Academic literature on the topic 'Three paradoxes of set theory'

AbstractDialetheism is the metaphysical claim that there are true contradictions. And based on this view, Graham Priest and his collaborators have been suggesting solutions to a number of paradoxes. Those paradoxes include Russell’s paradox in naive set theory. For the purpose of dealing with this paradox, Priest is known to have argued against the presence of classical negation in the underlying logic of naive set theory. The aim of the present paper is to challenge this view by showing that there is a way to handle classical negation.

This paper begins an axiomatic development of naive set theory—the consequences of a full comprehension principle—in a paraconsistent logic. Results divide into two sorts. There is classical recapture, where the main theorems of ordinal and Peano arithmetic are proved, showing that naive set theory can provide a foundation for standard mathematics. Then there are major extensions, including proofs of the famous paradoxes and the axiom of choice (in the form of the well-ordering principle). At the end I indicate how later developments of cardinal numbers will lead to Cantor’s theorem, the existence of large cardinals, and a counterexample to the continuum hypothesis.

Kant claimed that it is impossible for us to have a consistent notion of the infinite. I shall concentrate on three versions of the paradoxes of the infinite: Kant’s first antinomy, the paradoxes of Cantorian set theory, and applications of Cantorian arguments to the metaphysics of the world. I shall dare two side-glance looks at Ancient Chinese Philosophy, where analogies to the Western paradoxes can be found. I shall first discuss key passages from the Chinese sophists, and then consider the formulation of the Law of Non-Contradiction in the Moist Canons. I conclude that the paradoxes of the infinite remain a major challenge for reason.

Since the formulation of Russell’s paradox, many people have created more accessible models while trying to understand and solve the paradox. The Barber paradox is the most famous one, but it is not the case that this paradox was not proposed by Russell. This paper will demonstrate the nature of both paradoxes through truth-functional language and propose possible solutions (theory of types) for Russell’s paradox. The reason why the Barber paradox is a pseudo paradox will also be illustrated with a possible solution. There is a huge difference between the paradoxes due to the fundamental difference in the set, and more reasons will be clarified in the paper.

Cantor thought of the principles of set theory or intuitive principles as universal forms that can apply to any actual or possible totality. This is something, however, which need not be accepted if there are totalities which have a fundamental ontological value and do not conform to these principles. The difficulties involved are not related to ontological problems but with certain peculiar sets, including the set of all sets that are not members of themselves, the set of all sets, and the ordinal of all ordinals. These problematic totalities for intuitive theory can be treated satisfactorily with the Zermelo and Fraenkel (ZF) axioms or the von Neumann, Bernays, and Gödel (NBG) axioms, and the iterative conceptions expressed in them.

The St Petersburg Paradox revolves round the determination of a fair price for playing the St Petersburg Game. According to the original formulation, the price for the game is infinite, and, therefore, paradoxical. Although the St Petersburg Paradox can be seen as concerning merely a game, Paul Samuelson (1977) calls it a “fascinating chapter in the history of ideas”, a chapter that gave rise to a considerable number of papers over more than 200 years involving fields such as probability theory and economics. In a paper in this journal, Vivian (2013) undertook a numerical investigation of the St Petersburg Game. In this paper, the central issue of the paradox is identified as that of fair (risk-neutral) pricing, which is fundamental in economics and finance and involves important concepts such as no arbitrage, discounting, and risk-neutral measures. The model for the St Petersburg Game as set out in this paper is new and analytical and resolves the so-called pricing paradox by applying a discounting procedure. In this framework, it is shown that there is in fact no infinite price paradox, and simple formulas for obtaining a finite price for the game are also provided.

Purpose Paradox theory looks at ambidexterity as a set of paradoxical yet interrelated demands. A form of response to such paradoxes is transcendence. Currently, there is limited understanding of the concept among researchers. Using concepts from the Indian philosophy of Advaita Vedanta, this paper aims to provide a deeper understanding of transcendence, highlight some of the epistemological challenges it presents and suggest ways in which the concept can be used by practitioners and ambidexterity researchers. Design/methodology/approach The paper uses concepts and theories from advaitic episteme to look at concepts of paradox and transcendence. The method of adhyaropa–apavada is introduced as a way to help individuals get a transcendental perspective of paradoxes. The application of the method is demonstrated using secondary data from published research on ambidexterity management. Findings It is postulated that transcendence is an “intuitive experience” born out of reflexive thinking. The dialectic of adhyaropa–apavada (affirmation followed by recension) is suggested as a pedagogical tool that can promote reflexive thinking. Originality/value The paper significantly adds to the theoretical understanding of paradoxes and transcendence in ambidexterity literature. The paper also makes a strong pedagogical contribution to literature by suggesting the dialectic of adhyaropa–apavada that can be used by managers to promote reflexive thinking among subordinates when faced with paradoxical situations.

Integrated information theory (IIT) was initially proposed to describe human consciousness in terms of intrinsic-causal brain network structures. Particularly, IIT 3.0 targets the system’s cause–effect structure from spatio-temporal grain and reveals the system’s irreducibility. In a previous study, we tried to apply IIT 3.0 to an actual collective behaviour in Plecoglossus altivelis. We found that IIT 3.0 exhibits qualitative discontinuity between three and four schools of fish in terms of Φ value distributions. Other measures did not show similar characteristics. In this study, we followed up on our previous findings and introduced two new factors. First, we defined the global parameter settings to determine a different kind of group integrity. Second, we set several timescales (from Δ t = 5 / 120 to Δ t = 120 / 120 s). The results showed that we succeeded in classifying fish schools according to their group sizes and the degree of group integrity around the reaction time scale of the fish, despite the small group sizes. Compared with the short time scale, the interaction heterogeneity observed in the long time scale seems to diminish. Finally, we discuss one of the longstanding paradoxes in collective behaviour, known as the heap paradox, for which two tentative answers could be provided through our IIT 3.0 analysis.

Context Research increasingly suggests that the high school diploma has lost its meaning as a symbol of life preparation. Having faced economic struggles earlier and longer than most regions of the United States, the “Rust Belt” region offers important lessons for the broader nation regarding how high schools might prepare youth for stable futures. Much like in towns in India and China, communities in the United States’ Rust Belt experience a paradox of wanting youth to find successful careers but not leave the area. Focus of Study Recent research connections between high school and college have focused on the role of signaling strategies in preparing young people for postsecondary opportunities. High-quality signals that are clear, aligned, and consistent can positively influence student outcomes. This article examines the types of policy signals that local Rust Belt communities are trying to develop to both improve postsecondary attendance of young people and retain young people in their home communities as they choose career pathways. Research Design Three cases—”Steeltown,” “Milltown,” and “Railtown”—were chosen using a comparative case study design intended for the purpose of explanation building. Data-collection strategies consisted of a combination of semistructured interviews and document collection to ascertain the visions, intentions, and implementation of the reform efforts of the selected communities. Interview protocols explored the actors, problem definitions, collaborative patterns, and implementation of initiatives. Extensive written documentation from each city that served as validity checks of the interview data. Data analysis involved a grounded theory approach of moving from raw data to conclusions using a data reduction process that involved an extensive coding strategy and case histories. Findings The strategies of the three cases suggest that three specific signaling strategies were most often used to address individual and community policy needs in these Pennsylvania communities: achievement, alignment, and awareness. The focus on academic achievement was the most consistent strategy, but weak in terms of providing a connection to postsecondary signaling. Awareness strategies consisted of teaching youth and their families about the growing fields of industry in a local area. Alignment strategies provided a way for youth to see the full pathway to potential careers. They included a focus on creating easier transitions between traditional high school, vocational-technical high schools, community colleges, technical schools, and four-year institutions. Conclusions The alignment strategies presented in these cases were not always consistent with the awareness strategies that encouraged youth to stay local in job searches. Alignment strategies therefore often prioritized youth needs over community needs. If alignment efforts are paired with building awareness of local career opportunities, however, they could help to strengthen and rebuild Rust Belt communities. A combined strategy could both increase understanding of careers and provide a pathway to get the training necessary to compete for these available jobs.

Darwinian dynamics refers to the dynamical processes underlying natural selection that drives evolution. We are interested in the evolution of strategies used by biological entities. There are two dynamical processes involved, population dynamics (relationship between population density and the agents affecting density) and strategy dynamics (relationship between strategy values and the agents affecting these values). Darwinian dynamics is a total dynamic obtained through the coupling of these two processes, the modeling of which, involves dynamical systems, optimization, stability, and game theory. Using a method called the G-function approach, we explore how an evolutionary process can take place in a set of differential equations, and we examine some interesting links between evolutionary stability and optimization as embodied in the ESS maximum principle. One of the interesting paradoxes is how a "hill-climbing" algorithm can end up at a stable local minimum and why this might have important implications in understanding speciation (the creation of new species from a homogeneous population). Finally, we will examine how these concepts are currently being applied to model the development of tumors in humans.