User blog comment:PsiCubed2/Does anybody here know where I can have serious googology questions answered?/@comment-35470197-20181114222222/@comment-35470197-20181116020343

> (3) What are your proven lower and upper bounds of the ordinal representing the power of the notation you've shown to terminate?

First of all, I did not verify that the set of standard forms of PSS naturally forms an ordinal notation. I just proved the termination by sending them to the set OT of Buchholz's ordinal terms by an explicitly defined recursive map o.

In order to obtain an ordinal notation system, I need a recursive relation which is provably well-founded total order. For example, let us define a recursive relation on FSS as the pull-back of that of OT by o. After verifying that the expansion rule of PSS strictly reduces the corresponding ordinal terms, I obtain the well-foundedness of the pull-back.

On the other hand, the totality is not followed from this argument. Say, I need to verify the injectivity of the correspondence to ordinal terms.

Should I prove it in addition? Then I would get another claim. Is there a linear hierarchy on functions (f_a)_a such that the function F_M corresponding to a PSS M actually has a growth rate f_{o(M)}? Otherwise, the "correspondence" just gives the termination, and does not gives the "strength" of PSS itself.

Of course, I can change the definition of F_M from the original PSS so that it actually corresponds to psi_0(psi_omega(0)) with respect to FGH and the standard FS. But then, there is no correspondence to existing versions. It is not happy for me.

Well, for example, one reasonable way to defined (f_a)_a is to use the FGH-like recursion using n+1 or n^2:
 * 1) f_0(n) = n^2
 * 2) f_{a+1}(n) = f_a^n(n^2)
 * 3) f_{lim a_i}(n) = f_{a_n}(n^2)

Then you need an explicit FS. Buchholz's one is not suitable here, because it does not precisely correspond to the expansion rule of PSS. It is useless to estimate the precise growth rate of the functions in PSS. I usually like rough apprximation, but I emphasise that the approximation here would be too rough to justify even though it gives a correct estimation as a result.

So the most reasonable choice of an FS here is the one constructed from o and the expansion rule of PSS. Then, is (f_a)_a actually a linear hierarchy? As googologists usually expect, does the diagonalisation at the limit ordinal actually give a larger function? It is easy to see that diagonalisation is not necessarily give a larger function, because the relation given as the growth rate is not defined by domination, but eventual domination.

I need to emphasise that I am 99% sure that the diagonalisation appropriately works here, but in order to show this, I need to wright additional articles on generality of FS and FGH. This is just an example of what to do toward this project. What I would like to say is that I need to write down proofs for such very easy but tiresome arguments.

Well, then I doubt that someone would read my proof. Further, googologists here would say "We are 99% sure that PSS gives an ordinal notation system" without such arguments or even being unaware of the existence of problems.

After all, I would need to spend another couple of months if I try these additional works. Instead, I would like to enjoy other topics on googology. (For example, don't you think that my recent elementary function with growth rate larger than \(\varepsilon_0\) is good? I like to create more functions.)