User blog comment:Syst3ms/YAUD/@comment-30754445-20180810132619

You do realize that the way you define ψ is basically an OCF, right?

Yes, I realize the difference you're going for. I realize that you don't treat the I's and the M's as actual ordinals. But you manipulate them just like you would manipulate ordinals. You require them to have the exactly the properties(*) of the actual cardinals I and M (at least when it comes to collapsing them). Your rules explicitly talk about things like cofinality and fundamental sequences, which are clearly statements in "ordinalese".

So how is this any different than an actual OCF that uses the actual cardinals I and M?

Your ψ takes ordinals (or at least - something that looks suspiciously like ordinals and has properties of ordinals and behaves like ordinals) as input, and outputs countable ordinals. That's what OCFs do.

(*) Keep in mind that collapse functions usually use only a tiny fraction of the properties attached to the ordinals they collapse. An OCF that uses I and M doesn't really care about them being "weakly inacessible"/"weakly Mahlo". It just cares about them being large enough for the collapse to work properly.