User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-25418284-20130318051254/@comment-5529393-20130318053553

Just pick a standard enumeration for Turing machines. Here is one:  we first enumerate the 1-state Turing machines, then the 2-state TMs, then the 3-state TMs, etc.  To enumerate the n-state Turing machines, notice that a Turing machine can be coded up as a string of symbols, for example here is a four state Turing machine:

B1L D0R

C0R C1L

A0L B1L

H1R A1R

So simply order the n-state Turing machines lexicographically. This allows us to talk about the mth Turing machine.