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Metamathematics of first-order arithmetic

Author: Petr Hájek; Pavel Pudlák
Publisher: Berlin ; New York : Springer-Verlag, v1993.
Series: Perspectives in mathematical logic.
Edition/Format:   Print book : EnglishView all editions and formats
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This work covers some of the most important results in the study of a first order theory of the natural numbers, called "Peano arithmetic" and its fragments (subtheories). Also covered is  Read more...

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Additional Physical Format: Online version:
Hájek, Petr.
Metamathematics of first-order arithmetic.
Berlin ; New York : Springer-Verlag, v1993
(OCoLC)679312568
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Petr Hájek; Pavel Pudlák
ISBN: 3540506322 9783540506324 0387506322 9780387506326
OCLC Number: 26095691
Description: xiv, 460 pages ; 25 cm.
Contents: (A) Some Logic --
(b) The Language of Arithmetic, the Standard Model --
(c) Beginning Arithmetization of Metamathematics --
Ch. I. Arithmetic as Number Theory, Set Theory and Logic --
1. Basic Developments; Partial Truth Definitions --
(a). Properties of Addition and Multiplication, Divisibility and Primes --
(b). Coding Finite Sets and Sequences; the Theory I[Sigma [subscript 0]](exp) --
(c). Provably Recursive Functions; the Theory I[Sigma [subscript 1]] --
(d). Arithmetization of Metamathematics: Partial Truth Definitions --
2. Fragments of First-Order Arithmetic --
(a). Induction and Collection --
(b). Further Principles and Facts About Fragments --
(c). Finite Axiomatizability; Partial Truth Definitions for Relativized Arithmetical Formulas --
(d). Relativized Hierarchy in Fragments --
(e). Axiomatic Systems of Arithmetic with No Function Symbols --
3. Fragments and Recursion Theory --
(a). Limit Theorem --
(b). Low Basis Theorem --
(c). Infinite [delta] Subsets --
(d). Matiyasevic's Theorem in I[Sigma [subscript 1]] --
4. Elements of Logic in Fragments --
(a). Arithmetizing Provability --
(b). Arithmetizing Model Theory --
(c). Applications to Arithmetic --
Ch. II. Fragments and Combinatorics --
1. Ramsey's Theorems and Fragments --
(a). Statement of Results --
(b). Proofs (of 1.5, 1.7, 1.9) --
(c). Proofs (of 1.6, 1.8, 1.10) --
2. Instances of the Paris-Harrington Principle and Consistency Statements --
(a). Introduction and Statement of Results --
(b). Some Combinatorics --
(c). Proof of [actual symbol not reproducible] --
(d). Strong Indiscernibles --
(e). Final Considerations --
3. Schwichtenberg-Wainer Hierarchy and [alpha]-large Sets --
(a). Ordinals in I[Sigma [subscript 1]] --
(b). Transfinite Induction and Fragments --
(c). [alpha]-large Sets in I[Sigma [subscript 1]] --
(d). Schwichtenberg-Wainer Hierarchy --
Ch. III. Self-Reference --
1. Preliminaries --
(a). Interpretability and Partial Conservativity --
(b). Theories Containing Arithmetic; Sequential Theories; PA and ACA[subscript 0] --
(c). Numerations and Binumerations --
2. Self-Reference and Godel's Theorems, Reflexive Theories --
(a). Existence of Fixed Points --
(b). Godel's First Incompleteness Theorem and Related Topics --
(c). Godel's Second Incompleteness Theorem --
(d). Pure Extensions of PA --
(e). Interpretability in Pure Extensions of PA --
3. Definable Cuts --
(a). Definable Cuts and Their Properties --
(b). A Strong Form of Godel's Second Incompleteness Theorem --
(c). Herbrand Provability and Herbrand Consistency --
(d). Cuts and Interpretations --
4. Partial Conservativity and Interpretability --
(a). Some Prominent Examples --
(b). General Theorems on Partial Conservativity; Some Fixed-Point Theorems --
(c). Applications, Mainly to Interpret ability --
Ch. IV. Models of Fragments of Arithmetic --
1. Some Basic Constructions --
(a). Preliminaries --
(b). Definable Ultrapower of the Standard Model --
(c). On Submodels and Cuts --
(d). Models for the Hierarchy --
(e). Elementary End Extensions --
(f). A Conservation Result --
2. Cuts in Models of Arithmetic with a Top --
(a). Arithmetic with a Top and Its Models --
(b). Cuts --
(c). Extendable, Restrainable and Ramsey Cuts --
(d). Satisfaction in Finite Structures with an Application to Models of I[Sigma [subscript 1]] --
3. Provably Recursive Functions and the Method of Indicators --
(a). Provably Recursive Functions, Envelopes --
(b). Indicators and Paris Sequences --
(c). Paris Sequences of the First kind --
(d). Paris Sequences of the Second Kind --
(e). Further Consequences --
4. Formalizing Model Theory --
(a). Some Results on Satisfaction and Consistency --
(b). A Conservation Result in I[Sigma [subscript 1]] --
(c). Appendix: Another Conservation Result --
Ch. V. Bounded Arithmetic --
1. A Survey of Weak Fragments of Arithmetic --
(a). Fragnents of Arithmetic --
2. A Brief Introduction to Complexity Theory --
(a). Time and Space Complexity Classes --
(b). Nondeterministic Computations --
(c). Degrees and NP-completeness --
(d). Oracle Computations --
(e). The Linear Time Hierarchy and the Polynomial Hierarchy --
(f). Nepomnjascij's Theorem --
(g). The Diagonal Method for Separating Complexity Classes --
3. Exponentiation, Coding Sequences and Formalization of Syntax in I[Sigma [subscript 0]] --
(a). Introduction --
(b). Sets and Sequences --
(c). The Exponentiation Relation --
(d). Developing I[Sigma [subscript 0]] + [Omega [subscript 1]] --
(e). The Number of Ones in a Binary Expansion --
(f). Coding Sequences --
(g). Syntactical Concepts --
(h). Formalizations Based on Context-Free Grammars --
4. Witnessing Functions --
(a). Introduction --
(b). Fragments of Bounded Arithmetic --
(c). Definability of Turing Machine Computations in Fragments of Bounded Arithmetic --
(d). Witnessing Functions --
(e). On the Finite Axiomatizability of Bounded Arithmetic --
5. Interpretability and Consistency --
(a). Introduction --
(b). Truth Definitions for Bounded Formulae --
(c). An Interpretation of I[Sigma [subscript 0]] in Q --
(d). Cut-Elimination and Herbrand's Theorem in Bounded Arithmetic --
(e). The II[subscript 1] Theorems of I[Sigma [subscript 0]] + Exp --
(f). Incompleteness Theorems --
(g). On the Limited Use of Exponentiation --
Bibliographical Remarks and Further Reading --
Bibliography --
Index of Terms --
Index of Symbols.
Series Title: Perspectives in mathematical logic.
Responsibility: Petr Hájek, Pavel Pudlák.
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