The larger project broached here is to look at the generally Π 1 2 sentence " if X is well ordered then f (X) is well ordered", where f is a standard proof theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory T f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f . To illustrate this theme, we prove in this paper that the statement " if X is well ordered then εX is well ordered" is equivalent to ACA + 0 . This was first proved by Marcone and Montalban [7] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte's method of proof search (deduction chains) [11] and cut elimination for a (small) fragment of Lω 1 ,ω .
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.
We present variants of Goodstein's theorem that are in turn equivalent to respectively → in turn arithmetical comprehension and to arithmetical transfinite recursion over a weak base theory. These variants differ from the usual Goodstein theorem in that they (necessarily) entail the existence of complex infinite objects. As part of our proof, we show that the Veblen hierarchy of normal functions on the ordinals is closely related to an extension of the Ackermann function by direct limits.
This paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.
Abstract. One objective of this paper is the determination of the proof-theoretic strength of Martin-L6f's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with A~ comprehension and bar induction. As Martin-L6f intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a strong classical theory. Also the proof-theoretic strength of other inductive types like Aczel's type of iterative sets is investigated in various contexts.Further, we study metamathematical relations between type theories and other frameworks for formalizing constructive mathematics, e.g. Aczel's set theories and theories of operations and classes as developed by Feferman. IntroductionMartin-L6f's intuitionistic theory of types was originally introduced as a system for formalizing intuitionistic mathematics. It is a typed theory of constructions containing types which themselves depend on the constructions contained in previously constructed types. These dependent types enable one to express the general Cartesian product of the resulting families of types, as well as the disjoint union of such a famiIy.Using the Brouwer-Heyting semantics of the logical constants one sees how logical notions are obtained in this theory, This is done by interpreting propositions as types and proofs as constructions [ML 84]. In addition to ground types for the finite sets and the set of natural numbers the theory can be taken to contain types which play * The second author would like to thank the National Science Foundation of the USA for support by grant DMS-9203443
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