An intuitionistie proof of KruskaFs Theorem W im Veldman Subfaculteit W iskunde, Katholieke Universiteit Toernooiveld, 6525 ED Nijmegen, the N etherlands email: veldman@sei,kun.nl 0 Introduction In 1960, J.B. Kruskal published a proof of a conjecture due to A. Vazsonyi. Vazsonyi's conjecture, to be explained in detail in Section 8, says th a t the collection of all finite trees is well-quasi-ordered by the relation of embeddability, th a t is, for every infinite sequence a(0), a(l), a(2) ,. .. of finite trees there exist i, j such th a t i < j and a(i) embeds into a(j). Kruskal established an even stronger statem ent th a t he called the Tree Theorem. He proved it by a slight extension of an argum ent developed by G. Higman in 1952. In 1963, a short proof of Kruskal's Theorem was given by C .St.J.A Nash-Williams, who introduced the elegant and powerful but non-constructive minimal-bad-sequence argument. The purpose of this paper is to show th a t the argum ents given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistie point of view and th a t the later argum ent given by Nash-Williams is not. The paper consists of the following 11 Sections.

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