The genericity conjecture

1994 ◽  
Vol 59 (2) ◽  
pp. 606-614 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Generic Extension ◽  
Class Theory ◽  
First Order ◽  
Class A ◽  
Inner Model ◽  
Special Case

The Genericity Conjecture, as stated in Beller-Jensen-Welch [1], is the following:(*) If O# ∉ L[R], R ⊆ ω, then R is generic over L.We must be precise about what is meant by “generic”.Definition (Stated in Class Theory). A generic extension of an inner model M is an inner model M[G] such that for some forcing notion ⊆ M:(a) 〈M, 〉 is amenable and ⊩ is 〈M, 〉-definable for sentences.(b) G ⊆ is compatible, closed upwards, and intersects every 〈M, 〉-definable dense D ⊆ .A set x is generic over M if it is an element of a generic extension of M. And x is strictly generic over M if M[x] is a generic extension of M.Though the above definition quantifies over classes, in the special case where M = L and O# exists, these notions are in fact first order, as all L-amenable classes are definable over L[O#]. From now on assume that O# exists.Theorem A. The Genericity Conjecture is false.The proof is based upon the fact that every real generic over L obeys a certain definability property, expressed as follows.Fact. If R is generic over L, then for some L-amenable class A, Sat〈L, A〉 is not definable over 〈L[R],A〉, where Sat〈L,A〉 is the canonical satisfaction predicate for 〈L,A〉.

Proper classes via the iterative conception of set

1987 ◽  
Vol 52 (3) ◽  
pp. 636-650
Author(s):  
Mark F. Sharlow
Keyword(s):  
Order Theory ◽  
Inaccessible Cardinal ◽  
Proper Class ◽  
Similar Treatment ◽  
Class Theory ◽  
First Order ◽  
First Order Theory ◽  
Inner Model ◽  
Iterative Conception ◽  
Iterative Conception Of Set

AbstractWe describe a first-order theory of generalized sets intended to allow a similar treatment of sets and proper classes. The theory is motivated by the iterative conception of set. It has a ternary membership symbol interpreted as membership relative to a set-building step. Set and proper class are defined notions. We prove that sets and proper classes with a defined membership form an inner model of Bernays-Morse class theory. We extend ordinal and cardinal notions to generalized sets and prove ordinal and cardinal results in the theory. We prove that the theory is consistent relative to ZFC + (∃x) [x is a strongly inaccessible cardinal].

Relativized realizability in intuitionistic arithmetic of all finite types

1978 ◽  
Vol 43 (1) ◽  
pp. 23-44 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Extension Theorem ◽  
General Notion ◽  
Conservative Extension ◽  
First Order ◽  
New Information ◽  
Reflection Theorem ◽  
Order Arithmetic ◽  
Special Case ◽  
Dependent Choice ◽  
Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

Uniqueness, definability and interpolation

1988 ◽  
Vol 53 (2) ◽  
pp. 554-570 ◽  
Author(s):  
Kosta Došen ◽  
Peter Schroeder-Heister
Keyword(s):  
Proof Theory ◽  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
General Notion ◽  
Structural Part ◽  
First Order ◽  
First Case ◽  
Additional Constant ◽  
Special Case

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, Tα a set of sentences, α* an additional constant of the same syntactical category as α and Tα, a copy of Tα with α* instead of α, then for implicit definability of α in Tα one has, in the case of predicate constants, to derive α(x1,…,xn) ↔ α*(x1,…,xn) from Tα ∪ Tα*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata Sα and derivability from instances of Sα ∪ Sα* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

Basic Geometric Dispersion Theory of Decision Making Under Risk: Asymmetric Risk Relativity, New Predictions of Empirical Behaviors, and Risk Triad

2021 ◽  
Author(s):  
Behnam Malakooti ◽  
Mohamed Komaki ◽  
Camelia Al-Najjar
Keyword(s):  
Risk Behaviors ◽  
Empirical Studies ◽  
Dispersion Theory ◽  
Dispersion Models ◽  
Decision Making Under Risk ◽  
First Order ◽  
Order Stochastic Dominance ◽  
Out Of Sample ◽  
Mean Variance ◽  
Special Case

Many studies have spotlighted significant applications of expected utility theory (EUT), cumulative prospect theory (CPT), and mean-variance in assessing risks. We illustrate that these models and their extensions are unable to predict risk behaviors accurately in out-of-sample empirical studies. EUT uses a nonlinear value (utility) function of consequences but is linear in probabilities, which has been criticized as its primary weakness. Although mean-variance is nonlinear in probabilities, it is symmetric, contradicts first-order stochastic dominance, and uses the same standard deviation for both risk aversion and risk proneness. In this paper, we explore a special case of geometric dispersion theory (GDT) that is simultaneously nonlinear in both consequences and probabilities. It complies with first-order stochastic dominance and is asymmetric to represent the mixed risk-averse and risk-prone behaviors of the decision makers. GDT is a triad model that uses expected value, risk-averse dispersion, and risk-prone dispersion. GDT uses only two parameters, z and zX; these constants remain the same regardless of the scale of risk problem. We compare GDT to several other risk dispersion models that are based on EUT and/or mean-variance, and identify verified risk paradoxes that contradict EUT, CPT, and mean-variance but are easily explainable by GDT. We demonstrate that GDT predicts out-of-sample empirical risk behaviors far more accurately than EUT, CPT, mean-variance, and other risk dispersion models. We also discuss the underlying assumptions, meanings, and perspectives of GDT and how it reflects risk relativity and risk triad. This paper covers basic GDT, which is a special case of general GDT of Malakooti [Malakooti (2020) Geometric dispersion theory of decision making under risk: Generalizing EUT, RDEU, & CPT with out-of-sample empirical studies. Working paper, Case Western Reserve University, Cleveland.].

Reverberation Intensity Decaying in Range-Dependent Waveguide

2019 ◽  
Vol 27 (03) ◽  
pp. 1950007
Author(s):  
J. R. Wu ◽  
T. F. Gao ◽  
E. C. Shang
Keyword(s):  
Large Scale ◽  
Back Propagation ◽  
Normal Modes ◽  
Small Scale ◽  
Signal Pulse ◽  
First Order ◽  
Orthogonal Property ◽  
Short Signal ◽  
Scale Roughness ◽  
Special Case

In this paper, an analytic range-independent reverberation model based on the first-order perturbation theory is extended to range-dependent waveguide. This model considers the effect of bottom composite roughness: small-scale bottom rough surface provides dominating energy for reverberation, whereas large-scale roughness has the effect of forward and back propagation. For slowly varying bottom and short signal pulse, analytic small-scale roughness backscattering theory is adapted in range-dependent waveguides. A parabolic equation is used to calculate Green functions in range-dependent waveguides, and the orthogonal property of local normal modes is employed to estimate the modal spectrum of PE field. Synthetic tests demonstrate that the proposed reverberation model works well, and it can also predict the reverberation of range-independent waveguide as a special case.

scholarly journals Full Characterization of Act-and-wait Control for First-order Unstable Lag Processes

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1209-1233 ◽  
Author(s):  
T. Insperger ◽  
P. Wahi ◽  
A. Colombo ◽  
G. Stépán ◽  
M. Di Bernardo ◽  
...  
Keyword(s):  
Time Delay ◽  
Feedback Systems ◽  
Periodic Control ◽  
First Order ◽  
Full Characterization ◽  
The Stability ◽  
And Control ◽  
Time Periodic ◽  
Special Case

Act-and-wait control is a special case of time-periodic control for systems with feedback delay, where the control gains are periodically switched on and off in order to stabilize otherwise unstable systems. The stability of feedback systems in the presence of time delay is a challenging problem. In this paper, we show that the act-and-wait type time-periodic control can always provide deadbeat control for first-order unstable lag processes with any (large but) fixed value of the time delay in the feedback loop. A full characterization of this act-and-wait controller with respect to the system and control parameters is given based on performance and robustness against disturbances.

Transverse Impact of a Hyperelastic Stretched String

1985 ◽  
Vol 52 (1) ◽  
pp. 137-143 ◽  
Author(s):  
M. F. Beatty ◽  
J. B. Haddow
Keyword(s):  
Strain Energy Function ◽  
Similarity Solutions ◽  
Plane Motion ◽  
Governing Equations ◽  
Transverse Impact ◽  
Normal Impact ◽  
Wave Speeds ◽  
First Order ◽  
Special Case ◽  
Infinite String

Governing equations are derived for the plane motion of a stretched hyperelastic string subjected to a suddenly applied force at one end. These equations can be put in the form of a quasilinear system of first-order partial differential equations, which is totally hyperbolic for an admissible strain energy function. There are, in general, two wave speeds and two corresponding shock speeds. Special consideration is given to the jump relations across the shocks. Similarity solutions for a string moved at one end in loading or unloading are obtained for a general hyperelastic solid. The results are applicable to the familiar neo-Hookean or Mooney-Rivlin material, and the nature of the solution for another special hyperelastic material is discussed. These solutions are valid for a semi-infinite string, or until the first reflection occurs. It is shown that a special case of the similarity solution is valid for the normal impact of a stretched string by a constant speed, point application of load. Exact solution to the equations for the neo-Hookean model is derived in terms of elliptic integrals, and some numerical results are provided.

On precipitousness of the nonstationary ideal over a supercompact

1986 ◽  
Vol 51 (3) ◽  
pp. 648-662 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
General Theorem ◽  
Supercompact Cardinal ◽  
Generic Extension ◽  
Inner Model ◽  
Nonstationary Ideal

Namba [N] proved that the nonstationary ideal over a measurable (NSκ) cannot be κ+-saturated. Baumgartner, Taylor and Wagon [BTW] asked if it is possible for NSκ to be precipitous over a measurable κ. A model with this property was constructed by the author, and shortly after Foreman, Magidor and Shelah [FMS] proved a general theorem that after collapsing of a supercompact or even a superstrong to the successor of κ, NSκ became precipitous. This theorem implies that it is possible to have the nonstationary ideal precipitous over even a supercompact cardinal. Just start with a supercompact κ and a superstrong λ > κ. Make supercompactness of κ indistractible as in [L] and then collapse λ to be κ+.The aim of our paper is to show that the existence of a supercompact cardinal alone already implies the consistency of the nonstationary ideal precipitous over a supercompact. The proof gives also the following: if κ is a λ-supercompact for λ ≥ (2κ)+, then there exists a generic extension in which κ is λ-supercompact and NSκ is precipitous. Thus, for a model with NSκ precipitous over a measurable we need a (2κ)+-supercompact cardinal κ. Jech [J] proved that the precipitous of NSκ over a measurable κ implies the existence of an inner model with o(κ) = κ+ + 1. In §3 we improve this result a little by showing that the above assumption implies an inner model with a repeat point.The paper is organized as follows. In §1 some preliminary facts are proved. The model with NSκ precipitous over a supercompact is constructed in §2.

A nonlinear unsteady one-dimensional theory for wings in extreme ground effect

1980 ◽  
Vol 98 (1) ◽  
pp. 33-47 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Ground Effect ◽  
Vortex Sheet ◽  
Unsteady Motion ◽  
The Body ◽  
Order Ordinary Differential Equation ◽  
One Dimensional ◽  
First Order ◽  
Gap Flow ◽  
Normal Distance ◽  
Special Case

Flow induced by a body moving near a plane wall is analysed on the assumption that the normal distance from the wall of every point of the body is small compared to the body length. The flow is irrotational except for the vortex sheet representing the wake. The gap-flow problem in the case of unsteady motion is reduced to a nonlinear first-order ordinary differential equation in the time variable. In the special case of steady flow, some known results are recovered and generalized. As an illustration of the unsteady theory, the problem is solved of a flat plate falling toward the ground under its own weight, while moving forward at uniform speed.

Extending Gödel's negative interpretation to ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 221-229 ◽  
Author(s):  
William C. Powell
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Stable Sets ◽  
Class Theory ◽  
Heyting Arithmetic ◽  
Inner Model ◽  
Order Arithmetic ◽  
Definition Of ◽  
Excluded Middle ◽  
Negative Sense

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

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