On Hegel’s View of Dialectical Infinity

It is well known that the problem of finity and infinity is the basic problem of mathematics, and it is also the basic problem of Philosophy. From the perspective of philosophy and mathematics, this paper comprehensively reviews and analyzes Hegel’s view of dialectical infinity, introduces Engels’discussion on infinity, deeply analyzes the characteristics of the thought of actual infinity, and points out: Hegel’s thought of real infinity is completely different from the thought of actual infinity, the Being of infinity (objective infinity) is not equal to the completed infinity (subjective infinity), the mathematical limit is a real infinity, and real infinity is the inner law of infinite things and truth; the view of actual infinity views the objective material world from the viewpoint of static rather than motion, denying the contradiction between finity and infinity, so it is actually a downright idealist. In this paper, the author puts forward the Infinite Exchange Paradox, which strongly questions the idea of actual infinity in Hilbert Hotel Problem, and points out the internal irreconcilable contradiction in the idea of actual infinity. At the same time, we made a detailed comparison of Hegel’s view of infinity and the view of mathematical infinity, and on this basis, the author gives a complete definition of the view of dialectical infinity: abandoning the wrong aspects of the potential infinity and actual infinity, and actively absorbing correct aspects of both, that is, not only to recognize the existence and knowability of infinite objectivity, but also to admit the imcompletion of infinite process. The reexcavation of Hegel’s view of dialectical infinity and the criticism of the actual infinity thought aim to find possible philosophical solutions for Russell’s Paradox and the problem of Continuum Hypothesis

A Conversation on Divine Infinity and Cantorian Set Theory

This essay is written as a drama that opens with Aristotle, St. Augustine of Hippo, St. Thomas Aquinas, and Nicholas of Cusa debating the nature and reality of infinity, introducing historical concepts such as potential, actual, and divine infinity. Georg Cantor, founder of set theory, then gives a lecture on set theory and transfinite numbers. The lecture concludes with a discussion of the theological motivations and implications of set theory and Cantor\'s absolute infinity. The paradoxes inherent in analyzing absolute infinity seem to provide a useful analogy for understanding God\'s unknowable nature and the divine relation to creation

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

Energy and Information Near Black Hole Horizons

The central challenge in trying to resolve the firewall paradox is to identify excitations in the near-horizon zone of a black hole that can carry information without injuring a freely falling observer. By analyzing the problem from the point of view of a freely falling observer, I arrive at a simple proposal for the degrees of freedom that carry information out of the black hole. An infalling observer experiences the information-carrying modes as ingoing, negative energy excitations of the quantum fields. In these states, freely falling observers who fall in from infinity do not encounter a firewall, but freely falling observers who begin their free fall from a location close to the horizon are "frozen" by a flux of negative energy. When the black hole is "mined," the number of information-carrying modes increases, increasing the negative energy flux in the infalling frame without violating the equivalence principle. Finally, I point out a loophole in recent arguments that an infalling observer must detect a violation of unitarity, effective field theory, or free infall.Comment: 25 pages, 3 figures. v2: minor clarifications, references added; published versio

Zeno's Paradoxes. A Cardinal Problem 1. On Zenonian Plurality

In this paper the claim that Zeno's paradoxes have been solved is contested. Although no one has ever touched Zeno without refuting him (Whitehead), it will be our aim to show that, whatever it was that was refuted, it was certainly not Zeno. The paper is organised in two parts. In the first part we will demonstrate that upon direct analysis of the Greek sources, an underlying structure common to both the Paradoxes of Plurality and the Paradoxes of Motion can be exposed. This structure bears on a correct - Zenonian - interpretation of the concept of division through and through. The key feature, generally overlooked but essential to a correct understanding of all his arguments, is that they do not presuppose time. Division takes place simultaneously. This holds true for both PP and PM. In the second part a mathematical representation will be set up that catches this common structure, hence the essence of all Zeno's arguments, however without refuting them. Its central tenet is an aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some number theoretic and geometric implications will be shortly discussed. Furthermore, it will be shown how the Received View on the motion-arguments can easely be derived by the introduction of time as a (non-Zenonian) premiss, thus causing their collapse into arguments which can be approached and refuted by Aristotle's limit-like concept of the potentially infinite, which remained - though in different disguises - at the core of the refutational strategies that have been in use up to the present. Finally, an interesting link to Newtonian mechanics via Cremona geometry can be established.Comment: 41 pages, 7 figure

Decoupling a Fermion Whose Mass Comes from a Yukawa Coupling: Nonperturbative Considerations

Perturbative analyses seem to suggest that fermions whose mass comes solely from a Yukawa coupling to a scalar field can be made arbitrarily heavy, while the scalar remains light. The effects of the fermion can be summarized by a local effective Lagrangian for the light degrees of freedom. Using weak coupling and large N techniques, we present a variety of models in which this conclusion is shown to be false when nonperturbative variations of the scalar field are considered. The heavy fermions contribute nonlocal terms to the effective action for light degrees of freedom. This resolves paradoxes about anomalous and nonanomalous symmetry violation in these models. Application of these results to lattice gauge theory imply that attempts to decouple lattice fermion doubles by the method of Swift and Smit cannot succeed, a result already suggested by lattice calculations.Comment: 31 page