The uniform Kruskal theorem: between finite combinatorics and strong set existence

Abstract

The uniform Kruskal theorem extends the original result for trees to general recursive data types. As shown by Freund, Rathjen and Weiermann (Freund, Rathjen, Weiermann 2022 Adv. Math. 400 , 108265 ( doi:10.1016/j.aim.2022.108265 )), it is equivalent to Π 1 1 -comprehension, over R C A 0 with the ascending descending sequence principle ( A D S ). This result provides a connection between finite combinatorics and abstract set existence. The present article sheds further light on this connection. Firstly, we show that the original Kruskal theorem is equivalent to the uniform version for data types that are finitely generated. Secondly, we prove a dichotomy result for a natural variant of the uniform Kruskal theorem. On the one hand, this variant still implies Π 1 1 -comprehension over R C A 0 extended by the chain antichain principle ( C A C ). On the other hand, it becomes weak when C A C is removed from the base theory. This article is part of the theme issue ‘Modern perspectives in Proof Theory’.