Countable real analysis

TL;DR

This paper develops a form of real analysis restricted to hereditarily countable sets, demonstrating foundational results like the transcendence of e and constructing specific continuous functions within this framework.

Contribution

It introduces a countable real analysis framework using HMC sets, reworking classical analysis and proving key results like e's transcendence within this restricted setting.

Findings

Reformulation of real analysis using HMC sets

Proof of transcendence of e in countable real analysis

Construction of a specific uniformly continuous function with unique properties

Abstract

HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert's proof of transcendence of the number $\mathrm{e}$. We also construct a uniformly continuous function $f:[0,1]\cap\mathbb{Q}\to\mathbb{R}$ such that $f'=1$ on $[0,1]\cap\mathbb{Q}$ and $\lim_{\substack{a\to1/\sqrt{2}\\a\in\mathbb{Q}}}f(a)=\frac{1}{\sqrt{2}}>f(b)$ for every $b\in[0,1]\cap\mathbb{Q}$.