This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

A selection from the correspondence of the logician Wilhelm Ackermann (1896?1962) is presented in this article. The most significant letters were exchanged with Bernays, Scholz and Lorenzen, from which extensive passages are transcribed. Some remarks from other letters, with quotations, are also included.

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to Brouwer, Weyl, Bernays and Hilbert, and the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

The epsilon calculus is a logical formalism developed by David Hilbert in the service of his program in the foundations of mathematics. The epsilon operator is a term-forming operator which replaces quantifiers in ordinary predicate logic. Specifically, in the calculus, a term εx A denotes some x satisfying A(x), if there is one. In Hilbert's Program, the epsilon terms play the role of ideal elements; the aim of Hilbert's finitistic consistency proofs is to give a procedure which removes such terms (...) from a formal proof. The procedures by which this is to be carried out are based on Hilbert's epsilon substitution method. The epsilon calculus, however, has applications in other contexts as well. The first general application of the epsilon calculus was in Hilbert's epsilon theorems, which in turn provide the basis for the first correct proof of Herbrand's theorem. More recently, variants of the epsilon operator have been applied in linguistics and linguistic philosophy to deal with anaphoric pronouns. (shrink)

The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for ...

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...) University of Göttingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the -calculus, on which Hilbert's axiom systemswere based, and the development of the -substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics'' throughthe -notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time. (shrink)

In the 1920s, David Hilbert proposed a research program with the aim of providing mathematics with a secure foundation. This was to be accomplished by first formalizing logic and mathematics in their entirety, and then showing---using only so-called finitistic principles---that these formalizations are free of contradictions. ;In the area of logic, the Hilbert school accomplished major advances both in introducing new systems of logic, and in developing central metalogical notions, such as completeness and decidability. The analysis of unpublished material presented (...) in Chapter 2 shows that a completeness proof for propositional logic was found by Hilbert and his assistant Paul Bernays already in 1917--18, and that Bernays's contribution was much greater than is commonly acknowledged. Aside from logic, the main technical contribution of Hilbert's Program are the development of formal mathematical theories and proof-theoretical investigations thereof, in particular, consistency proofs. In this respect Wilhelm Ackermann's 1924 dissertation is a milestone both in the development of the Program and in proof theory in general. Ackermann gives a consistency proof for a second-order version of primitive recursive arithmetic which, surprisingly, explicitly uses a finitistic version of transfinite induction up to www . He also gave a faulty consistency proof for a system of second-order arithmetic based on Hilbert's &egr;-substitution method. Detailed analyses of both proofs in Chapter 3 shed light on the development of finitism and proof theory in the 1920s as practiced in Hilbert's school. ;In a series of papers, Charles Parsons has attempted to map out a notion of mathematical intuition which he also brings to bear on Hilbert's finitism. According to him, mathematical intuition fails to be able to underwrite the kind of intuitive knowledge Hilbert thought was attainable by the finitist. It is argued in Chapter 4 that the extent of finitistic knowledge which intuition can provide is broader than Parsons supposes. According to another influential analysis of finitism due to W. W. Tait, finitist reasoning coincides with primitive recursive reasoning. The acceptance of non-primitive recursive methods in Ackermann's dissertation presented in Chapter 3, together with additional textual evidence presented in Chapter 4, shows that this identification is untenable as far as Hilbert's conception of finitism is concerned. Tait's conception, however, differs from Hilbert's in important respects, yet it is also open to criticisms leading to the conclusion that finitism encompasses more than just primitive recursive reasoning. (shrink)

Some of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917-1923. The aim of this paper is to describe these results, focussing (...) primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic ("Post-") and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged. (shrink)

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are (...) generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman. (shrink)

What were the earliest reactions to Gödel's incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Gödel's groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to (...) the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought. (shrink)

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

Hilbert's plan for understanding the concept of infinity required the elimination of non‐finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy of progressively less elementary, but still constructive methods instead of finitist ones . A mathematical proof of this failure requires a definition of « finitist ».—The paper sketches the three principal methods for the syntactic analysis of non‐constructive mathematics, the resulting consistency proofs and constructive interpretations, modelled on Herbrand's theorem, and their (...) mathematical and logical consequences. A characterization of finitist proofs is sketched. A remark on the completeness of the predicate calculus concludes the paper. Throughout open problems and alternative approaches are emphasized.ZusammenfassungHilbert sah das Verstehen des Begriffs des Unendlichen in der Eliminierung nichtfiniter Methoden aus Beweisen finiter Sätze. Das Versagen dieses Unternehmens führt zu einer Hierarchic progressiv weniger elementarer, aber noch konstruktiver Methoden, statt der finiten . Ein Unmöglichkeitsbeweis des ursprünglichen Planes setzt eine Definition von « finit » voraus.—Die drei wichtigsten Methoden der syntaktischen Analyse nichtkonstruktiver mathematischer Theorien, die daraus gewonnenen Widerspruchsfreiheitsbeweise and konstruktiven Interpretationen and deren mathematische and logische Anwendungen werden besprochen. Eine Prazisierung des Begriffs « finit » wird skizziert. Eine Bemerkung zur Vollstandigkeit des engeren Funktionenkalküls beschliesst die Abhandlung.—Oflene Fragen and komplementare Fragestellungen werden betont. (shrink)

Hilbert's program attempts to show that our mathematical knowledge can be certain because we are able to know for certain the truths of elementary arithmetic. I argue that, in the absence of a theory of mathematical truth, Hilbert does not have a complete theory of our arithmetical knowledge. Further, while his deployment of a Kantian notion of intuition seems to promise an answer to scepticism, there is no way to complete Hilbert's epistemology which would answer to his avowed aims.

In the first part of this paper we discuss some aspects of Detlefsen's attempt to save Hilbert's Program from the consequences of Godel's Second Incompleteness Theorem. His arguments are based on his interpretation of the long standing and well-known controversy on what, exactly, finitistic means are. In his paper [1] Detlefsen takes the position that there is a form of the ω-rule which is a finitistically valid means of proof, sufficient to prove the consistency of elementary number theory Z. On (...) the other hand, he claims that Z with its first-order logic is not strong enough to allow a formalization of such an ω-rule. This would explain why the unprovability of $\operatorname{Con}(Z)$ in Z does not imply that the consistency of Z cannot be proved by finitistic means. We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggest leads to a partial program already studied for many years. We investigate the scope of such a program in terms of proof-theoretic reducibilities. We also show that this partial program encompasses mathematically important theories studied in the "Reverse Mathematics" program. In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones. We then investigate their properties, including a few that have no apparent philosophical significance but compare interestingly with the properties of the program based on the iteration of our ω-rule. We determine some of the limitations of such programs, pointing out that these limitations partly explain why partial programs that have been successfully carried out use quite different and substantially more radical extensions of finitistic methods with more general forms of restricted reasoning. (shrink)

Die Leitgedanken meiner Untersuchungen über die Grundlagen der Mathematik, die ich - anknüpfend an frühere Ansätze - seit 1917 in Besprechungen mit P. BERNAYS wieder aufgenommen habe, sind von mir an verschiedenen Stellen eingehend dargelegt worden. Diesen Untersuchungen, an denen auch W. ACKERMANN beteiligt ist, haben sich seither noch verschiedene Mathematiker angeschlossen. Der hier in seinem ersten Teil vorliegende, von BERNAYS abgefaßte und noch fortzusetzende Lehrgang bezweckt eine Darstellung der Theorie nach ihren heutigen Ergebnissen. Dieser Ergebnisstand weist zugleich die Richtung (...) für die weitere Forschung in der Beweistheorie auf das Endziel hin, unsere üblichen Methoden der Mathematik samt und sonders als widerspruchsfrei zu erkennen. Im Hinblick auf dieses Ziel möchte ich hervorheben, daß die zeit weilig aufgekommene Meinung, aus gewissen neueren Ergebnissen von GÖDEL folge die Undurchführbarkeit meiner Beweistheorie, als irrtüm lich erwiesen ist. Jenes Ergebnis zeigt in der Tat auch nur, daß man für die weitergehenden Widerspruchsfreiheitsbeweise den finiten Stand punkt in einer schärferen Weise ausnutzen muß, als dieses bei der Be trachtung der elementaren Formallsmen erforderlich ist. Göttingen, im März 1934 HILBERT Vorwort zur ersten Auflage Eine Darstellung der Beweistheorie, welche aus dem HILBERTschen Ansatz zur Behandlung der mathematisch-logischen Grundlagenpro bleme erwachsen ist, wurde schon seit längerem von HILBERT ange kündigt. (shrink)

A translation of the Écrits logiques, edited by Jean Van Heijenoort, published in 1968.

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

It is argued that an instrumentalist notion of proof such as that represented in Hilbert's viewpoint is not obligated to satisfy the conservation condition that is generally regarded as a constraint on Hilbert's Program. A more reasonable soundness condition is then considered and shown not to be counter-exemplified by Godel's First Theorem. Finally, attention is given to the question of what a theory is; whether it should be seen as a "list" or corpus of beliefs, or as a method for (...) selecting beliefs. The significance of this question for assessing "intensional" results like Godel's Second Theorem, and their bearing on Hilbert's Program are discussed. (shrink)

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing that (...) any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem. Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs. Originally published in 1978. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)