A Logical Calculus To Intuitively And Logically Denote Number Systems

TL;DR

This paper introduces a logical calculus to represent number systems, addressing limitations of traditional notations like continued fractions and Dedekind cuts by providing a more intuitive and logical framework.

Contribution

It develops a novel logical calculus that allows for the intuitive and logical representation of number systems, overcoming limitations of existing notations.

Findings

Continued fractions and base-b expansions fail to denote real numbers with proper logic.

Dedekind cuts and Cauchy sequences lack algebraic operation compatibility and intuition.

The proposed calculus successfully constructs a logical framework for number systems.

Abstract

Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, we construct a logical calculus and deduce numbers to intuitively and logically denote number systems.