The Foundations of Analysis

TL;DR

This paper offers a comprehensive categorical approach to foundational real analysis, starting from natural numbers and ratios to define positive real numbers, integrating classical Euclidean theory.

Contribution

It introduces a novel categorical framework for the real number system, connecting natural numbers, ratios, and real numbers in a unified manner.

Findings

Categorical definition of natural numbers

Categorical construction of positive real numbers

Integration of Euclidean ratio theory into modern foundations

Abstract

This is a detailed and self-contained introduction to the real number system from a categorical perspective. We begin with the categorical definition of the natural numbers, review the Eudoxus theory of ratios as presented in Book V of Euclid, and then use these classical results to define the positive real numbers categorically.