User blog comment:Deedlit11/Keith Ramsay on the largest definable number/@comment-5150073-20130322164719

Whatsoever, you can't get the function past w_1, the first uncountable ordinal. This is because, as Sbiis Saibian noted, every countable sequence of ordinals limits to a countable ordinal. This suggests me that the function f_w_1(n) cannot exist for the same reason that the number w cannot exist, so w_1 measures the strength of concept of the function! What next? Of course, hierarchies. I don't know how we can measure the strength of an hierarchy, but if that method exists, hierarchies would have the strength w_2. Then we can notice the sequence: numbers, functions, hierarchies, hierarchies of hierarchies. That is, we can create hierarchy of hierarchies. For example, the hierarchies in ascending order are slow-growing, Hardy, fast-growing, Bird, ...

Like functions can measure relationship between numbers and hierarchies can do the same with functions, we can, of course, create the super-hierarchy that diagonalizes over all these hyper-functions, and therefore, has limit ordinal w_w.

Some of that maybe not by subject, but that really gives me sense.