相关论文: The Meaning of Infinity

Without a doubt many problems in physics arise as a consequence of our philosophical conception of the world. In this contribution however we endeavor to alleviate this scenario by putting forward a philosophical approach under which some…

Although Bolzano's concept of the continuum has gradually evolved, the basis remained the same: the continuum as an infinite class of points arranged in such a way that the so-called \emph{Bolzano completeness} holds. Bolzano realized over…

We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of…

A proof of G\"odel's incompleteness theorem is given. With this new proof a transfinite extension of G\"odel's theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a…

It is well known that in Zermelo-Fraenkel (ZF) set theory any finite set is decidable. In this paper we discuss an extension of ZF where this result is no longer valid. Such an extension is quasi-set theory and it has its origin on problems…

We introduce a model of simple type theory with potential infinite carrier sets. The functions in this model are automatically continuous, as defined in this paper. This notion of continuity does not rely on topological concepts, including…

We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.

We introduce the notion of limiting theories, giving examples and providing a sufficient condition under which the first order theory of a structure is the limit of the first order theories of a collection of substructures. We also give a…

As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable.

The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The…

We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational…

In this article some difficulties are deduced from the set of natural numbers. By using the method of transfinite recursion we define an iterative process which is designed to deduct all the non-greatest elements of the set of natural…

Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…

We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one…

We describe a theory of finite sets, and investigate the analogue of Dedekind's theory of natural number systems (simply infinite systems) in this theory. Unlike the infinitary case, in our theory, natural number systems come in differing…

This paper introduces the concept of a generic finite set, and points out that a consistent and significant interpretation of the grossone notation of Yarolslav D. Sergeyev is that of a generic natural number. This means that the grossone…

Based upon the axiom of choice it is proved that the cardinality of the rational numbers is not less than the cardinality of the irrational numbers. This contradicts a main result of transfinite set theory and shows that the axiom of choice…

Some physical consequences of the negation of the continuum hypothesis are considered. It is shown that quantum and classical mechanics are component parts of the multicomponent description of the set of variable infinite cardinality.…

Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.…

We propose an interpretation of physics named potentiality realism. This view, which can be applied to classical as well as to quantum physics, regards potentialities (i.e. intrinsic, objective propensities for individual events to obtain)…