HMC real numbers in Countable Mathematical Analysis

TL;DR

This paper develops a theory of real numbers as rational Cauchy sequences within Countable Mathematical Analysis, emphasizing the use of hereditarily at most countable sets for foundational purposes.

Contribution

It introduces a novel framework for real numbers as rational Cauchy sequences tailored for Countable Mathematical Analysis using HMC sets.

Findings

Defines real numbers via rational Cauchy sequences with a new equality criterion.

Integrates the theory into Countable Mathematical Analysis context.

Provides a foundation compatible with hereditarily at most countable sets.

Abstract

We develop a theory of real numbers as rational Cauchy sequences, in which any two of them, $(a_n)$ and $(b_n)$, are equal iff $\lim\,(a_n-b_n)=0$. We need such reals in the Countable Mathematical Analysis ([4]) which allows to use only hereditarily at most countable (HMC) sets.