Cumulative Hierarchy Research Articles

Can the Cumulative Hierarchy Be Categorically Characterized

Mathematical realists have long invoked the categoricity of axiomatizations of arithmetic and analysis to explain how we manage to fix the intended meaning of their respective vocabulary. Can this strategy be extended to set theory? Although traditional wisdom recommends a negative answer to this question, Vann McGee (1997) has offered a proof that purports to show otherwise. I argue that one of the two key assumptions on which the proof rests deprives McGee's result of the significance he and the realist want to attribute to it. I consider two strategies to deal with the problem --- one of which is outlined by McGee himself (2000) --- and argue that both of them fail. I end with some remarks on the prospects for mathematical realism in the light of my discussion.

Algorithms for integration of stochastic differential equations using parallel optimized sampling in the Stratonovich calculus

A variance reduction method for stochastic integration of Fokker–Planck equations is derived. This unifies the cumulant hierarchy and stochastic equation approaches to obtaining moments, giving a performance superior to either. We show that the brute force method of reducing sampling error by just using more trajectories in a sampled stochastic equation is not the best approach. The alternative of using a hierarchy of moment equations is also not optimal, as it may converge to erroneous answers. Instead, through Bayesian conditioning of the stochastic noise on the requirement that moment equations are satisfied, we obtain improved results with reduced sampling errors for a given number of stochastic trajectories. The method used here converges faster in time-step than Ito–Euler algorithms. This parallel optimized sampling (POS) algorithm is illustrated by several examples, including a bistable nonlinear oscillator case where moment hierarchies fail to converge.

Establishment of the best maintenance practices for optimal reconfigurable vibrating screen management using decision techniques

Purpose Reconfigurable vibrating screen (RVS) is an innovative beneficiation machine designed at Tshwane University of Technology, Republic of South Africa (RSA); with adjustable screen structure to ensure sorting, sizing and screening of varying mineral particles (sizes and quantities) demanded by the customers in a cost-effective manner through the screen structure geometric transformation. In order to ensure that this machine is optimally maintained and managed when utilized in surface and underground mining industries, there is a need to establish or ascertain the best maintenance practices that would be used in optimally managing the RVS machine using decision making techniques. In view of this, the purpose of this paper is to ascertain the best maintenance practices that would be used to optimally maintain and manage the RVS machine when used in surface and underground mines. Design/methodology/approach Decision making techniques such as weighted decision matrix (WDM) and analytical hierarchy process (AHP) were used in this research work to establish the best maintenance practice for optimally maintaining and managing the RVS machine using relevant literature survey on maintenance management systems as well as the different maintenance criteria decision indices obtained from different conventional vibrating screen machine manufacturers and maintenance experts. Findings Based on the results obtained from the WDM analysis, it was anticipated that e-maintenance (e-M) system embedded with diagnosing and prognosing algorithms; with a cumulative weight score of 2.37 is the best maintenance practice for managing the RVS machine when used in surface mines, while AHP with deeper decision making analysis anticipated that the robotic-driven maintenance (RM) system with an important decision criteria; safety, and a cumulative hierarchy score of 28.6 percent, supported by e-M management system with a cumulative hierarchy score of 17.6 percent are the best maintenance mix that could be used in optimally maintaining and managing the RVS machine, when used in a craggy and hazardous underground mining environment. Practical implications To this effect, it could be anticipated that e-M management system (endowed with the ability to detect fault on the machine, diagnose and prognose the different subsystems of the RVS machine and ascertain the reconfiguration time and process of the RVS machine in recovering production loss during the maintenance of the machine as well as meeting customers demand, etc.) is the best maintenance practice for optimally maintaining the RVS machine when utilized in surface mines while both e-M management system and RM management system (endowed with the ability to carry out automated maintenance tasks achievement under little or no maintenance manager intervention) are also anticipated as the best customized maintenance practices mix that could be used in optimally maintaining the RVS machine, when used in dangerous and hazardous underground mining environment. Originality/value This maintenance management system evaluation and selection for optimal RVS machine functionality will serve as a useful information to different mining machines (and other related machines) maintenance managers, in selecting the best maintenance management system for ensuring optimal functionality, reliability and maintainability of machines used in their industries.

Optimal thermalization in a shell model of homogeneous turbulence

We investigate the turbulence-induced dissipation of the large scales in a statistically homogeneous flow using an ‘optimal closure,’ which one of us (BT) has recently exposed in the context of Hamiltonian dynamics. This statistical closure employs a Gaussian model for the turbulent scales, with corresponding vanishing third cumulant, and yet it captures an intrinsic damping. The key to this apparent paradox lies in a clear distinction between true ensemble averages and their proxies, most easily grasped when one works directly with the Liouville equation rather than the cumulant hierarchy. We focus on a simple problem for which the optimal closure can be fully and exactly worked out: the relaxation arbitrarily far-from-equilibrium of a single energy shell towards Gibbs equilibrium in an inviscid shell model of 3D turbulence. The predictions of the optimal closure are validated against DNS and contrasted with those derived from EDQNM closure.

A cumulative hierarchy of sets for constructive set theory

The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can be used to prove new theorems in .

THE POTENTIAL HIERARCHY OF SETS

AbstractSome reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

The Bernays-Schönfinkel-Ramsey class for set theory: decidability

AbstractAs proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays-Shönfinkel-Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.

On the number of infinite sequences with trivial initial segment complexity

The sequences which have trivial prefix-free initial segment complexity are known as K -trivial sets, and form a cumulative hierarchy of length ω . We show that the problem of finding the number of K -trivial sets in the various levels of the hierarchy is Δ 3 0 . This answers a question of Downey/Miller/Yu (see Downey (2010) [7, Section 10.1.4]) which also appears in Nies (2009) [17, Problem 5.2.16]. We also show the same for the hierarchy of the low for K sequences, which are the ones that (when used as oracles) do not give a shorter initial segment complexity compared to the computable oracles. In both cases the classification Δ 3 0 is sharp.

How to be a minimalist about sets

According to the iterative conception of set, sets can be arranged in a cumulative hierarchy divided into levels. But why should we think this to be the case? The standard answer in the philosophical literature is that sets are somehow constituted by their members. In the first part of the paper, I present a number of problems for this answer, paying special attention to the view that sets are metaphysically dependent upon their members. In the second part of the paper, I outline a different approach, which circumvents these problems by dispensing with the priority or dependence relation altogether. Along the way, I show how this approach enables the mathematical structuralist to defuse an objection recently raised against her view.

THE IDENTITY OF ARGUMENT-PLACES

Argument-places play an important role in our dealing with relations. However, that does not mean that argument-places should be taken as primitive entities. It is possible to give an account of ‘real’ relations in which argument-places play no role. But if argument-places are not basic, then what can we say about their identity? Can they, for example, be reconstructed in set theory with appropriate urelements? In this article, we show that for some relations, argument-places cannot be modeled in a neutral way in V[A], the cumulative hierarchy with basic ingredients of the relation as urelements. We argue that a natural way to conceive of argument-places is to identify them with abstract, structureless points of a derivative structure exemplified by positional frames. In case the relation has symmetry, these points may be indiscernible.

THE ITERATIVE CONCEPTION OF SET

The two expressions ‘The cumulative hierarchy’ and ‘The iterative conception of sets’ are usually taken to be synonymous. However, the second is more general than the first, in that there are recursive procedures that generate some ill-founded sets in addition to well-founded sets. The interesting question is whether or not the arguments in favour of the more restrictive version – the cumulative hierarchy – were all along arguments for the more general version.

Gödel and Set Theory

Kurt Gödel (1906–1978) with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice (AC) and the Continuum Hypothesis (CH). More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was.

Truth In V for ∃*∀∀-Sentences is Decidable

AbstractLet V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and. following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in ∀ but not by finite sets ([5]). and therefore of ∃∃∀∀ sentences of the same language, which are undecidable in ZFC without the Axiom of Infinity, truth in V for ∃*∀∀-sentences of the primitive language, is decidable ([1]). Completeness of ZF with respect to such sentences follows.

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.

Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets. What we give is a (critical)survey of quite a lot of such approaches which have been offered in the last approximately 35 years. The present Part I is devoted to model based and to axiomatic approaches; the forth-coming Part II will be devoted to category theoretic approaches.

Zermelo and Set Theory

Ernst Friedrich Ferdinand Zermelo (1871–1953) transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details.

An Axiomatization of 'Very' within systiems of Set Theory

A structural (as opposed to Zadeh’s quantitative) approach to fuzziness is given, based on the operator “very”, which is added to the language of set theory together with some elementary axioms about it. Due to the axiom of foundation and to a lifting axiom, the operator is proved trivial on the cumulative hierarchy of ZF. So we have to drop either foundation or lifting. Since fuzziness concerns complemented predicates rather than sets, a class theory is needed for the very operator. And of them the Kelley-Morse (KM) theory is more appropriate for reasons of class existence. Several definable realizations of the very-operator are presented in KM−. In the last section we consider the operator “very” without the lifting axiom on classes of urelements. To each structurally fuzzy set X a traditional quantitative fuzzy set X is assigned – its quantitative representation. This way we are able partly to recover ordinary fuzzy sets from the structurally fuzzy ones.

Cumulative Higher-Order Logic as a Foundation for Set Theory

The systems Kα of transfinite cumulative types up to α are extended to systems K∞α that include a natural infinitary inference rule, the so-called limit rule. For countable α a semantic completeness theorem for K∞α is proved by the method of reduction trees, and it is shown that every model of K∞α is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic first-order set theories can be interpreted in K∞α, for suitable α.

Models of Second-Order Zermelo Set Theory

In [12], Ernst Zermelo described a succession of models for the axioms of set theory as initial segments of a cumulative hierarchy of levelsUαVα. The recursive definition of theVα's is:Thus, a little reflection on the axioms of Zermelo-Fraenkel set theory (ZF) shows thatVω, the first transfinite level of the hierarchy, is a model of all the axioms ofZFwith the exception of the axiom of infinity. And, in general, one finds that ifκis a strongly inaccessible ordinal, thenVκis a model of all of the axioms ofZF. (For all these models, we take∈to be the standard element-set relation restricted to the members of the domain.) Doubtless, when cast as a first-order theory,ZFdoes not characterize the structures 〈Vκ,∈∩(Vκ×Vκ)〉 forκa strongly inaccessible ordinal, by the Löwenheim-Skolem theorem. Still, one of the main achievements of [12] consisted in establishing that a characterization of these models can be attained when one ventures into second-order logic. For let second-orderZFbe, as usual, the theory that results fromZFwhen the axiom schema of replacement is replaced by its second-order universal closure. Then, it is a remarkable result due to Zermelo that second-orderZFcan only be satisfied in models of the form 〈Vκ,∈∩(Vκ×Vκ)〉 forκa strongly inaccessible ordinal.

The linear logic of multisets

We consider a cumulative hierarchy of multisets over some set of urelements, equipped with additive union ] and a transform relation ¤, and investigate the Horn fragments of Intuitionistic Linear Logic (ILL) that are interpretable in it. The operator ! is defined in an asymptotic way which causes some deviations from the linear-theoretic behavior. Soundness, completeness and partial completeness results are proved for the various fragments. Certain processes of multisets suggest rules for the multiplicatives not compatible with full ILL. One such rule added to the Horn fragment makes the system sound and complete with respect to “coherent processes”.