Universal codes of the natural numbers

Yuval Filmus, S. Barry Cooper
2013 Logical Methods in Computer Science  
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.
doi:10.2168/lmcs-9(3:7)2013 fatcat:mjguhq6uefborefwp5btmlnyvy