Cumulative Hierarchy Research Articles

The purely iterative conception of set

Abstract According to the iterative conception of set, sets are formed in stages. According to the purely iterative conception of set, sets are formed by iterated application of a set-of operation. The cumulative hierarchy is a mathematical realization of the iterative conception of set. A mathematical realization of the purely iterative conception can be found in Peter Aczel’s type-theoretic model of constructive set theory. I will explain Aczel’s model construction in a way that presupposes no previous familiarity with the theories on which it is based.

Space–time structure of weak magnetohydrodynamic turbulence

The two-time energy spectrum of weak magnetohydrodynamic turbulence is found by applying a wave-turbulence closure to the cumulant hierarchy constructed from the dynamical equations. Solutions are facilitated via asymptotic expansions in terms of the small parameter $\varepsilon$ , describing the ratio of time scales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of nonlinearity at a given spatial scale is further quantified by an integration over all possible delta-correlated modes compliant in a given set of three-wave interactions that are associated with energy flux through the said scale. The wave-turbulence closure for the two-time spectrum uncovers a secularity occurring on a time scale of order $\varepsilon ^{-2}$ , and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that for the regime of stationary turbulence, the two-time energy spectrum exponentially decays on a lagged time scale $(\varepsilon ^2 \gamma _k^s)^{-1}$ in proportion to the strength of the associated three-wave interactions, characterized by nonlinear decorrelation frequency $\gamma _k^s$ . The scaling of the form $k_{\perp } v_0 \chi _0$ exhibited by this frequency is reminiscent of random sweeping by the outer scale with characteristic fluctuation velocity $v_0$ that is modified due to competition with Alfvénic propagation (characterized by $\chi _0$ ) at the said scale. A brief calculation of frequency broadening of the power spectrum due to nonlinear interactions is also presented.

Strongly compact cardinals and ordinal definability

This paper explores several topics related to Woodin’s HOD conjecture. We improve the large cardinal hypothesis of Woodin’s HOD dichotomy theorem from an extendible cardinal to a strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable [Formula: see text]-Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD.

Energy-Efficient LoRa Routing for Smart Grids.

Energy-efficient routing protocols in Internet of Things (IoT) applications are always of colossal importance as they improve the network's longevity. The smart grid (SG) application of the IoT uses advanced metring infrastructure (AMI) to read and record power consumption periodically or on demand. The AMI sensor nodes in a smart grid network sense, process, and transmit information, which require energy, which is a limited resource and is an important parameter required to maintain the network for a longer duration. The present work discusses a novel energy-efficient routing criterion in an SG environment realised using LoRa nodes. Firstly, a modified LEACH protocol-cumulative low-energy adaptive clustering hierarchy (Cum_LEACH) is proposed for cluster head selection among the nodes. It uses the cumulative energy distribution of the nodes to select the cluster head. Furthermore, for test packet transmission, multiple optimal paths are created using the quadratic kernelised African-buffalo-optimisation-based LOADng (qAB_LOADng) algorithm. The best optimal path is selected from these multiple paths using a modified version of the MAX algorithm called the SMAx algorithm. This routing criterion showed an improved energy consumption profile of the nodes and the number of active nodes after running for 5000 iterations compared to standard routing protocols such as LEACH, SEP, and DEEC.

Periodicity in the cumulative hierarchy

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in \mathrm{ZF} set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels V_\alpha are defined via iterated application of the power set operation, starting from V_0=\emptyset , setting V_{\alpha+1}=\mathcal{P}(V_\alpha) , and taking unions at limit stages. Assuming that j:V_{\alpha+1}\to V_{\alpha+1} is a (non-trivial) elementary embedding, we show that V_\alpha is fundamentally different from V_{\alpha+1} : we show that j is definable from parameters over V_{\alpha+1} iff \alpha+1 is an odd ordinal. The definability is uniform in odd \alpha+1 and j . We also give a characterization of elementary j:V_{\alpha+2}\to V_{\alpha+2} in terms of ultrapower maps via certain ultrafilters. For limit ordinals \lambda , we prove that if j:V_\lambda\to V_\lambda is \Sigma_1 -elementary, then j is not definable over V_\lambda from parameters, and if \beta<\lambda and j:V_\beta\to V_\lambda is fully elementary and \in -cofinal, then j is likewise not definable. If there is a Reinhardt cardinal, then for all sufficiently large ordinals \alpha , there is indeed an elementary j:V_\alpha\to V_\alpha , and therefore the cumulative hierarchy is eventually periodic (with period 2).

Direct statistical simulation of low-order dynamosystems

In this paper, we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third order and has cubic nonlinearities while the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be used to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics: timestepping, which is able to locate only stable solutions of the equations for the statistics, and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases, timestepping is a more robust protocol for finding meaningful solutions to DSS.

LEVEL THEORY, PART 1: AXIOMATIZING THE BARE IDEA OF A CUMULATIVE HIERARCHY OF SETS

AbstractThe following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: ‘Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S’. Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, Montague, Derrick, and Potter.

LEVEL THEORY, PART 3: A BOOLEAN ALGEBRA OF SETS ARRANGED IN WELL-ORDERED LEVELS

AbstractOn a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.

Analysing the Cumulative Hierarchy of the Taxonomy of Learning Objectives in Flipped Classroom

The emergence of digital learning formats influences the planning and structuring of digital teaching. Especially in times of the Corona Pandemic, when many universities remain closed, new digital learning concepts are emerging that can be integrated into face-to-face teaching in future. In this context, old teaching formats are often revised and questioned. But while technology only determines the form of collaboration, the real quality of learning depends on cognitive trials that the teacher addresses to the students. To classify these trials, a teacher can use Bloom's revised taxonomy, which ranks Learning Objectives in a six-level order and assumes a cumulative hierarchy: achieving a required Learning Objective level includes all lower levels. Especially in blended learning scenarios, such as a Flipped Classroom, this theory can be used to develop the course structure and to form exam questions. However, the applicability of the cumulative hierarchy is controversial in the literature and is rarely analysed in blended learning courses. Our goal is therefore to verify the cumulative hierarchy in a Flipped Classroom Course and derive recommendations for action. Therefore, we use a quantitative written survey. Since the analysis is based on the students' perceptions, these are verified by correlation analysis with the actual exam results and the awareness of contents and activities. Afterwards, the cumulative hierarchy is tested by regression analysis of the different levels of Learning Objectives. As a result, it could be confirmed for all levels, but not always by direct but often by indirect influences of other levels.

Abolishing Platonism in Multiverse Theories

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis (CH), by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily address ontological multiversism as advocated, for instance, by Hamkins or Vaatanen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter.

RANK-TO-RANK EMBEDDINGS AND STEEL’S CONJECTURE

AbstractThis paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a variant of the Mitchell order called the internal relation.

Why is Cantor’s Absolute Inherently Inaccessible?

In this article, as implied by the title, I intend to argue for the unattainability of Cantor’s Absolute at least in terms of the proof-theoretical means of set-theory and of the theory of large cardinals. For this reason a significant part of the article is a critical review of the progress of set-theory and of mathematical foundations toward resolving problems which to the one or the other degree are associated with the concept of infinity especially the one beyond that of the natural intuition of natural numbers. Naturally the review includes the foundation and development of the theory of large cardinals, especially after Cohen’s revolutionary introduction of the forcing method, insofar as it paved the way toward a transcendence of the delimitative character of Godel’s constructible universe L and further toward ever stronger infinity assumptions. Given that Cantor in his theory of transfinite numbers defined the mathematical absolute in ontological rather than concrete mathematical terms, I proceed in the last section to a philosophical discussion with certain prompts from phenomenology regarding the transposability of the formal conception of the infinite to the level of subjective constitution and argue in these terms on the infeasibility of acceding to an ‘absolute’ infinite cardinality. Of course the argumentation is strongly based on a view of the set-theoretical universe V of von Neumann’s cumulative hierarchy as a formal representative of the essential traits one would seek from a model of Cantor’s Absolute. It is also based on the conclusions reached from the whole discussion within the context of formal-theoretical structures as such.

Relationships Between the Distribution of Watanabe–Strogatz Variables and Circular Cumulants for Ensembles of Phase Elements

The Watanabe–Strogatz (WS) and Ott–Antonsen (OA) theories provided a seminal framework for rigorous and comprehensive studies of collective phenomena in a broad class of paradigmatic models for ensembles of coupled oscillators. Recently, a “circular cumulant” approach was suggested for constructing the perturbation theory for the OA approach. In this paper, we derive the relations between the distribution of WS phases and the circular cumulants of the original phases. These relations are important for the interpretation of the circular cumulant approach in the context of the WS and OA theories. Special attention is paid to the case of hierarchy of circular cumulants, which is generally relevant for constructing perturbation theories for the WS and OA approaches.

Adjoint sensitivity analysis of chaotic systems using cumulant truncation

We describe a simple and systematic method for obtaining approximate sensitivity information from a chaotic dynamical system using a hierarchy of cumulant equations. The resulting forward and adjoint systems yield information about gradients of functionals of the system and do not suffer from the convergence issues that are associated with the tangent linear representation of the original chaotic system. The functionals on which we focus are ensemble-averaged quantities, whose dynamics are not necessarily chaotic; hence we analyse the system’s statistical state dynamics, rather than individual trajectories. The approach is designed for extracting parameter sensitivity information from the detailed statistics that can be obtained from direct numerical simulation or experiments. We advocate a data-driven approach that incorporates observations of a system’s cumulants to determine an optimal closure for a hierarchy of cumulants that does not require the specification of model parameters. Whilst the sensitivity information from the resulting surrogate model is approximate, the approach is designed to be used in the analysis of turbulence, whose degrees of freedom and complexity currently prohibits the use of more accurate techniques. Here we apply the method to obtain functional gradients from low-dimensional representations of Rayleigh-Bénard convection.

О связи между распределением фаз Ватанабэ–Строгаца и круговыми кумулянтами

The theories of Watanabe–Strogatz and Ott–Antonsen served as the basis for rigorous and comprehensive investigations of collective phenomena in a broad class of paradigmatic models of en-sables of coupled oscillators. Recently, the “circular cumulant” approach was suggested for constructing a perturbation theory for the Ott–Anthonsen approach. In this paper we derived the relationship between the distribution of Watanabe–Strogatz phases and the circular cumulants of the original phases. These relationships are important for interpreting the approach of circular cumulants in the context of the Watanabe–Strogatz and Ott–Anthonsen theories. Particular attention is devoted to the case of the hierarchy of circular cumulants; this case is typical when constructing perturbation theories on top of the Watanabe–Strogats and Ott–Anthonsen theories.

Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory

We formalise second-order ZF set theory in the dependent type theory of Coq. Assuming excluded middle, we prove Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the categoricity of extended axiomatisations fixing the number of Grothendieck universes. These results are based on an inductive definition of the cumulative hierarchy eliminating the need for ordinals and set-theoretic transfinite recursion. Following Aczel’s sets-as-trees interpretation, we give a concise construction of an intensional model of second-order ZF with a weakened replacement axiom. Whereas this construction depends on no additional logical axioms, we obtain intensional and extensional models with full replacement assuming a description operator for trees and a weak form of proof irrelevance. In fact, these assumptions yield large models with n Grothendieck universes for every number n.

Finding closure: approximating Vlasov-Poisson using finitely generated cumulants

Since dark matter almost exclusively interacts gravitationally, the phase-space dynamics is described by the Vlasov-Poisson equation. A key characteristic is its infinite cumulant hierarchy, a tower of coupled evolution equations for the cumulants of the phase-space distribution. While on large scales the matter distribution is well described as a fluid and the hierarchy can be truncated, smaller scales are in the multi-stream regime in which all higher-order cumulants are sourced through nonlinear gravitational collapse. This regime is crucial for the formation of bound structures and the emergence of characteristic properties such as their density profiles. We present a novel closure strategy for the cumulant hierarchy that is inspired by finitely generated cumulants and hence beyond truncation. This constitutes a constructive approach for reducing nonlinear phase-space dynamics of Vlasov-Poisson to a closed system of equations in position space. Using this idea, we derive Schrödinger-Poisson as approximate quantal method for solving classical dynamics of Vlasov-Poisson with cold initial conditions. Our deduction complements the common reverse inference of the Schrödinger-Vlasov relation using a semi-classical limit of quantum mechanics and provides a clearer picture of the correspondence between classical and quantum dynamics. Our framework outlines an essential first step towards constructing approximate methods for Vlasov-like systems in cosmology and plasma physics with different initial conditions and potentials.

MODAL STRUCTURALISM AND REFLECTION

AbstractModal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (forModal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the article, I look at the prospects for supplementing MSST with a modal structuralreflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak.

Role of comprehension on performance at higher levels of Bloom's taxonomy: Findings from assessments of healthcare professional students.

The first four levels of Bloom's taxonomy were used to create quiz questions designed to assess student learning of the gross anatomy, histology, and physiology of the gastrointestinal (GI) system. Information on GI histology and physiology was presented to separate samples of medical, dental, and podiatry students in computer based tutorials where the information from the two disciplines was presented either separately or in an integrated fashion. All students were taught GI gross anatomy prior to this study by course faculty as part of the required curriculum of their respective program. Student responses to the quiz questions were analyzed to assess both the validity of Bloom's cumulative hierarchy and the effectiveness of an integrated curriculum. No statistically significant differences were found between quiz scores from students who received the integrated tutorial and from those who received the separate tutorials. Multiple regression analyses provided partial support for a cumulative hierarchy where scores on the lower levels of Bloom's taxonomy predicted scores on higher levels. Notably, in the majority of regression analyses, the comprehension score was the key foundational predictor for application and analysis scores. This study supports the suggestion that educators increase the number of comprehension level questions, even at the expense of knowledge level questions, in course assessments both to evaluate lower order cognitive skills and also as a predictor of success on questions requiring application and analysis levels of the higher order cognitive skills of Bloom's taxonomy. Anat Sci Educ 11: 433-444. © 2018 American Association of Anatomists.

Herbrand-satisfiability of a Quantified Set-theoretic Fragment*

In the last decades, several fragments of set theory have been studied in the context of the so-called Computable Set Theory. In general, the semantics of set-theoretical languages differs from the canonical first-order semantics in that the interpretation domain of set-theoretical terms is fixed to a given universe of sets, as for instance the von Neumann standard cumulative hierarchy of sets, i.e., the class consisting of all the pure sets. Because of this, theoretical results and various machinery developed in the context of first-order logic cannot be easily adapted to work in the set-theoretical realm. Recently, quantified fragments of set-theory which allow one to explicitly handle ordered pairs have been studied for decidability purposes, in view of applications in the field of knowledge representation. Among other results, a NexpTime decision procedure for satisfiability of formulae in one of these fragments, ∀0 , has been provided. In this paper we exploit the main features of such a decision procedure to reduce the satisfiability problem for the fragment ∀0 to the problem of Herbrand satisfiability for a first-order language extending it. In addition, it turns out that such reduction maps formulae of the Disjunctive Datalog subset of ∀0 into Disjunctive Datalog programs.