User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-30004975-20180104233320/@comment-30754445-20180105085740

It's a function that returns ordinals less than M.

Looking at the definitions, it seems - quite oddly - that:

ψM(0) = ψI(0) (the first omega fixed point).

This is really strange, but its what the definitions say. ψM(0) is the smallest ordinal which is closed under addition, veblen functions and a→Ωa.

Going to ψM(1), we get:

"the smallest ordinal which is closed under addition, veblen functions, a→Ωa, a→χ(0,a) and ψa(0)"

This seems to be the first fixed point of a→χ(0,a), which (by definition) is the first fixed point of a→Ia. So:

ψM(1) = lim (I, II, II I, ...)

And in general, for any integer n>0:

ψM(n) = the first fixed point of a→χ(n-1,a)

Which can easily be extended to any previously constructed countable ordinal (the rule above for successor ordinals, the usual limit rule for limit ordinals). The limit of all these would be ψM(Ω).

(I've deduced all this by going slowly and carefully throughout the definitions Deedlit has written, without any prior knowledge. I could have made some mistake, and I would appreciate it if Deedlit or somebody similarly well-versed in this notation would either correct or confirm my analysis)