Generalizations of the one-dimensional version of the Kruskal-Friedman theorems

The paper [Schutte + Simpson] deals with the following one-dimensional case of Friedman's extension (see in [Simpson 1]) of Kruskal's theorem ([Kruskal]). Given a natural number n , let S n +1 be the set of all finite sequences of natural numbers n + 1. If s 1 = ( a 0 ,…, a k ) ∈ S n +1 and s 2 = ( b 0 ,…, b m ) ∈ S n + 1 , then a strictly monotone function f : {0,…, k } → {0,…, m } is called an embedding of s 1 into s 2 if the following two assertions are satisfied: 1) a i , = b f(i) , for all i k ; 2) if f ( i ) j f ( i + 1) then b j > b f ( i +1) , for all i k , j m . Then for every infinite sequence s 1 , s 2 ,…, s k ,… of elements of S n + 1 there exist indices i j and an embedding of s i into S j . That is, S n +1 forms a well-quasi-ordering ( wqo ) with respect to embeddability. For each n , this statement W( S n +1 ) is provable in the standard second order conservative extension of Peano arithmetic. On the other hand, the proof-theoretic strength of the statements W( S n +1 ) grows so fast that this formal theory cannot prove the limit statement ∀ n W( S n +1 ). The appropriate first order -versions of these combinatory statements preserve their proof-theoretic strength, so that actually one can speak in terms of provability in Peano arithmetic. These are the main conclusions from [Schutte + Simpson]. We wish to extend this into the transfinite. That is, we take an arbitrary countable ordinal τ > 0 instead of n + 1 and try to obtain an analogous “strong” combinatory statement about finite sequences of ordinals τ .