Elementary Set Theory Research Articles

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ ОРГАНИЗАЦИИ ТЕХНИЧЕСКОГО СЕРВИСА

A wide range of agricultural machinery in need of high-quality maintenance throughout the entire operational period to ensure uninterrupted and efficient operation is one of the most important links in the agro-industrial complex. (Research purpose) The research purpose is developing a mathematical model showing the relationship between the conduct of maintenance activities and the unit of equipment of the agricultural fleet; within the framework of the obtained mathematical model, deriving a criterion assessment of the quantitative conduct of maintenance. (Materials and methods) Indicated that such estimates can be obtained by applying mathematical modeling. A mathematical model was constructed using a visual diagram reflecting the relationship between the machines of the agricultural park, types of maintenance, elements of set theory and mathematical analysis. We used the theory of sets, which allowed us to move from the units of equipment of the fleet of agricultural machines to an analytical representation of realities and derive a constant that gives an assessment of maintenance depending on the number of activities carried out during the time interval under consideration. (Results and discussion) It was noted that it is necessary to improve the quality of maintenance by improving the control of its implementation in real conditions. The novelty of this work consists in the originality of the obtained scheme linking the types of maintenance and the machines of the agricultural fleet, without being tied to the number of units of equipment; in the development of a mathematical model that allows, using theoretical calculations, contributing to improving the control of maintenance activities by obtaining a universal constant that quantifies for any type of equipment and maintenance. The theoretical results were tested using the Kirovets-739 tractor as an example. (Conclusions) We obtained a schematic representation of the relationship between the components of the machine and tractor fleet and a numerical estimate reflecting the number of completed scheduled maintenance steps during the time period under study when using mathematical methods in the organization and conduct of maintenance.

Обзор и классификация пространственно-временных векторов дискретных состояний семифазного преобразователя

The subject of the research is the control algorithms for a seven-phase converter that implement space-vector voltage modulation of a seven-phase motor as an alternative to a three-phase engine in modern electric traction. The study used elements of set theory, combinatorics, Fourier series expansion and vector analysis. Checking research results was implemented on a special stand for experimental studies of spatial vector voltage modulation of a seven-phase motor.

Review and classification of space-time vectors of discrete states ofa seven-phase converter

The subject of the research is the control algorithms for aseven-phase converter that implement space-vector voltage modulation of aseven-phase motor as an alternative to a three-phase engine in modernelectric traction. The study used elements of set theory, combinatorics,Fourier series expansion and vector analysis. Checking research results wasimplemented on a special stand for experimental studies of spatial vectorvoltage modulation of a seven-phase motor. Keywords: Power of the mathematical set of logical states of aseven-phase converter, vector space of generalized vectors of discretestate, combinations of generalized vectors of discrete voltage states of aseven-phase motor\

APPLICATION OF THE THEORY OF THE SETS TO REPRESENT THE LINGUISTIC OBJECTS IN ONTOLOGY

World experience shows that intelligent information systems for decision support are an integral part of modern society functioning. The basis of many tasks that lies in the development of decision support systems is the presentation of knowledge of a particular subject field. Each decision-support system, depending on the application field, has its own characteristics that characterize the main objectives of this subject area, among which, for example, the presentation of knowledge of multi-level administration systems for decision support purposes. In modern conditions, mathematical modeling is the most effective for the formal representation of knowledge in decision support systems which can be used to represent declarative knowledge of land relations. Therefore, urgent scientific practical problem is relevant now that lies in the representing contradictory knowledge in multilevel administration systems. The scientific and practical task envisages the transformation of declarative knowledge of the subject field in the form of mathematical and informational models using elements of set theory. The task envisages the transformation of declarative knowledge in the form of mathematical models with using set theory. The research used methods: analysis, set theory, mathematical modeling, corpus linguistics, ontological engineering. The results of the study are important for organizing activities in a variety of industries using a large body of documents and laws. The purpose of this work is to develop models of knowledge representation on the use of set theory in ontology. The object of research is the process of decision support in multilevel systems. Subject of research is the mathematical models of knowledge representation in multilevel administration systems using set theory. Keywords: formalization, theory, predicate logic, corpus linguistics, ontology, set theory, model.

SYNTHESIS OF ANALYTICAL DESCRIPTION OF THE AUTOCORRELATION FUNCTION FOR GENERALIZED BINARY BARKER SEQUENCES OF TYPE 1 BASED ON ITS DECOMPOSITION USING LINEAR COMPONENTS

Correlation functions (aperiodic and periodic auto- and cross-correlation) of sequences and codes are important for analysis and synthesis of signal-code constructions, because they define different characteristics (spectral, detection and others) in different radar, navigation, telecommunication and other kinds of signal processing and radiotechnical systems. The synthesis of mathematical expressions for the autocorrelation function for generalized binary Barker sequences of type 1, which based on the analysis and decomposition of a structure of the autocorrelation function using its linear components, is considered in the article. The hypothesis that the autocorrelation function for some kinds of sequences can be represented by some number of linear components and separated local points is possible due to an existence of periodical and recurrent constituents in structures of such sequences. The considered approach is characterized by a direct analysis of obtained structures of the autocorrelation function for described kind of sequences, separation from them related linear components, and generalization of the form of analytical description for the autocorrelation function of sequence of any possible length using elements of set theory. The approach substantially simplifies the process of solving a problem of synthesis of analytical description of the autocorrelation function. However, it requires a correct choice of the decomposition of structures of the autocorrelation function from all possible ones. The result of a synthesis shows that a general form of analytical description of the autocorrelation function, i.e. for any possible sequence length, for generalized binary Barker sequences of type 1 contains 19 components, of which 13 are linear components, and 6 are separated local points.

The method of iterative use of the ABC-XYZ analysis in the construction industry

The issue of marketing in the construction industry has always remained relevant. Indeed, it is not easy to design objects so that they bring high and stable income. The paper studies a solution to the problem, based on the use of the iteration method, the ABC - XYZ analysis, and digital filtering. It is based on the well-known principle of the study of complex systems: the sequential division of a complex task into a number of simpler ones. We would like to note that the “classic” ABC - XYZ analysis gives only general results about the profits from construction projects and does not study the reasons due to which these profits are received. In contrast, the proposed tooling allows taking into account the main technical, geographical and other factors typical for profitable construction projects. One of the advantages of the considered approach is that the conventional computer tools - a spreadsheet MS Excel, is used in solution. The paper discusses the advantages and disadvantages of the proposed approach, specifies in which cases it is advisable to use this approach and where to use more powerful and expensive tools - Big Data technology. For better visualization and clarity, the elements of set theory are used in the paper. This methodology is universal, and its use in the energy industry is possible when selecting more profitable types of products.

On the Methodology of the Analytic Method: Historical Account, Epistemological Suggestions, Stages

In this paper we attempt a good unearthing, sound tackling and effective pointing out of the critical key features permeating one of the two cornerstones supporting the total coordination and interplay underlying the complex epistemonic and scientific method: and that would be the constituent methodology concisely known under the cover name “analytic method”. In this attempt, however, it becomes evident very soon that the direct similarities, reciprocities and complementarities to the other cornerstone, concisely known as “abstractive method”, must also be laid out concurrently; the two share the same needs and behaviours, while they must also be mutually disambiguated and framed in inner consistencies and mutual contrast, to assert proper theoretical coverage, fruitful application and mutual impermeability overall.Thus, both of them end up being discussed in parallel. To that end, simple and intuitively appealing proposals are sought, expressed and projected, regarding their key attributes, range and boundaries, both generally and especially in between them; their defining points and most essential requirements are sought out and expressed, their functional standards are suggested and the stages of its application are prescribed, according to a shadowy yet rigorous gradual breakup and assembling procedure derived from elementary combinatorics and set theory. All this gets illustrated in the schematic stages to be pursued in two relevant enlightening analytic examples, chosen so as to also partake of a crucial share of an abstractive aspect.Two relevant issues are touched upon along the way in the vein of the present tone of quest, namely those concerning basic prerequisites of logic employed and the confusion and effective double-talk underlying the term “theory” (vs. “theories”) and the consequent term-induced ambiguity in the concepts. On the subject, analysis is the breakup of an “entity” into its “components” and synthesis is its retroactive operation, or “return”, from “components” to the “entity”. Similarly, “abstraction” is the course from an entity to a nesting of successive general categories, and “structure” is the other way around from more general categories towards smaller contained assemblages of entities, each uniformly exhibiting “partial manifestations” of the said categorical feature.The semantic context and its set-theoretical implications are of paramount importance here. What is definitively taken as a “unity”, in each case, is a matter of choice, and of mutual and binding agreement among the parties conducting the study and its discussion. Beyond that, figuratively speaking, the analytic method concerns “anatomical autopsies” digging into such unities constituting elements or members of sets, whereas the abstractive method concerns attributes connecting sets of such “unities” and their nested supersets and subsets. We close by sketching the stage-by-stage application as illustrated in two case-studies: one from Cultural Studies, regarding a model, focusing on the primitive ternary ritual of the performing arts [ΚΛΜ], discussed complete with some key side issues, and accompanied by considerations on the triple of the material arts [ΧΨΩ], and then one from Bioethics regarding all possible partial and combinatorial constituents in consent models.

Issues of Regulating Economic Agents Behavior in Territorial Development Management

One of the key problems in managing the development of territorial socio-economic systems is inconsistency and contradictory interests of various economic agents. In modern conditions, it is impossible to avoid the need to reconcile interests, even when one of the parties is the subject of management. The article examines the process of regulating behavior of economic agents in the system of strategic management of territorial development. The procedures of decision-making and the influence of various factors on them, including regulators are analyzed. Developing of an approach to managing the behavior of economic agents in the territorial socio-economic system based on the interests of not only management systems, but also the economic agents themselves.Methodology.The research is based on the principles of system and complexity, factor analysis, comparative analysis, grouping methods, economic and mathematical modeling, elements of set theory and fuzzy logic are also used.The general methodical scheme of behavior management of economic agents at the regional level, characterized by the presence of an iterative procedure of differential impact on the factors and deterministic properties of the behavior of agents with a mechanism for assessing the degree of influence of regulatory actions based on feedback, is proposed. The developed approach allows to form within the framework of market mechanisms, the target scenario of the behavior of economic agents in the process of developing a coherent strategy for the development of the regional socio-economic system.The development and adoption of managerial decisions should be based on taking into account the interests of all stakeholders. Therefore, this role of public authorities also includes the definition of regulators, allowing adjusting and coordinating the behavior of other economic agents. It is also should be mentioned that management bodies can also act as interacting agents. Accordingly, local self-government bodies act not only in the role of an element of management system, but also in the role of a legally independent counterparty when interacting with the governing bodies of a constituent entity of the Russian federation.

СЕТЕВАЯ МОДЕЛЬ БИЗНЕС-ПРОЦЕССОВ ИТ-ПРОВАЙДЕРА

The subject of the research is to improve the efficiency of business processes for various industries (for example, IT service provision). Constant involvement in the competition is forcing service providers and high-tech product manufacturers to optimize internal and external processes to achieve the desired result. The market information and other services need to obtain an integrated toolkit that allows to perform analysis of existing processes and their metrics and make management decisions. The paper focuses on variable procedural model or network model of the business process as one of the components of the assessment system and the possibility to improve the efficiency of enterprises and organizations. The main actively applied research methods are the system analysis of the elements of set theory and graph theory. Illustration of a business process model has been made with the assistance of a methodological framework of the best management practices in the IT sector. The following developing effective technologies of software systems have also been used: relational databases, object-oriented programming, and unit testing. The novelty of the research is to develop a changeable business process model, which if necessary can be adjusted for a particular subject area. The essence of the elements of the variable model is reflected in the possibility of a flexible dynamic binding of certain roles or assets to the processes of the network schedule or procedures. Thus, the proposed model is sufficiently versatile, which allows using it in a wide range of applications and systems for performance improvement with a comprehensive assessment of costs and resources.

A Note on Equivalence Among Various Scalar Field Models of Dark Energies

In this work, we have tried to find out similarities between various available models of scalar field dark energies (e.g., quintessence, k-essence, tachyon, phantom, quintom, dilatonic dark energy, etc). We have defined an equivalence relation from elementary set theory between scalar field models of dark energies and used fundamental ideas from linear algebra to set up our model. Consequently, we have obtained mutually disjoint subsets of scalar field dark energies with similar properties and discussed our observation.

Students’ Difficulties with Definitions in the Context of Proofs in Elementary Set Theory

In this paper we explore first-year students’ difficulties with the use and interpretation of definitions of mathematical objects as they attempt proof construction exercises in the area of elementary set theory. The participants are students at a historically disadvantaged university in South Africa. In this study the activities and utterances of 10 students who took part in consultative group sessions were observed and analysed. Consultative sessions were organised so as to encourage and develop students’ active participation while engaging in the task of proof construction. The framework that was used to analyse students’ proof comprehension and construction actions and contributions, particularly their interpretation and use of definitions, is described in the paper. The findings of the study resonate closely with those of researchers in the developed world. Students’ difficulties with definitions of mathematical objects include their misinterpretation of definitions of objects such as the union of sets and the Cartesian product and their association of mathematical objects with a word or symbol contained in their definitions.

Venn diagrams in cardiovascular disease: the Heart Team concept

Venn diagrams are illustrations composed of overlapping circles that demonstrate the relations between finite collections of things1 and are most useful in defining areas of commonality among different aggregations. Originally described by John Venn in the 1880s to teach elementary set theory, these diagrams are most often used to illustrate set relationships in such fields as probability, statistics and computer science. Venn diagrams can also be useful for understanding the roles of various stakeholders in the management of cardiovascular disease from its diagnosis through its treatment. As the field progresses, the area of overlap of the cardiovascular disease Venn diagram continues to expand. This is evident in many aspects of cardiovascular disease management, including individual diagnosticians and treatment specialists, diseases, technologies, institutions, payers and regulators. Despite a movement towards convergence in areas of disease management, however, professional societies remain in their individual silos. This article explores whether Venn diagrams of professional societies, which traditionally had little overlap or mutual engagement, should continue to converge until they merge into one. The well-recognized concept of atherosclerosis as a disease that affects all vascular beds has focused therapeutic strategies on the coexistence, for example, of coronary artery disease and peripheral arterial disease in the same patient. Whereas in the past there was a tendency to treat an acute coronary syndrome as an isolated event in a patient, attention now focuses on the fact that other manifestations of the disease, such as peripheral arterial disease and cerebral vascular disease, are likely to be present in this patient as well, and the involvement of these other vascular beds may affect treatment strategies. Thus, the evolution of the understanding about atherosclerosis has stimulated physicians and surgeons who focus on these different vascular beds to strategize together about the treatment of this patient. Another example of …

Venn diagrams in cardiovascular disease: the Heart Team concept

Venn diagrams are illustrations composed of overlapping circles that demonstrate the relations between finite collections of things1 and are most useful in defining areas of commonality among different aggregations. Originally described by John Venn in the 1880s to teach elementary set theory, these diagrams are most often used to illustrate set relationships in such fields as probability, statistics and computer science. Venn diagrams can also be useful for understanding the roles of various stakeholders in the management of cardiovascular disease from its diagnosis through its treatment. As the field progresses, the area of overlap of the cardiovascular disease Venn diagram continues to expand. This is evident in many aspects of cardiovascular disease management, including individual diagnosticians and treatment specialists, diseases, technologies, institutions, payers and regulators. Despite a movement towards convergence in areas of disease management, however, professional societies remain in their individual silos. This article explores whether Venn diagrams of professional societies, which traditionally had little overlap or mutual engagement, should continue to converge until they merge into one. The well-recognized concept of atherosclerosis as a disease that affects all vascular beds has focused therapeutic strategies on the coexistence, for example, of coronary artery disease and peripheral arterial disease in the same patient. Whereas in the past there was a tendency to treat an acute coronary syndrome as an isolated event in a patient, attention now focuses on the fact that other manifestations of the disease, such as peripheral arterial disease and cerebral vascular disease, are likely to be present in this patient as well, and the involvement of these other vascular beds may affect treatment strategies. Thus, the evolution of the understanding about atherosclerosis has stimulated physicians and surgeons who focus on these different vascular beds to strategize together about the treatment of this patient. Another example of …

Mathematics in games, sports, and gambling: the games people play

Basic Probability Introduction Of Dice and Men Probability The Laws That Govern Us Poker Hands versus Batting Orders Let's Play for Money! Is That Fair? The Odds Are against Us Things Vary Conditional Expectation The Game's Afoot Applications to Games Counting and Probability in Poker Hands Roulette Craps Let's Make a Deal - The Monty Hall Problem Carnival Games Other Casino Games Backgammon Repeated Play Introduction Binomial Coefficients The Binomial Distribution The Poisson Distribution Streaks-Are They Real? Betting Strategies The Gambler's Ruin Card Tricks and More Introduction The Five-Card Trick The Two-Deck Matching Game More Tricks The Paintball Wars Dealing with Data Introduction Batting Averages and Simpson's Paradox NFL Passer Ratings Viewing Data - Simple Graphs Confidence in Our Estimates Measuring Differences in Performance Testing and Relationships Introduction Suzuki versus Pujols I'll Decide If I Believe That Are the Old Adages True? How Good Are Certain Measurements? Arguing over Outstanding Performances A Last Look at Comparisons Games and Puzzles Introduction Number Arrays The Tower of Hanoi Instant Insanity Lights Out Peg Games Puzzles on the Chessboard Guarini's Problem Martin Gardner's No 3-in-a-Line Problem The Knight's Tour Domination and Independence Attacking Placements and Independence Combinatorial Games Introduction to Combinatorial Games Subtraction Games Nim Games as Digraphs Blue-Red Hackenbush Green Hackenbush Games as Numbers More about Nimbers Appendix Review of Elementary Set Theory Relations and Functions Standard Normal Distribution Table Student's t-Distribution Solutions to Problems Solutions to Selected Exercises Bibliography Index

The mathematics of infinity: a guide to great ideas

Preface. 1. Elementary Set Theory. 1.1 Sets. 1.2 Cartesian Products. 1.3 Power Sets. 1.4 Something From Nothing. 1.5 Indexed Families of Sets. 2. Functions. 2.1 Functional Preliminaries. 2.2 Images and Preimages. 2.3 One-to-one and Onto . 2.4 Bijections. 2.5 Inverse Functions. 3. Counting Infinite Sets. 3.1 Finite Sets. 3.2 Hilbert's Infinite Hotel. 3.3 Equivalent Sets and Cardinality. 4. Infinite Cardinals. 4.1 Countable Sets. 4.2 Uncountable Sets. 4.3 Two Infinities. 4.4 Power Sets. 4.5 The Arithmetic of Cardinals. 5. Well Ordered Sets. 5.1 Successors of Elements. 5.2 The Arithmetic of Ordinals. 5.4 Magnitude versus Cardinality. 6. Inductions and Numbers. 6.1 Mathematical Induction. 6.2 Transfinite Induction. 6.3 Mathematical Recursion . 6.4 Number Theory. 6.5 The Fundamental Theorem of Arithmetic. 6.6 Perfect Numbers. 7. Prime Numbers. 7.1 Prime Number Generators. 7.2 The Prime Number Theorem. 7.3 Products of Geometric Series. 7.4 The Riernann Zeta Function. 7.5 Real Numbers. 8. Logic and Meta-Mathematics. 8.1 The Collection of All Sets. 8.2 Other Than True or False. Bibliography. Index.

The History of Set Theory from a European Point of View

The History of Set Theory from a European Point of View L__7fa L-? ? Luigi Verdi The use of elementary set theory inmusic is usually identifiedwith developments in speculative compositional theory in the United States since the early 1950s. However, before 1950 composers in both America and Europe invented various forms of pitch theory that are clearly early manifestations of the later "set theory." In Europe versus America, the pioneering stages of what has become known as "set theory" is a good deal more complicated and diverse. In this paper, we trace the development of European set theory in some detail. One of the typical features of earlyEuropean set theory is itsasystematic nature, consisting of empirical research oftenmixed with extra-scientific or even esoteric elements. No single school emerged as definitive, and the var ious theories remained isolated indifferent geographical locations. Thanks to research groups connected with universities or cultural centers such as the IRCAM1 Music Representation Group, recendy some common lines History of SetTheory froma European PointofView 155 of development invarious European countries have been identified, result ing in amore complete and integrated understanding. The use of simple arithmetic to define and describe musical processes is typical ofWestern culture.With the dissolution of the tonal system, the use of numbers for defining pitch and time relations inmusic substan tially increased. On the one hand, arithmetic was used to identify syntac tic rules tomodel the development of the past musical language. On the other hand, arithmetic implemented a search fornew compositional rules to guide themusic of the future. Very early in the twentieth century, several European theorists and composers were simultaneously considering issues such as the structural equivalence between minor and major chords under inversion or the construction of scales dividing the octave into equal parts. For example, see Vincent d'lndy (1851-1931), Cours de composition (1909), Hermann Schroder (1843-1909), Die symmetrischeUmkehrung in der Musik (1902), or later, Sigfrid Karg-Elert (1877-1933), Polaristische Klang- und Tonalitdtslehre (19 31). The basic methodology was to iden tifygeometrical patterns that could provide insight into the features of a musical language. A pioneering work by theUkrainian theorist Boleslav Javorsky (1877 1942) analyzed Scriabin's music and pointed out thatmany of theRussian composer's most innovative works were based upon patterns dividing the octave into equal parts. The use of whole numbers proved particularly effective to the identification of the properties of such scales and therefore the intervals of the octatonic (tone/semitone) scale began to be indicated as 1-2-1-2-1-2-1-2. Those of the whole-tone scale were identified as 2 2-2-2-2-2, and so on. This is one of the earliest examples of the use of interval strings to describe the structure of sets of tones. The study of the division of the octave into equal parts was of such considerable interest in theUkraine that it stimulated thework of two renowned Ukrainian com posers, Nicolas Slonimsky and Joseph Schillinger. The work of these authors, however, ispurely empirical. In the case of Slonimsky, scale sym metry is almost treated as a curiosity. Schillinger's theories were based on esoteric elements that are completely arbitrary and lacking scientific basis. Both authors moved to America early in their careers and became out standing personalities inAmerican musical culture. Yet their influence as theorists remained limited. Their most important theoretical works are Schillinger's TheMathematical Basis of theArts (1948), and Slonimsky's Thesaurus ofScales and Melodic Patterns (1947). However, the work of Slonimsky and Schillinger was the result of a European variant of set theory; this is confirmed by many early twentieth-century Russian avant-garde composers who segmented the 156 PerspectivesofNew Music chromatic scale in various ways. JefimGolyscheff (b. Kherson, Ukraine 1897, d. Paris 1970), Nicolas Obuchov (b.Moscow 1892, d. Paris 1954) Nikolaj Roslavec (b. Duschatin, Ukraine 1881, d. Moscow 1944), Alexander Mosolov (b. Kiev 1900, d. Moscow 1973), and Ivan Wyschnegradsky (b. Saint-Petersburg 1893, d. Paris 1979) were the "avant-garde pioneers" (Gojowy 1972). A similar trend to the Ukrainian-Russian development arose in Italy, particularly by Domenico Alaleona (1881-1928), whose study J moderni orizzonti delta tecnica musicale is unanimously recognized as a...

Quelques Erreurs Pouvant Être Liées à Une Difficulté à Concevoir un Ensemble Comme un Objet Distinct De Ses Éléments Chez Des Étudiants Et Des Étudiantes Universitaires

As a first step of a project on learning difficulties in elementary set theory, we carried out a study aimed at identifying some of the difficulties encountered by a group of 21 mathematics majors taking a course on logic and set theory. In this paper, we shall discuss one of the difficulties that we observed, namely a difficulty in conceiving a set as an object distinct from its elements, i.e. failing to fully grant sets the status of objects. This difficulty may be related to the following three types of errors that we observed in the students' first assignment: (1) confusing belonging and inclusion, (2) confusing the union of sets A, B, C, ... with the set whose elements are A, B, C, … and (3) adding or deleting curly brackets. We shall present excerpts from the students' work illustrating these three types of errors and explain how they may be related to a difficulty in conceiving a set as an object distinct from its elements.

Fuzzy class theory

The paper introduces a simple, yet powerful axiomatization of Zadeh's notion of fuzzy set, based on formal fuzzy logic. The presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics. Its essence is elementary fuzzy set theory, cast as two-sorted first-order theory over fuzzy logic, which is generalized to simple type theory. We show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism. In this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed; however, we also demonstrate how to add structure to the universe of discourse within our framework.

Mereological sets of distributive classes

We will present an elementary theory in which we can speak of mereological sets composed of distributive classes. Besides the concept of a distributive class and the membership relation, it will possess the notion of a mereological set and the relation of being a mereological part. In this theory we will interpret Morse’s elementary set theory (cf. Morse [11]). We will show that our theory has a model, if only Morse’s theory has one.

Parametric corecursion

This paper gives a treatment of substitution for “parametric” objects in final coalgebras, and also presents principles of definition by corecursion for such objects. The substitution results are coalgebraic versions of well-known consequences of initiality, and the work on corecursion is a general formulation which allows one to specify elements of final coalgebras using systems of equations. One source of our results is the theory of hypersets, and at the end of this paper we sketch a development of that theory which calls upon the general work of this paper to a very large extent and particular facts of elementary set theory to a much smaller extent.