# Axiomatic Origins of Mathematical Entropy: Grading Ordered Sets

## TL;DR

This paper introduces a generalized concept of entropy based on relative divergence of grading functions on ordered sets, extending Shannon's entropy and enabling new applications in information theory.

## Contribution

It develops a new axiomatic framework for entropy derived from grading functions, generalizing Shannon's entropy and broadening its applicability.

## Key findings

- Shannon's entropy formulas are derived from the new divergence concept.
- The framework extends entropy-based methods to more general ordered sets.
- New applications of entropy in different contexts are demonstrated.

## Abstract

Shannon's entropy and other entropy-based concepts are derived from the new, more general concept of relative divergence of one "grading' function on a linearly ordered set from another such function. The definition of relative divergence is derived based on "common sense' assumptions about comparing grading functions. Shannon's entropy formulas emerge from the respective relative divergence ones, entropy based methods are extended to more general cases and some new applications.