User blog comment:Username5243/Hedrondude just updated his large number pages/@comment-27173506-20160202124141/@comment-1605058-20160202135720

Something pretty much along the lines of the K function has already been considered before by one of the users, and prevailing view on this was that the use of English or any other natural language (which would most likely be also implicit in the "bytes" defining the number for Bowers' new function) would give us too much vagueness or ill-definedness that it's not worthwhile to try to formalize it, because you will fail at doing that.

The iota function and this other attempt are actually mildly different, as their basic idea is not diagonalization over everything expressible, but everything that "humans have ever percieved or considered". Whether that is ill-defined is beyond the point right now, I just wanted to point out the difference.

To add to your point, I see at least two major flaws in this function (not even counting what it means that system is "well-defined" and what does it mean for something to be "described"): first, Bowers did not specify what is meant with "system of mathematics". Are only formal theories considered, or does, for example, set of points on some elliptic curve also count as a system?

Secondly, some "systems" (let's leave out for a second the above issue) might be able to define arbitrarily large numbers in bounded number of characters, provided we have infinitely many of them. For example consider the system "language of arithmetic together with a constant for every natural number". Here every number can be defined in one number.