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Countable real analysis

Martin Klazar

TL;DR

The work develops a countable, $ ext{HMC}$-based framework for univariate real analysis, using uniformly continuous countable functions and uniform derivatives to replicate core tools while avoiding uncountable objects. It reworks essential results—suprema, series, integrals, the Fundamental Theorem of Analysis, and Euler’s identity—in a countable setting, culminating in a Hilbert–Hermite-style transcendence proof for $e$ that relies solely on countable constructs. Additionally, the paper constructs a counterexample UC function demonstrating that the uniform-derivative hypothesis cannot be omitted in the global-extremum arguments, underscoring the boundaries of the framework. The results suggest that many central theorems of real analysis admit countable analogues, with implications for analytic number theory and foundational perspectives on the necessity of uncountable sets.

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 $b\in[0,1]\cap\mathbb{Q}$.

Countable real analysis

TL;DR

The work develops a countable, -based framework for univariate real analysis, using uniformly continuous countable functions and uniform derivatives to replicate core tools while avoiding uncountable objects. It reworks essential results—suprema, series, integrals, the Fundamental Theorem of Analysis, and Euler’s identity—in a countable setting, culminating in a Hilbert–Hermite-style transcendence proof for that relies solely on countable constructs. Additionally, the paper constructs a counterexample UC function demonstrating that the uniform-derivative hypothesis cannot be omitted in the global-extremum arguments, underscoring the boundaries of the framework. The results suggest that many central theorems of real analysis admit countable analogues, with implications for analytic number theory and foundational perspectives on the necessity of uncountable sets.

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 . We also construct a uniformly continuous function such that on and for every .
Paper Structure (15 sections, 68 theorems, 143 equations)

This paper contains 15 sections, 68 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Countable real analysis
  3. Real numbers as cuts
  4. Real sequences
  5. Real series
  6. Countable functions
  7. Derivatives of countable functions
  8. The countable exponential function
  9. Integrals of countable functions
  10. Improper integrals of countable functions
  11. The FTA for countable functions
  12. Countable Euler's identity
  13. Transcendence of $\mathrm{e}$ via countable functions
  14. A counter-intuitive countable real function
  15. Concluding remarks

Key Result

Lemma 2.4

If $(a_n)\subset\mathbb{Q}$ and $b$ are fractions such that $a_1\ge a_2\ge\dots\ge a_n\ge\dots\ge b$, then $(a_n)$ is a Cauchy sequence. The same holds if all inequalities $\ge$ are reversed.

Theorems & Definitions (88)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 78 more