User blog comment:Denis Maksudov/M-notation (new version)/@comment-10262436-20160620212735/@comment-24920136-20160621162455

My problem is that with the functional power being an ordinal, intuitively i only know that the functional power is the amount of nestings of the function, and amounts relate to cardinal numbers.

ie. f_1(w) for example = (f_0)^w(w), how many nestings do you make of f_0?. w is a limit ordinal but here it appears in two places, so which sequence do you follow?

(f_0)^1(w), (f_0)^2(w), (f_0)^3(w)...

or

(f_0)^w(1), (f_0)^w(2), (f_0)^w(3)...

also your f_3 is kinda dubious, you go from >  2^^n to w^^w, shouldnt it be 2^^w at best?