Mathematical Intuitionism

Mathematical Intuitionism
Title Mathematical Intuitionism PDF eBook
Author Carl J. Posy
Publisher Cambridge University Press
Total Pages 116
Release 2020-11-12
Genre Science
ISBN 1108593259

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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.

Mathematical Intuitionism

Mathematical Intuitionism
Title Mathematical Intuitionism PDF eBook
Author Alʹbert Grigorʹevich Dragalin
Publisher
Total Pages 241
Release 1988
Genre Intuitionistic mathematics
ISBN 9781470444815

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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.

An Introduction to Proof Theory

An Introduction to Proof Theory
Title An Introduction to Proof Theory PDF eBook
Author Paolo Mancosu
Publisher Oxford University Press
Total Pages 336
Release 2021-08-12
Genre Philosophy
ISBN 0192649299

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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.

Principles of Intuitionism

Principles of Intuitionism
Title Principles of Intuitionism PDF eBook
Author Anne S. Troelstra
Publisher Springer
Total Pages 114
Release 2006-11-14
Genre Mathematics
ISBN 3540361308

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Intuitionism and Proof Theory

Intuitionism and Proof Theory
Title Intuitionism and Proof Theory PDF eBook
Author Conference On Intuitionism And Proof Theory. 1968. Buffalo
Publisher
Total Pages 0
Release 1970
Genre Logic, Symbolic and mathematical
ISBN 9780720422573

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Proof Theory and Intuitionistic Systems

Proof Theory and Intuitionistic Systems
Title Proof Theory and Intuitionistic Systems PDF eBook
Author Bruno Scarpellini
Publisher Springer
Total Pages 298
Release 2006-11-15
Genre Mathematics
ISBN 3540368752

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Proof Theory

Proof Theory
Title Proof Theory PDF eBook
Author Vincent F. Hendricks
Publisher Springer Science & Business Media
Total Pages 345
Release 2013-03-09
Genre Philosophy
ISBN 9401727961

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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.