An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

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.

Politics & Social Sciences

Mancosu, Paolo

Galvan, Sergio

Zach, Richard

Philosophy

Oxford University Press

Product 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.\nAbout the Author 
Paolo Mancosu, 
UC Berkeley,Sergio Galvan, 
Catholic University of Milan,Richard Zach, 
University of Calgary\nPaolo Mancosu is Willis S. and Marion Slusser Professor of Philosophy at the University of California at Berkeley. He is the author of numerous articles and books in logic and philosophy of mathematics. During his career he has taught at Stanford, Oxford, and Yale. He was awarded a fellowship at the 
Wissenschaftskolleg zu Berlin in 1997-1998, a stipendiary position as 
Directeur de recherche invité; au CNRS in Paris in 2004-2005, a Guggenheim Fellowship in 2008-2009, a position as member at the Institute of Advanced Study in Princeton in 2009, a visiting professorship as LMU-UCB Research in the 
Humanities at LMU in 2014, and a Humboldt Research Award in 2017-2018.\nSergio Galvan is emeritus Professor of Logic at the Catholic University of Milan, Italy. His main areas of research are proof-theory and philosophical logic. In the first area he focuses on sequent calculi and natural deduction, the metamathematics of arithmetic systems (from Q to PA), Gödel's 
incompleteness theorems, and Gentzen's cut-elimination theorem. In the second area, his major interest is in the philosophical interpretations (deontic, epistemic and metaphysical) of modal logic. Recently, he has been working on the relationships between formal proof and intuition in mathematics, 
and on the metaphysics of essence and the ontology of possibilia.\nRichard Zach is Professor of Philosophy at the University of Calgary, Canada. He works in logic, history of analytic philosophy, and the philosophy of mathematics. In logic, his main interests are non-classical logics and proof theory. He has also written on the development of formal logic and 
historical figures associated with this development such as Hilbert, Gödel, and Carnap. He has held visiting appointments at the University of California, Irvine, McGill University, and the University of Technology, Vienna.

An introduction to formal logic, covering truth-functional (propositional) and first-order (predicate) logic. It deals with logical notions such as entailment and satisfiablity, symbolization of English sentences using connectives and quantifiers (including identity, e.g., definite descriptions), the semantics of TFL (truth tables) and FOL (first-order interpretations), and proofs (using and natural deduction). Advanced topics discussed are the truth-functional adequacy of propositional logic, normal forms, the soundness of the proof system, and the basics of modal logic. Available for free download at forallx.openlogicproject.org.

forall x: Calgary: An Introduction to Formal Logic

Button, Tim

Magnus, P.D.

Trueman, Robert

Thomas-Bolduc, Aaron

Independently published

A textbook on modal and other intensional logics. It covers normal modal logics, relational semantics, axiomatic and tableaux proof systems, intuitionistic logic, and counterfactual conditionals. It is based on the Open Logic Project and available for free download at openlogicproject.org.

Boxes and Diamonds: An Open Introduction to Modal Logic (Open Logic Project Textbooks)

A textbook on the semantics, proof theory, and metatheory of first-order logic. Covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. It is based on the Open Logic project, and available for free download at slc.openlogicproject.org.

Sets, Logic, Computation: An Open Introduction to Metalogic (Open Logic Project Textbooks)

This book is an introduction to metamathematics and Gödel's theorems. It covers recursive function theory, arithmetization of syntax, the first and second incompleteness theorem, models of arithmetic, second-order logic, and the lambda calculus. It is based on the Open Logic Project, and available for free download at ic.openlogicproject.org.

Incompleteness and Computability: An Open Introduction to Gödel's Theorems (Open Logic Project Textbooks)

An introduction to the metatheory of first-order logic. Covers naive set theory, semantics and proof theory of first-order logic (sequent calculus and natural deduction), the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines and the undecidability of first-order logic.

Sets, Logic, Computation: An Open Logic Text

Science & Math

Mathematics

CreateSpace Independent Publishing Platform

Loftis, J. Robert

See the best price to sell, buy, or rent books by Zach, Richard. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Books by Zach, Richard

See the best price to sell, buy, or rent books by Zach, Richard. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

An Introduction to Proof Theory: Normalization, Cut-Elimination, and Consistency Proofs

forall x: Calgary: An Introduction to Formal Logic

Boxes and Diamonds: An Open Introduction to Modal Logic (Open Logic Project Textbooks)

Sets, Logic, Computation: An Open Introduction to Metalogic (Open Logic Project Textbooks)

Incompleteness and Computability: An Open Introduction to Gödel's Theorems (Open Logic Project Textbooks)

Sets, Logic, Computation: An Open Logic Text

forall x: Calgary: An Introduction to Formal Logic

Sets, Logic, Computation: An Open Introduction to Metalogic (Open Logic Project Textbooks)

forall x: Calgary: An Introduction to Formal Logic