User blog comment:Alemagno12/Making an OCF, attempt 2/@comment-5529393-20170803204913/@comment-5529393-20170803221526

Your comparison rule only works if each expression has the diagonalizer of a unique function. However, this cannot be the case, since according to your definition of diagonalizer, if X is a diagonalizer of F, it must be the diagonalizer of any function H that is faster growing than F.

I do not know what you mean by "G does not contain a rule that makes X not a diagonalizer of F". Functions are just mappings from their domains to their codomains, they don't have to have any rules. Also, I think this definition still has the problem that it will never apply, since again there is nothing that will force G(X) to be sup{G(0),G(G(0)),...}.

There is also the problem that your definition of "eventually outgrows" uses the term "diagonalizer", and your definition of "diagonalizer" uses the term "eventually outgrows", so we have circularity.