User blog comment:Syst3ms/A sketch for an — actually — formal definition of UNOCF/@comment-35470197-20180803231131/@comment-30754445-20180807154249

@Syst3ms

"If you have an alternative definition I'll take it, but the C function is meant to go much, much further than just Mahlos."

Yeah, with a lot of help from the main ψ functions.

C, on it's own, doesn't get there. More specifically, this statement by you:

"C(1;0) = M, it diagonalizes over all C(a,b,...,n)"

Is not true.

If you diagonalize over C(a,b,...,n) you merely get ψ(M^M^ω). This cannot be expressed with the C function in any way. Niether can ψ(M^M^(ω+1)), ψ(M^M^Ω), ψ(M^M^I), ψ(M^M^M),, U(M^M^M^M) and so on.

The next cardinal expressible with the C function is C(1;0)=M, which is a huge skip. This is true in both UNOCF and the standard OCF, because at this level the two notations are still of the same strength.

Had we needed the C function to bridge this gap, we would be stuck. Fortunately, we don't. As the UNOCF experts here already pointed out, the C-function is just an added convinience, which is why it shouldn't appear in the formal definitions at all. For example, you don't need to construct the expression C(1,4,2), when you can construct the equivalent expression ψ(MM+3×3). The former may look prettier, but this isn't relevant to the actual collapsing process. The proper place to define the C function is later, in an addenum that lists various shorthands for certain expressions.