User blog comment:Deedlit11/Keith Ramsay on the largest definable number/@comment-5150073-20130322164719/@comment-5529393-20130323015810

Here is a question on Math Overflow that seems pertinent to this discussion:

http://mathoverflow.net/questions/3057/is-there-a-topology-on-growth-rates-of-functions

In particular note the answer by Joel David Hamkins. To summarize:

By "function" we mean a function on the natural numbers.

a function f dominates a function g if f(n) > g(n) for all n.

a function f eventually dominates a function g if f(n) > g(n) for all n > N for some N.

The dominating number is the smallest cardinality of a set S such that, for any function f, there is a function g in S such that g dominates f.

The bounding number is the smallest cardinality of the set S such that there is no function f that eventually dominates all g in S.

The bounding number must be at least aleph_1, since for any countable collection of functions we can enumerate them as f_i and take g(n) = sup_{k < n} {f_k} + 1, and g will eventually dominate all the f_i. Any dominating collection of sets is a bounding collection of sets, so the bounding number of sets must be less than or equal to the dominating number, and the dominating number must be less than or equal to the total number of functions, which is the continuum.

It turns out that this is basically all that can be said about the bounding and dominating numbers in ZFC. That is, it is consistent with ZFC that the bounding number is as low as aleph_1, and it is consistent with ZFC that the bounding number is as high as the continuum, no matter what that is. And it is an old result of Solovay that the continuum can be arbitrarily high (to be precise, the continuum can be any cardinal with uncountable cofinality - so for example, it can be any cardinal of the form aleph_{alpha + 1} for any alpha.).

How does this relate the fast-growing hierarchy? Well, if we have any ordinal hierarchy of functions that only goes up to an ordinal less than the bounding number, than there will be some function that eventually dominates all functions in the hierarchy. So if the bounding number is greater than aleph_1, than the fast-growing hierarchy, no matter how we define it for ordinals less than omega_1, can be continued past omega_1, all the way up to the bounding number, whatever that is.

Now if the Continuum Hypothesis is true, then the dominating number will be aleph_1, so there will exist an ordinal hierarchy such that for any function f there is a function g in the hierarchy that dominates it. However, it is possible that a particular ordinal hierarchy will not go "all the way up" by omega_1, so it would still be possible to go past omega_1. However, you could not reach omega_2, since you would run out of functions!

Of course, the problem remains how to _define_ a particular ordinal hierarchy up to omega_1. I don't think this is possible without using the axiom of choice, since we cannot even name all the countable ordinals. But it's intriguing that such ordinal hierarchies will exist nonetheless.