User blog comment:PsiCubed2/How to make Deedlit's Mahlo-level notation more intuitive/@comment-35470197-20180807000338/@comment-30754445-20180807124532

"Therefore in order to say X_M(0) > I(1,0,0), it suffices to show the existence of an a < M = X_M(0) with X_a(0) = a."

What?

Where on earth did you get the idea that X(M,0) = M? This is not true. In fact, M is bigger than any ordinal of the form X(α,β), regardless of how big α is (and β is always less than M).

Now, obviously, you are right that there are cardinals below M which are (1,0)-inaccessible. Anybody with even a cursory understanding of Deedlit's notation could tell you that, without needing all the heavy set-theoretical stuff you've mentioned. That's the whole point of using a Mahlo: It is big enough to form the basis of a collapsing function that generates the various kinds of inaccessibles.

"Maybe you did the same mistake as what I did: X(X(...(X(0,0)...,0),0),0) is not the first fixed point of a -> X(a,0) because X is not Scott continuous. "

No. The only mistake I've done was to use "Λ₀" as shorthand for Deedlit's I(1,0,0). I now understand from you that the symbol means something else (and that you misused it in your previous post). This doesn't change anything about my arguments, though. Just replace every instance of Λ₀ with I(1,0,0) - which is what I meant to write in the first place.

And yes, obviously, X(X(...(X(0,0)...,0),0),0) cannot written as X(a,b), because the former has cofinality ω and the later has cofinality X(a,b). In Deedlit's definitions, X(X(...(X(0,0)...,0),0),0) would be written as ψM(something) (I'm guessing ψM(M) but don't bet on that guess being right).