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 A. Freund, M. Rathjen and A. Weiermann, it is equivalent to $\Pi^1_1$-comprehension, over $\mathsf{RCA_0}$ with the chain antichain principle ($\mathsf{CAC}$). This result provides a connection between finite combinatorics and abstract set existence. The present paper sheds further light on this connection. First, 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 $\Pi^1_1$-comprehension over $\mathsf{RCA}_0+\mathsf{CAC}$. On the other hand, it becomes weak when $\mathsf{CAC}$ is removed from the base theory.