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Author: Carl J. Posy Publisher: Cambridge University Press ISBN: 1108593259 Category : Science Languages : en Pages : 116
Book Description
L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.
Author: Carl J. Posy Publisher: Cambridge University Press ISBN: 1108593259 Category : Science Languages : en Pages : 116
Book Description
L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.
Author: Alʹbert Grigorʹevich Dragalin Publisher: ISBN: 9781470444815 Category : Intuitionistic mathematics Languages : en Pages : 241
Book Description
This monograph is intended to present the most important methods of proof theory in intuitionistic logic, assuming the reader to have mastered an introductory course in mathematical logic. The book starts with purely syntactical methods based on Gentzen's cut-elimination theorem, followed by intuitionistic arithmetic where Kleene's realizability method plays a central role. The author then studies algebraic models and completeness theorems for them. After giving a survey on the principles of intuitionistic analysis, the last part of the book presents the cut-elimination theorem in intuitionist.
Author: Paolo Mancosu Publisher: Oxford University Press ISBN: 0192649299 Category : Philosophy Languages : en Pages : 336
Book Description
An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the Gödel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.
Author: Conference On Intuitionism And Proof Theory. 1968. Buffalo Publisher: ISBN: 9780720422573 Category : Logic, Symbolic and mathematical Languages : en Pages : 0
Author: Vincent F. Hendricks Publisher: Springer Science & Business Media ISBN: 9401727961 Category : Philosophy Languages : en Pages : 345
Book Description
hiS volume in the Synthese Library Series is the result of a conference T held at the University of Roskilde, Denmark, October 31st-November 1st, 1997. The aim was to provide a forum within which philosophers, math ematicians, logicians and historians of mathematics could exchange ideas pertaining to the historical and philosophical development of proof theory. Hence the conference was called Proof Theory: History and Philosophical Significance. To quote from the conference abstract: Proof theory was developed as part of Hilberts Programme. According to Hilberts Programme one could provide mathematics with a firm and se cure foundation by formalizing all of mathematics and subsequently prove consistency of these formal systems by finitistic means. Hence proof theory was developed as a formal tool through which this goal should be fulfilled. It is well known that Hilbert's Programme in its original form was unfeasible mainly due to Gtldel's incompleteness theorems. Additionally it proved impossible to formalize all of mathematics and impossible to even prove the consistency of relatively simple formalized fragments of mathematics by finitistic methods. In spite of these problems, Gentzen showed that by extending Hilbert's proof theory it would be possible to prove the consistency of interesting formal systems, perhaps not by finitis tic methods but still by methods of minimal strength. This generalization of Hilbert's original programme has fueled modern proof theory which is a rich part of mathematical logic with many significant implications for the philosophy of mathematics.
Author: K. Schütte Publisher: Springer Science & Business Media ISBN: 3642664733 Category : Mathematics Languages : en Pages : 309
Book Description
This book was originally intended to be the second edition of the book "Beweis theorie" (Grundlehren der mathematischen Wissenschaften, Band 103, Springer 1960), but in fact has been completely rewritten. As well as classical predicate logic we also treat intuitionistic predicate logic. The sentential calculus properties of classical formal and semiformal systems are treated using positive and negative parts of formulas as in the book "Beweistheorie". In a similar way we use right and left parts of formulas for intuitionistic predicate logic. We introduce the theory of functionals of finite types in order to present the Gi:idel interpretation of pure number theory. Instead of ramified type theory, type-free logic and the associated formalization of parts of analysis which we treated in the book "Beweistheorie", we have developed simple classical type theory and predicative analysis in a systematic way. Finally we have given consistency proofs for systems of lI~-analysis following the work of G. Takeuti. In order to do this we have introduced a constni'ctive system of notation for ordinals which goes far beyond the notation system in "Beweistheorie."
Author: Carl J. Posy
Author: Carl J. Posy
Author: Alʹbert Grigorʹevich Dragalin
Author: Paolo Mancosu
Author: Anne S. Troelstra
Author: Lev D. Beklemishev
Author: Bruno Scarpellini
Author: Conference On Intuitionism And Proof Theory. 1968. Buffalo
Author: Vincent F. Hendricks
Author: K. Schütte
Author: Arend Heyting