User blog comment:Flitri/An ordinal Collapsing up to the Least weakly Mahlo Cardinal/@comment-35470197-20190409053305/@comment-35470197-20190411072632

Good. To be more precise, in order to define an ordinal notation system, what you need to do is the following: In our context, you want to define an ordinal notation system associated to an OCF. In this case, you additionally need to define the following data: I note that if you have x < y <--> o(x) < o(y), the second condition of < (i.e. the existence of a least element in any non-empty subset) is automatic, and hence this condition is very strong.
 * 1) Define a set S of all alphabets which you use in your notation (e.g. 0,ω, Ω, specific brackets, ψ, M, and so on).
 * 2) Define a recursive subset T of the set S* of formal strings consisting of alphabets in S. In other word, define a subset T and a recursive function f : S* -> {0,1} such that f(x) = 0 if and only if x is an element of T for any element x of S*. (Namely, f is an "algorithm" to determine whether a formal string is an element of S or not.) The existence of f is often trivial for a given T. In such a case, you can skip the definition of f.
 * 3) Define a primitive recursive function g : T^2 \to {0,1} satisfying the following:
 * 4) The binary relation a < b <--> g(a,b) = 1 on T is a total order.
 * 5) Any non-empty subset of T admits a least element with respect to <. (Yeah, this is what you mentioned.)
 * 1) A subset P of S cap T consisting of alphabets (e.g. 0, ω, M, and so on) which you want to interprete into ordinals, and a map o_1 : P -> Ord realising the interpretation. (If you directly use ordinals as alphabets, then o' can be just the identity.)
 * 2) A subset F of S consisting of alphabets (e.g. +, ^, ψ, and so on) which you want to interprete into ordinals, and a map o_2 from F to the class of (possibly multivariable) ordinal functions.
 * 3) A map o : T -> Ord satisfying the following:
 * 4) For any x in P, o(x) coincides with o_1(x).
 * 5) For any x,y,... in T and any s in F, o(s(x,y,...)) coincides with o_2(s)(o(x),o(y),...)
 * 6) For any x,y in T, x < y <--> o(x) < o(y)

The main difficulty to construct an ordinal notation is to define T and g and to verify x < y <--> o(x) < o(y). On the other hand, many googologists often use the terminology "ordinal notation" without the knowledge of the precise definition, and state "I constructed an ordinal notation" by referring to none of T, g, and the reasoning. So you might have unfortunately caught a wrong definition written by them. If you understand the precise definition of the notion of an ordinal notation, then I hope that you will tell other googologists the definition when they study an ordinal notation, in order to prevent spreading this heavy comfounding which has already been widely spread in googology communities. (OCFs without such interpretations into ordinal notations do not yield a computable large number through FGH, and hence this is a very serious problem in googology.)