Book contents
- Frontmatter
- Contents
- PREFACE
- CHAPTER I INTRODUCTION
- CHAPTER II THE PREDICATE CALCULUS AND THE SYSTEM LK
- CHAPTER III PEANO ARITHMETIC AND ITS SUBSYSTEMS
- CHAPTER IV TWO-SORTED LOGIC AND COMPLEXITY CLASSES
- CHAPTER V THE THEORY V0 AND AC0
- CHAPTER VI THE THEORY V1 AND POLYNOMIAL TIME
- CHAPTER VII PROPOSITIONAL TRANSLATIONS
- CHAPTER VIII THEORIES FOR POLYNOMIAL TIME AND BEYOND
- CHAPTER IX THEORIES FOR SMALL CLASSES
- CHAPTER X PROOF SYSTEMS AND THE REFLECTION PRINCIPLE
- APPENDIX A COMPUTATION MODELS
- BIBLIOGRAPHY
- INDEX
CHAPTER III - PEANO ARITHMETIC AND ITS SUBSYSTEMS
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- PREFACE
- CHAPTER I INTRODUCTION
- CHAPTER II THE PREDICATE CALCULUS AND THE SYSTEM LK
- CHAPTER III PEANO ARITHMETIC AND ITS SUBSYSTEMS
- CHAPTER IV TWO-SORTED LOGIC AND COMPLEXITY CLASSES
- CHAPTER V THE THEORY V0 AND AC0
- CHAPTER VI THE THEORY V1 AND POLYNOMIAL TIME
- CHAPTER VII PROPOSITIONAL TRANSLATIONS
- CHAPTER VIII THEORIES FOR POLYNOMIAL TIME AND BEYOND
- CHAPTER IX THEORIES FOR SMALL CLASSES
- CHAPTER X PROOF SYSTEMS AND THE REFLECTION PRINCIPLE
- APPENDIX A COMPUTATION MODELS
- BIBLIOGRAPHY
- INDEX
Summary
Peano Arithmetic is the first order theory of ℕ with simple axioms for +, ·, ≤, and the induction axiom scheme. Here we focus on the subsystem IΔ0 of Peano Arithmetic, in which induction is restricted to bounded formulas. This subsystem plays an essential role in the development of the theories in later chapters: All (two-sorted) theories introduced in this book extend V0, which is a conservative extension of IΔ0. At the end of the chapter we briefly discuss Buss's hierarchy. These single-sorted theories establish a link between bounded arithmetic and the polynomial time hierarchy, and have played a central role in the study of bounded arithmetic. In later chapters we introduce their twosorted versions, including V1, a theory that characterizes P. The theories considered in this chapter are singled-sorted, and the intended domain is ℕ = {0, 1, 2, …}.
Subsection III.3.3 shows that the relation y = 2x is definable by a bounded formula in the vocabulary of IΔ0, and in Section III.4 this is used to show that bounded formulas represent precisely the relations in the Linear Time Hierarchy (LTH).
Peano Arithmetic
See Section II.2 for notions such as vocabulary, formula, and logical consequence.
Definition III.1.1. A theory over a vocabulary L is a set T of formulas over L which is closed under logical consequence and universal closure.
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- Logical Foundations of Proof Complexity , pp. 39 - 72Publisher: Cambridge University PressPrint publication year: 2010