Universal codes of the natural numbers

A code of the natural numbers is a uniquely-decodable binary code of thenatural numbers with non-decreasing codeword lengths, which satisfies Kraft'sinequality tightly. We define a natural partial order on the set of codes, andshow 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 ascale of codes (a well-ordered set of codes which contains a code better thanany given code) is independent of ZFC.Comment: 11 page