Constructive Infinity, Actual Infinity, and the Recursive-Explosion Blind Spot

This paper distinguishes two kinds of infinity: constructive infinity and actual infinity. Constructive infinity is defined as the infinity reached from 0 by the traversal rule n \mapsto n+1; actual infinity is defined as an infinity strictly greater than all finite integers. This paper points out that there exist integers that are finite but generated by recursive-explosion structures, whose generation method does not belong to the traversal closure, thereby strictly exceeding the coverage of constructive infinity. It follows that constructive infinity does not constitute actual infinity. Further, this paper reveals that the number line and the traditional concept of \aleph_0 implicitly contain a traversal assumption, which fails in the presence of recursive-explosion structures. Finally, from a unified hierarchical generative viewpoint, it shows the consistency of the conceptual core between set-theoretic cardinal hierarchies and large-number growth hierarchies, and discusses in the appendix the hierarchical meaning of the Axiom of Choice and the continuum.