The pure part of HYP(ℳ)

1977 ◽  
Vol 42 (1) ◽  
pp. 33-46 ◽  
Author(s):  
Mark Nadel ◽  
Jonathan Stavi
Keyword(s):  
Admissible Set ◽  
Admissible Sets ◽  
Negative Results

AbstractLet ℳ be a structure for a language ℒ on a set M of urelements. HYP(ℳ) is the least admissible set above ℳ. In §1 we show that pp(HYP(ℳ)) [= the collection of pure sets in HYP(ℳ)] is determined in a simple way by the ordinal α = ° (HYP(ℳ)) and the ℒxω theory of ℳ up to quantifier rank α. In §2 we consider the question of which pure countable admissible sets are of the form pp(HYP(ℳ)) for some ℳ and show that all sets Lα (α admissible) are of this form. Other positive and negative results on this question are obtained.

Σ1 compactness for next admissible sets

1974 ◽  
Vol 39 (1) ◽  
pp. 105-116 ◽  
Author(s):  
Judy Green
Keyword(s):  
Compactness Theorem ◽  
Admissible Set ◽  
Compact Sets ◽  
Admissible Sets ◽  
New Class ◽  
Negative Results ◽  
Power Set ◽  
Consistency Property ◽  
Definition Of

Let σ be any sequence B0, B1 …, Bn, … of transitive sets closed under pairs with for each n. In this paper we show that the smallest admissible set Aσ with σ ∈ Aσ is Σ1 compact. Thus we have an entirely new class of explicitly describable uncountable Σ1 compact sets.The search for uncountable Σ1 compact languages goes back to Hanf's negative results on compact cardinals [7]. Barwise first showed that all countable admissible sets were Σ1 compact [1] and then went on to give a characterization of the Σ1 compact sets in terms of strict reflection [2]. While his characterization has been of interest in understanding the Σ1 compactness phenomenon it has led to the identification of only one class of uncountable Σ1 compact sets. In particular, Barwise showed [2], using the above notation, that if ⋃nBn is power set admissible it satisfies the strict reflection principle and hence is Σ1 compact. (This result was obtained independently by Karp using algebraic methods [9].)In proving our compactness theorem we follow Makkai's approach to the Barwise Compactness Theorem [12] and use a modified version of Smullyan's abstract consistency property [14]. A direct generalization of Makkai's method to the cofinality ω case yields a proof of the Barwise-Karp result mentioned above [6]. In order to obtain our new result we depart from the usual definition of language and use instead the indexed languages of Karp [9] in which a conjunction is considered to operate on a function whose range is a set of formulas rather than on a set of formulas itself.

Anti-admissible sets

1999 ◽  
Vol 64 (2) ◽  
pp. 407-435
Author(s):  
Jacob Lurie
Keyword(s):  
Canonical Extension ◽  
Axiom System ◽  
Admissible Set ◽  
Standard View ◽  
Modal Language ◽  
Admissible Sets ◽  
Interesting Alternative

AbstractAczel's theory of hypersets provides an interesting alternative to the standard view of sets as inductively constructed, well-founded objects, thus providing a convienent formalism in which to consider non-well-founded versions of classically well-founded constructions, such as the “circular logic” of [3], This theory and ZFC are mutually interpretable; in particular, any model of ZFC has a canonical “extension” to a non-well-founded universe. The construction of this model does not immediately generalize to weaker set theories such as the theory of admissible sets. In this paper, we formulate a version of Aczel's antifoundation axiom suitable for the theory of admissible sets. We investigate the properties of models of the axiom system KPU−, that is, KPU with foundation replaced by an appropriate strengthening of the extensionality axiom. Finally, we forge connections between “non-wellfounded sets over the admissible set A” and the fragment LA of the modal language L∞.

Multistage Suboptimal Model Predictive Control With Improved Computational Efficiency

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Xiaotao Liu ◽  
Daniela Constantinescu ◽  
Yang Shi
Keyword(s):  
Model Predictive Control ◽  
Predictive Control ◽  
The State ◽  
Admissible Set ◽  
Admissible Sets ◽  
Prediction Horizon ◽  
Terminal Set ◽  
Minimum Number ◽  
System States ◽  
The Stability

This paper proposes a multistage suboptimal model predictive control (MPC) strategy which can reduce the prediction horizon without compromising the stability property. The proposed multistage MPC requires a precomputed sequence of j-step admissible sets, where the j-step admissible set is the set of system states that can be steered to the maximum positively invariant set in j control steps. Given the precomputed admissible sets, multistage MPC first determines the minimum number of steps M required to drive the state to the terminal set. Then, it steers the state to the (M – N)-step admissible set if M > N, or to the terminal set otherwise. The paper presents the offline computation of the admissible sets, and shows the feasibility and stability of multistage MPC for systems with and without disturbances. A numerical example illustrates that multistage MPC with N = 5 can be used to stabilize a system which requires MPC with N ≥ 14 in the absence of disturbances, and requires MPC with N ≥ 22 when affected by disturbances.

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.

Ordinal spectra of first-order theories

1977 ◽  
Vol 42 (4) ◽  
pp. 492-505 ◽  
Author(s):  
John Stewart Schlipf
Keyword(s):  
Recent Work ◽  
Information Content ◽  
Recursion Theory ◽  
Admissible Set ◽  
First Order ◽  
Admissible Sets ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Basic Treatment

The notion of the next admissible set has proved to be a very useful notion in definability theory and generalized recursion theory, a unifying notion that has produced further interesting results in its own right. The basic treatment of the next admissible set above a structure ℳ of urelements is to be found in Barwise's [75] book Admissible sets and structures. Also to be found there are many of the interesting characterizations of the next admissible set. For further justification of the interest of the next admissible set the reader is referred to Moschovakis [74], Nadel and Stavi [76] and Schlipf [78a, b, c].One of the most interesting single properties of is its ordinal (ℳ). It coincides, for example, with Moschovakis' inductive closure ordinal over structures ℳ with pairing functions—and over some, such as algebraically closed fields of characteristic 0, without pairing functions (by recent work of Arthur Rubin) (although a locally famous counterexample of Kunen, a theorem of Barwise [77], and some recent results of Rubin and the author, show that the inductive closure ordinal may also be strictly smaller in suitably pathological structures). Further justification for looking at (ℳ) alone may be found in the above-listed references. Loosely, we can consider the size of to be a useful measure of the complexity of ℳ. One of the simplest measures of the size of —and yet a very useful measure—is its ordinal, (ℳ). Keisler has suggested thinking of (ℳ) as the information content of a model—the supremum of lengths of wellfounded relations characterizable in the model.

HC of an admissible set

1979 ◽  
Vol 44 (1) ◽  
pp. 95-102
Author(s):  
Sy D. Friedman
Keyword(s):  
Set Theory ◽  
Admissible Set ◽  
The Real ◽  
Admissible Sets

AbstractIf A is an admissible set, let HC(A) = {x∣x ∈ A and x is hereditarily countable in A}. Then HC(A) is admissible. Corollaries are drawn characterizing the “real parts” of admissible sets and the analytical consequences of admissible set theory.

Impairment Questions and Answers

1999 ◽  
Vol 4 (4) ◽  
pp. 4-4
Keyword(s):  
Symptom Validity ◽  
Validity Testing ◽  
Symptom Validity Testing ◽  
Negative Results ◽  
Tunnel Vision ◽  
Questions And Answers ◽  
Blurry Vision ◽  
Fatigue Evaluation ◽  
Choice Testing ◽  
Rule Out

Abstract Symptom validity testing, also known as forced-choice testing, is a way to assess the validity of sensory and memory deficits, including tactile anesthesias, paresthesias, blindness, color blindness, tunnel vision, blurry vision, and deafness—the common feature of which is a claimed inability to perceive or remember a sensory signal. Symptom validity testing comprises two elements: A specific ability is assessed by presenting a large number of items in a multiple-choice format, and then the examinee's performance is compared with the statistical likelihood of success based on chance alone. Scoring below a norm can be explained in many different ways (eg, fatigue, evaluation anxiety, limited intelligence, and so on), but scoring below the probabilities of chance alone most likely indicates deliberate deception. The positive predictive value of the symptom validity technique likely is quite high because there is no alternative explanation to deliberate distortion when performance is below the probability of chance. The sensitivity of this technique is not likely to be good because, as with a thermometer, positive findings indicate that a problem is present, but negative results do not rule out a problem. Although a compelling conclusion is that the examinee who scores below probabilities is deliberately motivated to perform poorly, malingering must be concluded from the total clinical context.

Needed: A publication of negative results: Comment.

1959 ◽  
Vol 14 (9) ◽  
pp. 598-598 ◽  
Author(s):  
Leroy Wolins
Keyword(s):  
Negative Results
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