User blog comment:PsiCubed2/How to make Deedlit's Mahlo-level notation more intuitive/@comment-35470197-20180807000338/@comment-35470197-20180807041529

Ok, I write down how I computed. I guess that there is some mistake, because I am cofused in the differences of conventions as I said.

First of all, I used Lambda_0 in the last reply, but it was wrong. I foolishly mistook X(X(...(X(0,0)...,0),0),0) with the first fixed point of a -> X(a,0). (Unlike I(a,b), X(a,b) is not Scott continuous separatedly on each variable, and hence such a computation is invalid.) I noticed this failure when I was writing a proof of the wrong equality X_{Lambda_0}(0) = Lambda_0.

Nevertheless, I think that the equality X(M,0) = I(1,0,0) is unprovable under ZFC + M if we assume that ZFC + M is consistent. In order to show it, I verify that the equality is disprovable under ZFC + M + CH, which is relatively consistent with ZFC + M by Theorem 3.1 in the following: http://logika.ff.cuni.cz/radek/papers/failureCHandLargecardinals.pdf

Before that, I recall conventions. An a-weakly inaccessible cardinal in the sense of Deedlit is (1+a)-weakly inaccessible cardinal in the sense of Rathjen. The difference has no essential effect here. (But a little confusing for me.)

After that, I(1,0,0) in the sense of Deedlit is the first ordinal a satisfying I(a,0) = a in the sense of Deedlit. It is easy to show that the equality is equivalent to the one in the sense of Rathjen, because a is not finite in this case. So I(1,0,0) in the sense of Deedlit is the first ordinal a satisfying X_a(0) = a, where X_a(b) is the b-th a-weakly inaccessible cardinal in the sense of Rathjen.

Therefore in order to say X_M(0) > I(1,0,0), it suffices to show the existence of an a < M (leq X_M(0)) with X_a(0) = a. This is essentially due to the following: https://mathoverflow.net/questions/224247/mahlo-cardinal-and-hyper-k-inaccessible-cardinal

By the weak inaccessiblity of M, V_M forms a model of ZFC + I + hyper-I + hyper-hyper-I + ... . By the additional assumption of CH, M is actually a strongly Mahlo cardinal. Then by reflection principle for a strongly inaccesible cardinal, the subset C of M consisting of an a < M such that V_M is an elementary extension of V_a is club. Then by the weak Mahloness, there is a weakly inaccessible cardinal a in C.

On the other hand, the equality X_b(0) = b for a regular cardinal b is expressible by a formula F relative to V_b in the first order set theory. Namely, F is the sentence "For any ordinal d and e, there is a tower of d-weakly inaccessible cardinals of ordinal type e", which implies the unboundedness of the hierarchy of d-weakly inaccessible cardinals in V_b.

Then we have V_M |= F. Since V_M is an elementary extension of V_a, we obtain V_a |= F. It implies X_a(0) = a.

> Can it be any larger?

Right. It is Corollary 4.2 (i) in Rathjen's paper.