On the original gentzen consistency proof for number theory pdf

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. The proofs are completely unformalized and gentzen does not say anything specific about formalization. The story of gentzens original consistency proof for. Contentual and formal aspects of gentzens consistency. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms.

However, gentzen did not present his finitist interpretation explicitly. Intuitionism and proof theory, proceedings of the summer conference at buffalo n. An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate. Gentzens original consistency proof and the bar theorem w. Gentzen consistency proof for the formal system of first order number theory, including standard logic, the peano axioms and recursive definitions is considered. Estimates of some functions on primes and stirlings formula 15 part 1. The goal of this paper then, is to investigate whether gentzens and bernayss suggestions that. Already in 1936, however, gerhard gentzen found a way out of this dilemma. Gerhard gentzen proved the consistency of peano axioms. The author comments on gentzens steps which are supplemented with exact calculations and parts of formal derivations. Gentzens centenary, the quest for consistency reinhard. This book explains the first published consistency proof of pa. Number symbol meaning so this proves the easy half of the theorem.

Pdf gentzens original consistency proof and the bar theorem. Gentzen inherited the research on the consistency of elementary number theory from. The epistemological gain, if there is one, rests in the evidence for the consistency of spectors quanti. In what sense is the proof based on primitive recursive arithmetic. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Contentual and formal aspects of gentzens consistency proofs. For example, here are some problems in number theory that remain unsolved. Consistency proof an overview sciencedirect topics. In gentzens thesis there is a conjecture about the normalization theorem for derivation in intuitionistic natural deduction, then transformed into a proof. 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. The contributions range from philosophical reflections and reevaluations of gentzens original consistency proofs to the most recent developments in proof theory.

Initial sequents are used in order to replace logical rules and dis junction. The present paper is intended to change this unsatisfactory situation by presenting ge36, iv. Gentzens proof of normalization for natural deduction. In this paper, first we formulate an interpretation for the implicationformulas in firstorder arithmetic by using gentzens 1935 consistency proof.

Arithmetic elementary number theory pa cannot prove its own consistency. Gentzens 1936 consistency proof for firstorder arithmetic gentzen, math ann, 112. Gentzen did some work in this direction, but was then assigned to military service in the fall of 1939. Gentzens original papers prove the consistency of peano arithmetic albeit using the axioms of pra in the 1938 version. Today, proof theory is a wellestablished branch of mathematical and philosophical logic. Interpretational proof theory compares formalisms via syntactic translations or interpretations. A philosophical significance of gentzens 1935 consistency.

Hilbert was a german mathematician and significantly con. If nowadays gentzens consistency proof for arithmetic is mentioned, one usually refers to ge38 while gentzens. 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. These notes were prepared by joseph lee, a student in the class, in collaboration with prof. The course was designed by susan mckay, and developed by stephen donkin, ian chiswell, charles. Each formal theory has a signature that specifies the nonlogical symbols in the language of the theory. Olympiad number theory through challenging problems. These deduction trees are wellknown objects, namely cutfree deductions in a formalization of firstorder number theory in the sequent calculus with the. Then, as gentzen showed, that is best possible in ordinal terms, since pa proves trans.

Gentzens centenary the quest for consistency reinhard. Stillwell is a master expositor and does a very good job explaining and. Gentzens first version of his consistency proof can be formulated as a game. If the number 253 is composite, for example, it must have a factor less than or equal to 15. 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. It covers the basic background material that an imo student should be familiar with. On gentzens rst consistency proof for arithmetic introduction. We next show that all ideals of z have this property. Find materials for this course in the pages linked along the left. Pdf basic proof theory download full pdf book download. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. The story of gentzens original consistency proof for firstorder number theory 9, as told by paul bernays 1, 9, 11, letter 69, pp. Moreover, he gave no argument for its noncircularity. 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.

We hope to appreciate the conception and realization of proof theory as deeply. Gentzens original consistency proof and the bar theorem. From traditional set theory that of cantor, hilbert, g. Let me begin with a description of gentzens consistency proof. An irrational number is a number which cannot be expressed as the ratio of two integers. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers. Topics in logic proof theory university of notre dame.

Basic proof theory download ebook pdf, epub, tuebl, mobi. Gentzens consistency proof is a result of proof theory in mathematical logic, published by. Underclassical mathematics one meansthe mathematics in the sense in which it was understood before the begin of the criticism of set theory. As had already been noted in 5, we may express it in terms of a game. It aroused immediate interest, especially through bernays who stayed at the institute for advanced study in princeton in 193536. The development of proof theory stanford encyclopedia of. Vesley, studies in logic and the foundations of mathematics, northholland publishing company, amsterdam and london1970, pp.

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. The next obvious task in proof theory, after the proof of the consistency of arithmetic, was to prove the consistency of analysis, 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. First of all one wants to give a proof of the consistency of the classical mathematics. Bernays, p 1970, on the original gentzen consistency proof for number theory. It contains the original gentzens proof, but it uses modern terminology and examples to illustrate the essential notions. Although the main elements of the result were essentially already present in 1936, they were re. This work comprises articles by leading proof theorists, attesting to gentzens enduring legacy to mathematical logic and beyond. David hilberts program of recovering the consistency of math.

On the original gentzen consistency proof for number theory. A plausible candidate for such a consistency proof is gentzens second proof of the consistency of pure number theory. It is worth remarking that this settheoretic proof of the consistency of pa. 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. The proof of spector was published posthumously in 1962 spector died. 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 ideals that are listed in example 4 are all generated by a single number g. A rational number is a number which can be expressed as the ratio a b of two integers a,b, where b 6 0. Its focus has expanded from hilberts program, narrowly construed, to a more general study of proofs and their properties. Thus we need only check the primes 2, 3, 5, 7, 11, and.