Abstract not found

This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact, unbounded) proof speedup of (i + 1)-st order arithmetic over i-th order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic; this allows all tautologies as axioms and allows all generalizations of axioms as axioms.

The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the so-called limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic rst-order set theories can be interpreted in K 1 , for suitable . Keywords: cumulative types, innitary inference rule, logical foundations of set theory. MSC: 03B15 03B30 03E30 03F25 1 Introduction The idea of founding mathematics on a theory of types was rst proposed by Russell [20] (foreshadowed already in [19]), and subsequently implemented by Whitehead and Russell [26]. The formal systems presented in these works were later simplied and cast into their modern shape by Ramsey [18]. Godel [9] and Tarski [25] were the rst to restrict the type structure to types of unary predi...

This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further references. Since Godel's main achievement concerns the problem of consistency and some of the problems that I am going to describe had been considered by him, I think that it is appropriate to publish this article in Godel Society. 1 Historical remarks The question that we are going to consider in is interesting per se and is related to some more practical questions, especially in complexity theory, but the original motivation for it comes from foundational studies. Among the variety of streams in foundations of mathematics, the one which had the biggest influence and which very much determined later development of mathematical logic was Hilbert's<F1

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in these systems compared to lengths in Frege proof systems. As an application we give a near-linear simulation of the propositional Gentzen sequent calculus by Frege proofs. The length of a proof is the number of steps or lines in the proof. A general