfect and eternal, and the final goal of man's intellectual activity? As the author states in the Preface that "one of the leading thoughts of the commentary is that there is a real continuity between Academic theories and Aristotle's doctrine of the Unmoved Mover" he devotes considerable attention to various aspects of Plato's thought. Here he frequently succumbs to the temptation to read a number of Neoplatonic, Pythagorean, and even Christian ideas into the Greek philosopher, and fails to distinguish Plato from "the Platonists" mentioned by Aristotle (el. pp. 276, 291, and 292). Such passages as Metaphysics 1090b20-24 and 1090b31 ft. do not necessarily refer to Plato himself but could easily mean some unidentified members of his school. In the introductory chapters especially, Elders attributes to Plato a number of theories which derive from other sources, frequently with little or no direct textual justification for so doing. For instance, he says that "a great number of texts indicate that Plato made the One a principle beyond being, and thus beyond thinking" (p. 16). His only corroborating evidence is that this is "implicitly contained in the famous text of Rep. 509B" (footnote 7, p. 16). Earlier he has said that "Plato developed a theory of principles according to which all essences are connected with the supreme principle, which, in his later years, Plato called the One" (pp. 8-9). Again, the only textual evidence he mentions is several passages from the Republic, which have at best a tenuous connection with such a theory and which were certainly not written during Plato's "later years." The book also suffers from a number of organizational problems. In an attempt to bring together the opinions of other scholars on the issues at hand, the author continually mixes up secondary literature and interpretations with the problems he poses... (shrink)

Involuntary civil commitment to treatment for opioid use disorder prevents imminent overdose, but also restricts autonomy and raises other ethical concerns. Using the Kass Public Health Ethics Framework, we identified ICC benefits and harms. Benefits include: protection of vulnerable, underserved patients; reduced legal consequences; resources for families; and “on-demand” treatment access. Harms include: stigmatizing and punitive experiences; heightened family conflict and social isolation; eroded patient self-determination; limited or no provision of OUD medications; and long-term overdose risk. To use ICC ethically, (...) it should be recognized as comprising vulnerable patients worthy of added protections; be a last resort option; utilize consensual, humanizing processes; provide medications and other evidence-based-treatment; integrate with existing healthcare systems; and demonstrate effective outcomes before diffusion. ICC to OUD treatment carries significant potential harms that, if unaddressed, may outweigh its benefits. Findings can inform innovations for ensuring that ICC is used in an ethically responsible way. (shrink)