User blog comment:! cook the lettuce/Question about ordinal tetration/@comment-30004975-20200221210307/@comment-35470197-20200221234000

By the way, if we allow w^^(a+1) ≠ (w^^a)^w, what is the agreed-upon restriction on tetration? For example, if we define w^^a in the following way, is it suitable?
 * 1) If a=0, then w^^a = 1.
 * 2) If 0 w^^(w+1) = ψ_0(ψ_1(ψ_1(ψ_0(0)))) = φ_w(0). The issue is that this tetration together with +, ×, and power does not cover all ordinals below its limit. Do other formulations satisfy the covering property?