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57
Accessible independence results for Peano arithmetic
 Bulletin of the London Mathematical Society
, 1982
"... Recently some interesting firstorder statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely numbertheoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are c ..."
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Recently some interesting firstorder statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely numbertheoretic 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
Kreisel's `Unwinding Program
 In Odifreddi [53
, 1996
"... 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 fortyodd years. My purpose here is to give ..."
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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 fortyodd years. My purpose here is to give
Bounded Arithmetic and Propositional Proof Complexity
 in Logic of Computation
, 1995
"... 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 t ..."
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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 cutfree 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.
Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... 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 A ..."
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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
Ordinal arithmetic in ACL2
 In ACL2 Workshop 2003
, 2003
"... 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 tha ..."
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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
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... 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 meet ..."
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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.
Efficient execution in an automated reasoning environment
 Journal of Functional Programming
, 2006
"... 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 ot ..."
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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 logicallydefined 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 Ramsey Theorem in BoyerMoore Logic
 Journal of Automated Reasoning
, 1995
"... We use the BoyerMoore Prover, Nqthm, to verify the ParisHarrington 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. Introduct ..."
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We use the BoyerMoore Prover, Nqthm, to verify the ParisHarrington 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 wellknown 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 firstorder logic, and contains the induction schema, which can apply to arbitrary firstorder formulas. The logic of PRA allows only quantifierfree formulas, which are thought of as being universally quantified, and PRA has the induction scheme for quantifierfree 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...
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we sho ..."
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.