User blog comment:Wythagoras/Friendship function/@comment-1605058-20140717170937/@comment-1605058-20140718160833

I have found this paper in which, in introduction, it's said that Schutte and Simpson have proved Higman's lemma in system of notations up to e_0. This is a solid reason for us to think that Higman's lemma certainly cannot bring us beyond e_0, so, as initial well-ordering of natural numbers is far below e_0, double application of Higman's lemma still leaves us at level below e_0. The question of exact strength of Friend function is not resolved, but that's certainly something.