A successivity in a linear order is a pair of elements with no other elements between them. A recursive linear order with recursive successivities U is recursively categorical if every recursive linear order with recursive successivities isomorphic to U is in fact recursively isomorphic to U . We characterize those recursive linear orders with recursive successivities that are recursively categorical as precisely those with order type k 1 + g 1 + k 2 + g 2 +…+ g n -1 (...) + k n where each k n is a finite order type, non-empty for i ϵ{2,…, n -1} and each g i is an order type from among {ω,ω * ,ω+ω * }∪{ k ·η: k. (shrink)

It has been known for more than thirty years that the degree of a non-standard model of true arithmetic is a subuniform upper bound for the arithmetic sets. Here a notion of generic enumeration is presented with the property that the degree of such an enumeration is an suub but not the degree of a non-standard model of true arithmetic. This answers a question posed in the literature.

We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

In this paper we investigate computable models of -categorical theories and Ehrenfeucht theories. For instance, we give an example of an -categorical but not -categorical theory such that all the countable models of except its prime model have computable presentations. We also show that there exists an -categorical but not -categorical theory such that all the countable models of except the saturated model, have computable presentations.

This book describes a program of research in computable structure theory. The goal is to find definability conditions corresponding to bounds on complexity which persist under isomorphism. The results apply to familiar kinds of structures (groups, fields, vector spaces, linear orderings Boolean algebras, Abelian p-groups, models of arithmetic). There are many interesting results already, but there are also many natural questions still to be answered. The book is self-contained in that it includes necessary background material from recursion theory (ordinal notations, (...) the hyperarithmetical hierarchy) and model theory (infinitary formulas, consistency properties). (shrink)

With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is (...) intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum. (shrink)

In this paper we consider three potential simplifications to a result of Solovay’s concerning the Turing degrees of nonstandard models of arbitrary completions of first-order Peano Arithmetic (PA). Solovay characterized the degrees of nonstandard models of completions T of PA, showing that they are the degrees of sets X such that there is an enumeration R ≤T X of an “appropriate” Scott set and there is a family of functions (tn)n∈ω, ∆0 n(X) uniformly in n, such that lim tn(s) s→∞.

Slaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable Δ 0 2 degree, but not 0. Since our argument requires the technique of permitting below a (...) Δ 0 2 set, we include a detailed explanation of the mechanics and intuition behind this type of permitting. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

We describe a strongly minimal theory S in an effective language such that, in the chain of countable models of S, only the second model has a computable presentation. Thus there is a spectrum of an -categorical theory which is neither upward nor downward closed. We also give an upper bound on the complexity of spectra.

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)

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)

У статті розглядається питання авторства та його природи у зв’язку з подією Герострата. Проблемною стає ситуація забуття творців храму Артеміди і безсмертна слава його палія. У цьому ключі пропонується концепція деструктивного авторства як форма відображення авторства, заснованого не на звичному акті творення, а на протилежному йому – акті деструкції. При цьому проводиться розмежування деструктивного авторства як реалізації певних атрибутів авторства з використанням деструкції в якості інструмента й можливого його перевороту – авторства деструктивності як феномена, заснованого на автономності та першочерговості деструкції (...) по відношенню до авторства. Також розроблюються поняття «антиавтора» та «антидару», запропоновані як опозиції класичним поняттям «автора» й «дару». Герострат представлений як центральна й структуротворча фігура культурно-історичного наративу, яка досягає зворотного боку турботи про себе через акт деструкції на шляху до безсмертя. При цьому мова вже йде про антиавтора як про індивідуалізацію через авторство без творіння і замкнення індивіда на самому собі. Для історії Герострата неможливим є гомерівське питання через те, що сама природа його ситуації є гіпертрофацією індивідуального авторського витоку, котрий є несумісним зі спів-авторством. Не маючи творіння, він існує лише в якості прагнення здобути авторські дивіденди, а отже, в якості прагнення залишатись єдиним. Не менш важливим є звернення до питання дару в контексті авторства, який перевертається Геростратом у бік крадіжки. Якщо будь-яке авторство нерозривно пов’язане з даруванням, то деструктивне авторство містить у собі антидар – культурну крадіжку об’єкта чи суб’єкта, що піддається деструкції. Саме антидар дозволяє антиавтору реалізувати себе не лише як того, хто здійснює деструктивний акт, але й того, хто викрадає увагу та пам’ять, відокремлюючи їх від попереднього носія і застосовуючи до себе. Таким чином, у статті на прикладі Герострата аналізується розгортання логік, що йдуть паралельно або наперекір усталеній моделі авторства, заснованої на творчому акті, з пропозицією введення концептуальних понять, здатних розширити розуміння авторства та дослідити його в контексті деструктивних аспектів. (shrink)

A linear order is -decidable if its universe is and the relations defined by formulas are uniformly computable. This means that there is a computable procedure which, when applied to a formula and a sequence of elements of the linear order, will determine whether or not is true in the structure. A linear order is decidable if the relations defined by all formulas are uniformly computable. These definitions suggest two questions. Are there, for each , -decidable linear orders that are (...) not -decidable? Are there linear orders that are -decidable for all but not decidable? The former was answered in the positive by Moses in 1993. Here we answer the latter, also positively. (shrink)

We extend results of Harizanov and Barker. For a relation R on a recursive structure /oA, we give conditions guaranteeing that the image of R in a recursive copy of /oA can be made to have arbitrary ∑α0 degree over Δα0. We give stronger conditions under which the image of R can be made ∑α0 degree as well. The degrees over Δα0 can be replaced by certain more general classes. We also generalize the Friedberg-Muchnik Theorem, giving conditions on a pair (...) of relations R and S under which the images of R and S can be made ∑α0 and independent over Δα0 in a recursive copy of /oA. (shrink)