User blog comment:Xtrasnack/The paradox function./@comment-35470197-20200115004525

The notion of "fastest growing function" is ill-defined because functions are not linearly ordered. Instead, you can use the sum of all such functions.

Also, "the describability" is too ambiguous, because first order arithmetic is unable to directly refer to functions themselves. You need to specify the way to convert strings in a certain class, e.g. function formulae in arithmetic or pairs of a variable symbol and a formulae which plays a role of a "function" on the fixed variable symbol, into functions.