In this paper we calibrate the exact proof--theoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents proof--theoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the proof--theoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of \Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...

The following known observation is useful in establishing program termination: if a transitive relation R is covered by finitely many well-founded relations U1,..., Un then R is well-founded. A question arises how to bound the ordinal height |R | of the relation R in terms of the ordinals αi = |Ui|. We introduce the notion of the stature ‖P ‖ of a well partial ordering P and show that |R | ≤ ‖α1 × · · · × αn ‖ and that this bound is tight. The notion of stature is of considerable independent interest. We define ‖P ‖ as the ordinal height of the forest of nonempty bad sequences of P, but it has many other natural and equivalent definitions. In particular, ‖P ‖ is the supremum, and in fact the maximum, of the lengths of linearizations of P. And ‖α1 × · · · × αn ‖ is equal to the natural product α1 ⊗ · · · ⊗ αn.

this paper is to show that the arguments given by Higman and Kruskal are essentially constructive and acceptable from an intuitionistic point of view and that the later argument given by Nash-Williams is not. The paper consists of the following 11 Sections. 1. Dickson's Lemma 2. Almost full relations 3. Brouwer's Thesis 4. Ramsey's Theorem 5. The Finite Sequence Theorem 6. Vazsonyi's Conjecture for binary trees 7. Higman's Theorem 8. Vazsonyi's Conjecture and the Tree Theorem 9. Minimal-Bad-Sequence Arguments 10. The Principle of Open Induction 11. Concluding Remarks Except for Section 9, we will argue intuitionistically. 1 1 Dickson's Lemma

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

. In rewriting theory, termination of a rewrite system by Kruskal's theorem implies a theoretical upper bound on the complexity of the system. This bound is, however, far from having been reached by known examples of rewrite systems. All known orderings used to establish termination by Kruskal's theorem yield a multiply recursive bound. Furthermore, the study of the order types of such orderings suggests that the class of multiple recursive functions constitutes the least upper bound. Contradicting this intuition, we construct here a rewrite system which reduces by Kruskal's theorem and whose complexity is not multiply recursive. This system is even totally terminating. This leads to a new lower bound for the complexity of totally terminating rewrite systems and rewrite systems which reduce by Kruskal's theorem. Our construction relies on the Hydra battle using classical tools from ordinal theory and subrecursive functions. Introduction One of the main questions in rewriting theory i...

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is well-ordered ” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is well-ordered”. 1