Recently some interesting first-order statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely number-theoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are combinatorial. We also give another independence result (unashamedly combinatorial in character) proved by the same methods. The first result is an improvement of a theorem of Goodstein [2]. Let m and n be natural numbers, n> 1. We define the base n representation of m as follows: First write m as the sum of powers of n. (For example, if m = 266, n = 2, write 266 = 2 8 + 2 3 + 2 1.) Now write each exponent as the sum of powers of n. (For example, 266 = 2 23 + 2 2 + 1 +2 1.) Repeat with exponents of exponents and so on until the representation stabilizes. For example, 266 stabilizes at the representation 2 * +l + 2 2 + l +2 l. We now define the number Gn(m) as follows. If m = 0 set Gn(m) = 0. Otherwise set Gn(m) to be the number produced by replacing every n in the base n representation of m by n +1 and then subtracting 1. (For example, G2(266) = 3 33+1 + 3 3 + 1 +2). Now define the Goodstein sequence for m starting at 2 by So, for example, m0 = m, mx = G2{m0), m2 = G^mJ, m3 = G^m2),.... 266O = 266 = 2 22+1 + 2 2+1 + 2 X = 3 33+1 + 3 3+1 + 2 ~ 1O 38 2662 = 4 44+1 + 4 4+1 + l ~ 10 616 2663 = 5 s5+1 + 5 5+1 ~ 10 10- 000. Similarly we can define the Goodstein sequence for m starting at n for any n> 1. THEOREM 1. (i) (Goodstein [2]) Vm 3/c mk = 0. More generally for any m, n> 1 the Goodstein sequence for m starting at n eventually hits zero. (ii) Vm 3k mk = 0 (formalized in the language of first order arithmetic) is not provable

Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1

This is a survey of basic facts about bounded arithmetic and about the relationships between bounded arithmetic and propositional proof complexity. We introduce the theories S 2 of bounded arithmetic and characterize their proof theoretic strength and their provably total functions in terms of the polynomial time hierarchy. We discuss other axiomatizations of bounded arithmetic, such as minimization axioms. It is shown that the bounded arithmetic hierarchy collapses if and only if bounded arithmetic proves that the polynomial hierarchy collapses. We discuss Frege and extended Frege proof length, and the two translations from bounded arithmetic proofs into propositional proofs. We present some theorems on bounding the lengths of propositional interpolants in terms of cut-free proof length and in terms of the lengths of resolution refutations. We then define the RazborovRudich notion of natural proofs of P NP and discuss Razborov's theorem that certain fragments of bounded arithmetic cannot prove superpolynomial lower bounds on circuit size, assuming a strong cryptographic conjecture. Finally, a complete presentation of a proof of the theorem of Razborov is given. 1 Review of Computational Complexity 1.1 Feasibility This article will be concerned with various "feasible" forms of computability and of provability. For something to be feasibly computable, it must be computable in practice in the real world, not merely e#ectively computable in the sense of being recursively computable.

Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation than the one used by ACL2. The algorithms have been implemented and numerous properties of the arithmetic operators have been mechanically verified, thereby greatly extending ACL2’s ability to reason about the ordinals. We describe how to use the libraries containing these results, which are currently distributed with ACL2 version 2.7. 1

We use the Boyer-Moore Prover, Nqthm, to verify the Paris-Harrington version of Ramsey's Theorem. The proof we verify is a modification of the one given by Ketonen and Solovay. The Theorem is not provable in Peano Arithmetic, and one key step in the proof requires ffl 0 induction. x0. Introduction. The most well-known formalizations of finite mathematics are PA (Peano Arithmetic) and PRA (Primitive Recursive Arithmetic). In both, the "intended" domain of discourse is the set of natural numbers. PA is formalized in standard first-order logic, and contains the induction schema, which can apply to arbitrary first-order formulas. The logic of PRA allows only quantifier-free formulas, which are thought of as being universally quantified, and PRA has the induction scheme for quantifier-free formulas, expressed as a proof rule. Also, for each primitive recursive function f , PRA contains a function symbol for f and has the recursive definition of f as an axiom. Clearly, PRA is much weaker tha...

Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last forty-odd years. My purpose here is to give

This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

Abstract We describe a method to permit the user of a mathematical logic to write elegant logical definitions while allowing sound and efficient execution. We focus on the ACL2 logic and automated reasoning environment. ACL2 is used by industrial researchers to describe microprocessor designs and other complicated digital systems. Properties of the designs can be formally established with the theorem prover. But because ACL2 is also a functional programming language, the formal models can be executed as simulation engines. We implement features that afford these dual applications, namely formal proof and execution on industrial test suites. In particular, the features allow the user to install, in a logically sound way, alternative executable counterparts for logically-defined functions. These alternatives are often much more efficient than the logically equivalent terms they replace. We discuss several applications of these features. 1 Introduction This paper is about a way to permit the functional programmer to prove efficientprograms correct. The idea is to allow the provision of two definitions of the program: an elegant definition that supports effective reasoning by a mechanizedtheorem prover, and an efficient definition for evaluation. A bridge of this sort,

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...

. After a brief flirtation with logicism in 1917--1920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the e-calculus, on which Hilbert's axiom systems were based, and the development of the e-substitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the e-notation and provides an analysis of Ackermann's consisten...