User blog comment:Alemagno12/How strong is this function? (2)/@comment-25337554-20171111093003/@comment-25601061-20171112022250

Here's how an ordinal pair system for f(2) exceeding ω might start: But actually, I realized that f(2) and beyond don't actually exist, since there can be infinitely many ordinal pairs that can be labelled with ω.
 * 0 = 0
 * 1 = (0,0)
 * 2 = (0,(0,0))
 * 3 = (0,(0,(0,0)))
 * n+1 = (0,n)
 * ω = ((0,0),0)
 * ω+1 = (0,((0,0),0))
 * ω+2 = (0,(0,((0,0),0)))
 * ω2 = ((0,0),((0,0),0))
 * ω3 = ((0,0),((0,0),((0,0),0)))
 * ω2 = ((0,(0,0)),0)