User blog comment:Ikosarakt1/Uncountable function cannot exists/@comment-1605058-20130707061148/@comment-1605058-20130707113141

We can show that f_alpha(1)=2. If we take f(n)=3, all ordinals have complexity at most 1. As far as I know existence of function you want to disprove is independent of ZFC, so if you really find proof, ZFC may turn out to be incomplete.