R. e. presented linear orders

1983 ◽  
Vol 48 (2) ◽  
pp. 369-376 ◽  
Author(s):  
Dev Kumar Roy
Keyword(s):  
Linear Order ◽  
Order Relation ◽  
Recursion Theory ◽  
Linear Orders ◽  
Preliminary Results ◽  
Negative Results ◽  
Recursively Enumerable ◽  
Equality Relation ◽  
The Law

This paper looks at linear orders in the following way. A preordering is given, which is linear and recursively enumerable. By performing the natural identification, one obtains a linear order for which equality is not necessarily recursive. A format similar to Metakides and Nerode's [3] is used to study these linear orders. In effective studies of linear orders thus far, the law of antisymmetry (x ≦ y ∧ y ≦ x ⇒ y) has been assumed, so that if the order relation x ≦ y is r.e. then x < y is also r.e. Here the assumption is dropped, so that x < y may not be r.e. and the equality relation may not be recursive; the possibility that equality is not recursive leads to new twists which sometimes lead to negative results.Reported here are some interesting preliminary results with simple proofs, which are obtained if one looks at these objects with a view to doing recursion theory in the style of Metakides and Nerode. (This style, set in [3], is seen in many subsequent papers by Metakides and Nerode, Kalantari, Remmel, Retzlaff, Shore, and others, e.g. [1], [4], [6], [7], [8], [11]. In a sequel, further investigations will be reported which look at r.e. presented linear orders in this fashion and in the context of Rosenstein's comprehensive work [10].Obviously, only countable linear orders are under consideration here. For recursion-theoretic notation and terminology see Rogers [9].

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Two results on borel orders

1989 ◽  
Vol 54 (3) ◽  
pp. 865-874 ◽  
Author(s):  
Alain Louveau
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Lexicographic Ordering ◽  
Countable Ordinal

AbstractWe prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

Types of simple α-recursively enumerable sets

1976 ◽  
Vol 41 (3) ◽  
pp. 681-694
Author(s):  
Anne Leggett ◽  
Richard A. Shore
Keyword(s):  
Lattice Theory ◽  
Lattice Structure ◽  
Sufficient Conditions ◽  
Recursion Theory ◽  
Recursively Enumerable ◽  
Countable Ordinal ◽  
Congruence Relations ◽  
General Program ◽  
Simple Sets

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

2016 ◽  
Vol 73 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Juan A. Nido Valencia ◽  
Julio E. Solís Daun ◽  
Luis M. Villegas Silva
Keyword(s):  
Recursion Theory ◽  
Recursively Enumerable ◽  
Horn Formulas ◽  
Recursively Enumerable Sets

On Π1-automorphisms of recursive linear orders

1987 ◽  
Vol 52 (3) ◽  
pp. 681-688
Author(s):  
Henry A. Kierstead
Keyword(s):  
Linear Order ◽  
Rational Numbers ◽  
Order Type ◽  
Linear Orders ◽  
Arithmetical Hierarchy ◽  
Linear Orderings ◽  
Nontrivial Automorphism ◽  
Order Types ◽  
Element Chain

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ. For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ1-automorphism and showing that every recursive linear order of type ζ has a nontrivial Π1-automorphism. Rosenstein also considered linear orders of order type 2 · η, where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿2-automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ1-automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿2. The main result of this article is that :

scholarly journals The infinite injury priority method

1976 ◽  
Vol 41 (2) ◽  
pp. 513-530 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursion Theory ◽  
Powerful Method ◽  
Recursively Enumerable ◽  
Priority Method

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {Rn}n∈ω called requirements. If n < m, requirement Rn is given priority over Rm and action taken for Rm at some stage s may at a later stage t > s be undone for the sake of Rn thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often.Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity.The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own.

Micellar and Molecular Solutions of Rubber

1932 ◽  
Vol 5 (2) ◽  
pp. 123-128
Author(s):  
Paul Bary
Keyword(s):  
The State ◽  
The Other ◽  
Solvent Concentration ◽  
Dilute Solutions ◽  
Preliminary Results ◽  
Viscosity Measurements ◽  
Molecular Dispersion ◽  
The Law ◽  
Chief Interest

Abstract The chief interest in studying the degradation of rubber lies in the fact that it makes possible an investigation of the manner in which substances first swell in a liquid before they disperse. There are two opinions current about the state of rubber in its solutions; one, for several years the most popular, is that rubber consists of micelles in the liquid; the other is that rubber is a molecular dispersion with the molecules in the form of chains which may be very long. This latter hypothesis has been taken up comparatively recently by Staudinger, and is supported by a large number of experiments. In the investigation being carried out by the author and E. Fleurent on the degradation of rubber, of which the preliminary results have already been published, the object has been first of all to determine the law according to which the degradation progresses with time under different conditions of solvent, concentration and temperature. When one operates with dilute solutions (0.4 to 2%), the law of degradation is easily proved to be proportional to the time, but these solutions have no appreciable rigidity until they are homogeneous, and all the viscosity measurements of the solutions can be made with a single viscosimeter, without waiting for too long a period of flow.

scholarly journals On cuts in ultraproducts of linear orders I

2016 ◽  
Vol 16 (02) ◽  
pp. 1650008 ◽  
Author(s):  
Mohammad Golshani ◽  
Saharon Shelah
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Regular Cardinals

For an ultrafilter [Formula: see text] on a cardinal [Formula: see text] we wonder for which pair [Formula: see text] of regular cardinals, we have: for any [Formula: see text]-saturated dense linear order [Formula: see text] has a cut of cofinality [Formula: see text] We deal mainly with the case [Formula: see text]

scholarly journals Using Fishburne’s sequences in suitable modeling used for sample data

2021 ◽  
Vol 15 (4) ◽  
pp. 50-60
Author(s):  
Anatoliy Sigal
Keyword(s):  
Decision Making ◽  
Probability Distribution ◽  
Decision Maker ◽  
Linear Order ◽  
Order Relation ◽  
Economic Environment ◽  
Statistical Estimates ◽  
The Third ◽  
Unknown Probability ◽  
Sample Data

This article deals with probabilistic and statistical modeling of managerial decision-making in the economy based on sample data for the previous periods of time. For better definition, the study is limited to Markowitz’s models in the problem of finding an effective portfolio of the field in the third information situation. The third information situation is a widespread decision-making situation and is characterized by the fact that the decision-maker sets, according to his opinion, are a linear order relation on the components of an unknown probabilistic distribution of the states of the economic environment. Often, from the point of view of the decision-maker, the components of an unknown probability distribution of the states of the economic environment must satisfy a partially reinforced linear order relation. As a result, the use of traditional statistical estimates turns out to be impossible, while the following question arises, which is practically not studied in the scientific literature. In this case, what formulas should be used to find statistical estimates and, above all, estimates of unknown probabilities of the state of the economic environment? As an estimate of an unknown probability distribution, we proposed to use the Fishburne sequence that satisfies all available constraints, while corresponding to the opinion of the decision maker and the linear order relation given by him. Fishburne sequences are a generalization of the well-known Fishburne formulas. It is fundamentally important that any Fishburne sequence satisfies a simple linear order relation, and under certain conditions, a partially strengthened linear order relation. Particular attention is paid to the entropic properties of generalized Fishburne progressions, which represent the most important class of Fishburne sequences, as well as the use of generalized Fishburne progressions to take into account the opinion of the decision maker. Such a scheme for estimating an unknown probability distribution has been developed, which makes it possible to achieve the correctness of probabilistic and statistical modeling, as well as appropriate consideration of the opinion of the decision-maker, uncertainty and risk.

Recursion theory on orderings. I. A model theoretic setting

1979 ◽  
Vol 44 (3) ◽  
pp. 383-402 ◽  
Author(s):  
G. Metakides ◽  
J.B. Remmel
Keyword(s):  
Algebraic Closure ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Formal Definition ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Partial Orderings ◽  
Definition Of ◽  
Algebraically Closed Fields

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a setS, denoted cl(S), is given in §1, however in vector spaces, cl(S) is just the subspace generated byS, in Boolean algebras, cl(S) is just the subalgebra generated byS, and in algebraically closed fields, cl(S) is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl(S) = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings.

Sign in / Sign up

Export Citation Format

Share Document