It aroused immediate interest, especially through bernays who stayed at the institute for advanced study in princeton in 193536. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Gentzen sent it off to mathematische annalen in august of 1935 and then withdrew it in december after receiving criticism and, in particular, the criticism that the proof used the fan theorem, a criticism that, as the. Moreover, he gave no argument for its noncircularity. Contentual and formal aspects of gentzens consistency proofs. Gentzen consistency proof for the formal system of first order number theory, including standard logic, the peano axioms and recursive definitions is considered. The next obvious task in proof theory, after the proof of the consistency of arithmetic, was to prove the consistency of analysis, i. Although the main elements of the result were essentially already present in 1936, they were re. The proofs are completely unformalized and gentzen does not say anything specific about formalization. The epistemological gain, if there is one, rests in the evidence for the consistency of spectors quanti. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms. As had already been noted in 5, we may express it in terms of a game. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. Today, proof theory is a wellestablished branch of mathematical and philosophical logic.
Bernays, p 1970, on the original gentzen consistency proof for number theory. On gentzens rst consistency proof for arithmetic introduction. Interpretational proof theory compares formalisms via syntactic translations or interpretations. Proof theory came into being in the twenties of the last century, when it was inaugurated by david hilbert in order to secure the foundations of mathematics. Arithmetic elementary number theory pa cannot prove its own consistency. Each formal theory has a signature that specifies the nonlogical symbols in the language of the theory. This work comprises articles by leading proof theorists, attesting to gentzens enduring legacy to mathematical logic and beyond. Topics in logic proof theory university of notre dame. In what sense is the proof based on primitive recursive arithmetic. Gentzen did some work in this direction, but was then assigned to military service in the fall of 1939. A philosophical significance of gentzens 1935 consistency. Gentzens consistency proof is a result of proof theory in mathematical logic, published by.
An irrational number is a number which cannot be expressed as the ratio of two integers. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Estimates of some functions on primes and stirlings formula 15 part 1. The course was designed by susan mckay, and developed by stephen donkin, ian chiswell, charles.
Number symbol meaning so this proves the easy half of the theorem. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. The contributions range from philosophical reflections and reevaluations of gentzens original consistency proofs to the most recent developments in proof theory. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. The development of proof theory stanford encyclopedia of. This book explains the first published consistency proof of pa. The story of gentzens original consistency proof for.
Gentzens first version of his consistency proof can be formulated as a game. Already in 1936, however, gerhard gentzen found a way out of this dilemma. Stillwell is a master expositor and does a very good job explaining and. We hope to appreciate the conception and realization of proof theory as deeply. Thus we need only check the primes 2, 3, 5, 7, 11, and.
Gentzens quest for consistency a gentzenstyle proof without heightlines gentzens programme gentzens four proofs the earliest proofs of the consistency of peano arithmetic were presented by gentzen, who worked out a total of four proofs between 1934 and 1939. Intuitionism and proof theory, proceedings of the summer conference at buffalo n. Then, as gentzen showed, that is best possible in ordinal terms, since pa proves trans. Gentzens centenary the quest for consistency reinhard. It is worth remarking that this settheoretic proof of the consistency of pa. Hilbert was a german mathematician and significantly con. Gentzen inherited the research on the consistency of elementary number theory from. These deduction trees are wellknown objects, namely cutfree deductions in a formalization of firstorder number theory in the sequent calculus with the. Gentzens original papers prove the consistency of peano arithmetic albeit using the axioms of pra in the 1938 version. Preface these are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. However, gentzen did not present his finitist interpretation explicitly.
Its focus has expanded from hilberts program, narrowly construed, to a more general study of proofs and their properties. Gerhard gentzen proved the consistency of peano axioms. Find materials for this course in the pages linked along the left. It contains the original gentzens proof, but it uses modern terminology and examples to illustrate the essential notions. A rational number is a number which can be expressed as the ratio a b of two integers a,b, where b 6 0. Initial sequents are used in order to replace logical rules and dis junction. Gentzens original consistency proof and the bar theorem w. Gentzens original consistency proof and the bar theorem. Gentzens 1936 consistency proof for firstorder arithmetic gentzen, math ann, 112. The ideals that are listed in example 4 are all generated by a single number g.
Gentzens centenary, the quest for consistency reinhard. The present paper is intended to change this unsatisfactory situation by presenting ge36, iv. We next show that all ideals of z have this property. On gentzens rst consistency proof for arithmetic wilfried buchholz ludwigmaximilians universit at munc hen february 14, 2014 introduction if nowadays \gentzens consistency proof for arithmetic is mentioned, one usually refers to ge38 while gentzens rst published consistency proof, i. Tait the story of gentzens original consistency proof for rstorder number theory gentzen 1974,1 as told by paul bernays gentzen 1974, bernays 1970, g odel 2003, letter 69, pp.
From traditional set theory that of cantor, hilbert, g. Contentual and formal aspects of gentzens consistency. If nowadays gentzens consistency proof for arithmetic is mentioned, one usually refers to ge38 while gentzens. The story of gentzens original consistency proof for firstorder number theory 9, as told by paul bernays 1, 9, 11, letter 69, pp. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. The goal of this paper then, is to investigate whether gentzens and bernayss suggestions that. In this paper, first we formulate an interpretation for the implicationformulas in firstorder arithmetic by using gentzens 1935 consistency proof. First of all one wants to give a proof of the consistency of the classical mathematics. Let me begin with a description of gentzens consistency proof. It is surprising that there is lack of information on gentzens consistency proof sure, there are some contents on gentzens first consistency proof of peano axioms, but not on what we usually say gentzens consistency proof. It covers the basic background material that an imo student should be familiar with.
In gentzens thesis there is a conjecture about the normalization theorem for derivation in intuitionistic natural deduction, then transformed into a proof. For example, here are some problems in number theory that remain unsolved. David hilberts program of recovering the consistency of math. Olympiad number theory through challenging problems. Underclassical mathematics one meansthe mathematics in the sense in which it was understood before the begin of the criticism of set theory. Pdf gentzens original consistency proof and the bar theorem. Consistency proof an overview sciencedirect topics. A plausible candidate for such a consistency proof is gentzens second proof of the consistency of pure number theory. On the original gentzen consistency proof for number theory. Basic proof theory download ebook pdf, epub, tuebl, mobi. The proof of spector was published posthumously in 1962 spector died. These notes were prepared by joseph lee, a student in the class, in collaboration with prof. Vesley, studies in logic and the foundations of mathematics, northholland publishing company, amsterdam and london1970, pp. Pdf basic proof theory download full pdf book download.