Next Article in Journal
A Quantum Adiabatic Algorithm for Multiobjective Combinatorial Optimization
Previous Article in Journal
On the Linear Quadratic Optimal Control for Systems Described by Singularly Perturbed Itô Differential Equations with Two Fast Time Scales
Article Menu

Export Article

Open AccessArticle
Axioms 2019, 8(1), 31; https://doi.org/10.3390/axioms8010031

A New Set Theory for Analysis

Centro de Investigación en Matemáticas (CIMAT), Guanajuato 36023, Mexico
Received: 8 January 2019 / Revised: 20 February 2019 / Accepted: 1 March 2019 / Published: 6 March 2019
  |  
PDF [15560 KB, uploaded 6 March 2019]
  |     |  

Abstract

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets. View Full-Text
Keywords: general topology; axiomatic set theory; real analysis; continuum; graph theory; benacerraf’s identification problem; mathematical structuralism general topology; axiomatic set theory; real analysis; continuum; graph theory; benacerraf’s identification problem; mathematical structuralism
Figures

Graphical abstract

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Ramírez, J.P. A New Set Theory for Analysis. Axioms 2019, 8, 31.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Axioms EISSN 2075-1680 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top