The next admissible set

1971 ◽  
Vol 36 (1) ◽  
pp. 108-120 ◽  
Author(s):  
K. J. Barwise ◽  
R. O. Gandy ◽  
Y. N. Moschovakis
Keyword(s):  
Admissible Set ◽  
Admissible Ordinals ◽  
Theoretic Function ◽  
Hyperarithmetic Hierarchy

In this paper we describe generalizations of several approaches to the hyperarithmetic hierarchy, show how they are related to the Kripke-Platek theory of admissible ordinals and sets, and study conditions under which the various approaches remain equivalent.To put matters in some perspective, let us first review various approaches to the theory of hyperarithmetic sets. For most purposes, it is convenient to first define the semi-hyperarithmetic (semi-HA) subsets of N. A set is then said to be hyperarithmetic (HA) if both it and its complement are semi-HA. A total number-theoretic function is HA if its graph is HA.

Simple r. e. degree structures

1987 ◽  
Vol 52 (1) ◽  
pp. 208-213
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Recursion Theory ◽  
Turing Degrees ◽  
Local Version ◽  
Admissible Set ◽  
Jump Operator ◽  
Admissible Sets ◽  
Closed Sets ◽  
Locally Countable ◽  
To Come ◽  
Admissible Ordinals

Much of recursion theory centers on the structures of different kinds of degrees. Classically there are the Turing degrees and r. e. Turing degrees. More recently, people have studied α-degrees for α an ordinal, and degrees over E-closed sets and admissible sets. In most contexts, deg(0) is the bottom degree and there is a jump operator' such that d' is the largest degree r. e. in d and d' > d. Both the degrees and the r. e. degrees usually have a rich structure, including a relativization to the cone above a given degree.A natural exception to this pattern was discovered by S. Friedman [F], who showed that for certain admissible ordinals β the β-degrees ≥ 0′ are well-ordered, with successor provided by the jump.For r. e. degrees, natural counterexamples are harder to come by. This is because the constructions are priority arguments, which require only mild restrictions on the ground model. For instance, if an admissible set has a well-behaved pair of recursive well-orderings then the priority construction of an intermediate r. e. degree (i.e., 0 < d < 0′) goes through [S]. It is of interest to see just what priority proofs need by building (necessarily pathological) admissible sets with few r. e. degrees.Harrington [C] provides an admissible set with two r. e. degrees, via forcing. A limitation of his example is that it needs ω1 (more accurately, a local version thereof) as a parameter. In this paper, we find locally countable admissible sets, some with three r. e. degrees and some with four.

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.

Σ-predicates of finite types over an admissible set

1985 ◽  
Vol 24 (5) ◽  
pp. 327-351 ◽  
Author(s):  
Yu. L. Ershov
Keyword(s):  
Admissible Set

Uncountable models and infinitary elementary extensions

1973 ◽  
Vol 38 (3) ◽  
pp. 460-470 ◽  
Author(s):  
John Gregory
Keyword(s):  
Countable Subset ◽  
Elementary Extension ◽  
Admissible Set ◽  
Natural Numbers ◽  
Elementary Equivalence ◽  
Countable Structure

Let A be a countable admissible set (as defined in [1], [3]). The language LA consists of all infinitary finite-quantifier formulas (identified with sets, as in [1]) that are elements of A. Notationally, LA = A ∩ Lω1ω. Then LA is a countable subset of Lω1ω, the language of all infinitary finite-quantifier formulas with all conjunctions countable. The set is the set of Lω1ω sentences defined in 2.2 below. The following theorem characterizes those A-Σ1 sets Φ of LA sentences that have uncountable models.Main Theorem (3.1.). If Φ is an A-Σ1set of LA sentences, then the following are equivalent:(a) Φ has an uncountable model,(b) Φ has a model with a proper LA-elementary extension,(c) for every , ⋀Φ → C is not valid.This theorem was announced in [2] and is proved in §§3, 4, 5. Makkai's earlier [4, Theorem 1] implies that, if Φ determines countable structure up to Lω1ω-elementary equivalence, then (a) is equivalent to (c′) for all , ⋀Φ → C is not valid.The requirement in 3.1 that Φ is A-Σ1 is essential when the set ω of all natural numbers is an element of A. For by the example of [2], then there is a set Φ LA sentences such that (b) holds and (a) fails; it is easier to show that, if ω ϵ A, there is a set Φ of LA sentences such that (c) holds and (b) fails.

A lift of a theorem of Friedberg: A Banach-Mazur functional that coincides with no α-recursive functional on the class of α-recursive functions

1981 ◽  
Vol 46 (2) ◽  
pp. 216-232 ◽  
Author(s):  
Robert A. di Paola
Keyword(s):  
Recursive Functions ◽  
Admissible Ordinals

AbstractR. M. Friedberg demonstrated the existence of a recursive functional that agrees with no Banach-Mazur functional on the class of recursive functions. In this paper Friedberg's result is generalized to both α-recursive functionals and weak α-recursive functionals for all admissible ordinals α such that λ < α*, where α* is the Σ1-projectum of α and λ is the Σ2-cofinality of α. The theorem is also established for the metarecursive case, α = ω1, where α* = λ = ω.

An α-finite injury method of the unbounded type

1976 ◽  
Vol 41 (1) ◽  
pp. 1-17
Author(s):  
C. T. Chong
Keyword(s):  
Fine Structure ◽  
Initial Segment ◽  
Fundamental Importance ◽  
Upper Semilattice ◽  
The Past ◽  
Recursively Enumerable ◽  
Background Material ◽  
A Priority ◽  
Admissible Ordinals ◽  
Priority Argument

Let α be an admissible ordinal. In this paper we study the structure of the upper semilattice of α-recursively enumerable degrees. Various results about the structure which are of fundamental importance had been obtained during the past two years (Sacks-Simpson [7], Lerman [4], Shore [9]). In particular, the method of finite priority argument of Friedberg and Muchnik was successfully generalized in [7] to an α-finite priority argument to give a solution of Post's problem for all admissible ordinals. We refer the reader to [7] for background material, and we also follow closely the notations used there.Whereas [7] and [4] study priority arguments in which the number of injuries inflicted on a proper initial segment of requirements can be effectively bounded (Lemma 2.3 of [7]), we tackle here priority arguments in which no such bounds exist. To this end, we focus our attention on the fine structure of Lα, much in the fashion of Jensen [2], and show that we can still use a priority argument on an indexing set of requirements just short enough to give us the necessary bounds we seek.

Generalized reduction theorems for model-theoretic analogs of the class of coanalytic sets

1993 ◽  
Vol 58 (1) ◽  
pp. 81-98
Author(s):  
Shaughan Lavine
Keyword(s):  
Pairwise Disjoint ◽  
Admissible Set ◽  
Effective Version ◽  
Reduction Property ◽  
Uniformization Property ◽  
Weakly Stable

AbstractLet be an admissible set. A sentence of the form is a sentence if φ ∈ (φ is ∨ Φ where Φ is an -r.e. set of sentences from ). A sentence of the form is an , sentence if φ is a sentence. A class of structures is, for example, a ∀1 class if it is the class of models of a ∀1() sentence. Thus ∀1() is a class of classes of structures, and so forth.Let i, be the structure 〈i, <〉, for i > 0. Let Γ be a class of classes of structures. We say that a sequence J1, …, Ji,…, i < ω, of classes of structures is a Γ sequence if Ji ∈ Γ, i < ω, and there is I ∈ Γ such that ∈ Ji, if and only if [],i, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence J1,…, Ji,…, i < ω, there is a Γ sequence J′t, …, J′i, …, i < ω, such that J′i ⊆ Ji, i < ω, ⋃J′i = ⋃Ji, and the J′i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property.We show over countable structures that ∀1() and ∃2() have the easy uniformization property if is a countable admissible set with an infinite member, that and have the easy uniformization property if α is countable, admissible, and not weakly stable, and that and have the easy uniformization properly. The results proved are more general. The result for answers a question of Vaught(1980).

Constrained Control of Input Delayed Systems With Partially Compensated Input Delays

2020 ◽  
Author(s):  
Imoleayo Abel ◽  
Mrdjan Janković ◽  
Miroslav Krstić
Keyword(s):  
Nonlinear Systems ◽  
Input Delay ◽  
Constrained Control ◽  
Admissible Set ◽  
Safe Operation ◽  
Barrier Functions ◽  
Delayed Systems ◽  
Free System ◽  
Single Input ◽  
Input Delays

Abstract Control Barrier Functions (CBFs) have become popular for enforcing — via barrier constraints — the safe operation of nonlinear systems within an admissible set. For systems with input delay(s) of the same length, constrained control has been achieved by combining a CBF for the delay free system with a state predictor that compensates the single input delay. Recently, this approach was extended to multi input systems with input delays of different lengths. One limitation of this extension is that barrier constraint adherence can only be guaranteed after the longest input delay has been compensated and all input channels become available for control. In this paper, we consider the problem of enforcing constraint adherence when only a subset of input delays have been compensated. In particular, we propose a new barrier constraint formulation that ensures that when possible, a subset of input channels with shorter delays will be utilized for keeping the system in the admissible set even before longer input delays have been compensated. We include a numerical example to demonstrate the effectiveness of the proposed approach.

scholarly journals Continuity and monotonicity of solutions to a greedy maximization problem

2020 ◽  
Vol 92 (1) ◽  
pp. 33-76
Author(s):  
Łukasz Kruk
Keyword(s):  
Real Time ◽  
Network Topology ◽  
Resource Sharing ◽  
Time Parameter ◽  
Maximization Problem ◽  
Fluid Models ◽  
Admissible Set ◽  
Network Modelling ◽  
Underlying Network ◽  
Real Time Transmission

Abstract Motivated by an application to resource sharing network modelling, we consider a problem of greedy maximization (i.e., maximization of the consecutive minima) of a vector in $${\mathbb {R}}^n$$ R n , with the admissible set indexed by the time parameter. The structure of the constraints depends on the underlying network topology. We investigate continuity and monotonicity of the resulting maximizers with respect to time. Our results have important consequences for fluid models of the corresponding networks which are optimal, in the appropriate sense, with respect to handling real-time transmission requests.

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