User blog comment:Billicusp/So I don't know if anybody noticed.../@comment-27173506-20160318160910/@comment-27173506-20160319082416

That's exactly the reason why  X(M M ) is so huge: the  X function  always collapses to inaccessible ordinals. This is why for example, X(M) is equal to \(I(1,0)\) (the smallest ordinal greater than any recursive extension of \(\alpha \maps to I_\alpha\) and not to \(\psi_{I(1,0)}(0)\) (the first fixed point of \(\alpha \maps to I_\alpha\)). If we do use the second option, than the result of collapsing any ordinal smaller than M_2 in the X function will be smaller than I(1,0), since we are just collapsing using a recursive definition, and I(1,0) is greater than any recursive definition.

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