Axioms for the Real Numbers: A Constructive Approach

TL;DR

This paper develops a constructive set of axioms for real numbers, deriving their field properties without assuming the traditional field axioms, and proves all theorems constructively.

Contribution

It introduces a novel constructive axiomatic framework for real numbers that omits the standard field axioms and derives their properties from these axioms.

Findings

Constructive proofs of real number properties.

Axioms that do not include field axioms.

Derivation of field properties from alternative axioms.

Abstract

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.