Universal codes of the natural numbers
Yuval Filmus (University of Toronto)
TL;DR
This paper explores the structure of universal codes for natural numbers, introduces a method to improve codes, and discusses the independence of code scales from ZFC set theory.
Contribution
It defines a partial order on codes, provides a construction method for better codes, and proves the independence of code scales from ZFC.
Findings
A natural partial order on codes is established.
A method to construct better codes than given sequences is developed.
The existence of a scale of codes is independent of ZFC.
Abstract
A code of the natural numbers is a uniquely-decodable binary code of the natural numbers with non-decreasing codeword lengths, which satisfies Kraft's inequality tightly. We define a natural partial order on the set of codes, and show how to construct effectively a code better than a given sequence of codes, in a certain precise sense. As an application, we prove that the existence of a scale of codes (a well-ordered set of codes which contains a code better than any given code) is independent of ZFC.
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