User blog:Lepton Adapter/Meta: What a growth rate is

In some articles (and there's most likely more than few of them) I've noted that the community speaks about a given function by saying "this function has a growth rate of \(\gamma\)", where \(\gamma\) is some ordinal.

In other articles however the community is more specific and says "this function has a growth rate of \(\gamma\) in the Fast-growing hierarchy", or, "the function is about \(f_{\gamma}(n)\)". This seems to me a more accurate description of a function's growth rate.

If we carry over the practice of just listing an ordinal to describe a function's growth rate for finite subscripts of the fast-growing hierarchy it leads to some awkward phrasing. We'd have to say that the function \(j(n)=n+1\) has a growth rate of 0 which makes it sound like the function has no increase when it does!

So it is important to denote a specific hierarchy when bringing up ordinals in growth-rates, otherwise saying a function "has a growth rate of \(\gamma\)" does not have that much meaning.

There is another issue though. Is anyone absolutely sure that the people who analyzed these functions used the same fundamental sequences as the ones we're assuming them to be based on (i.e. the Wainer hierarchy)?

Couldn't \(f_{\omega}(n) > BB(n)\) if our system of fundamental sequences said that \(\omega[n]=BB(n)\)?

This is the odd thing. Many articles don't say if the Wainer hierarchy is being used, let alone with what ordinal hierarchy in general, so someone who has assumed the Wainer hierarchy is being used may end up getting the wrong idea about the true strength of a notation. Maybe this will hardly ever happen, but the times when it will should motivate everyone to look carefully at what is being taught here and to really look into matters.

That is all.