Related papers: The grounding for Continuum
The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural…
Sequences diverge either because they head off to infinity or because they oscillate. Part 1 constructs a non-Archimedean framework of infinite numbers that is large enough to contain asymptotic limit points for non-oscillating sequences…
In this work we study the space complexity of computable real numbers represented by fast convergent Cauchy sequences. We show the existence of families of trascendental numbers which are logspace computable, as opposed to algebraic…
A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…
We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete…
Given a Dedekind incomplete ordered field, a pair of convergent nets of gaps which are respectively increasing or decreasing to the same point is used to obtain a further equivalent criterion for Dedekind completeness of ordered fields:…
In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that…
The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincar\'e, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan…
We present axioms for the real numbers by omitting the field axioms and then derive the field properties of the real numbers. We prove all our theorems constructively.
This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…
The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a…
This paper examines the completion of an w-ordered sequence of recursive definitions which on the one hand defines an increasing sequence of nested set and on the other redefines successively a numeric variable as the cardinal of the…
Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first…
In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy's ideas of function, continuity, limit and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the…
The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can…
The Endpoint Theorem links the existence of a sequence (curve), without accumulation points, in a manifold to the existence of an open embedding of that manifold so that the image of the given sequence (curve) has a unique endpoint. It…
We study a metric-like structure on categories, showing that the concept of the limit of a sequence in a metric space and the concept of the colimit of a sequence in a category have a common generalization. The main concept is a norm on a…
The hypothesis concerning the off-site continuum existence is investigated from the point of view of the mathematical theory of sets. The principles and methods of the mathematical description of the physical objects from different off-site…
Brief development of the idea of the very important notion of continuity is given. Continuity is often confused with contiguity, "drawing the graph in one go," "no gaps," etc. The author argues in support of using correct notions of…
In a recent paper as an alternative to models based on the notion of ideal mathematical point, characterized by a property of separatedness, we considered a viewpoint based on the notion of continuous change, making use of elements of a…