## Handbook of Proof Theory (Google eBook)This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth. The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science. |

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### Contents

Chapter II FirstOrder Proof Theory of Arithmetic | 79 |

Chapter III Hierarchies of Provably Recursive Functions | 149 |

Chapter IV Subsystems of Set Theory and Second Order Number Theory | 209 |

Chapter V Gödels Functional Dialectica Interpretation | 337 |

Chapter VI Realizability | 407 |

Chapter VII The Logic of Provability | 475 |

Chapter VIII The Lengths of Proofs | 547 |

Chapter IX A ProofTheoretic Framework for Logic Programming | 639 |

Chapter X Types in Logic Mathematics and Programming | 683 |

Name Index | 787 |

797 | |

### Common terms and phrases

analysis apply assignment assume axiomatized boolean bounded arithmetic Buss classical clauses completeness Computer Science consistent constructive contains Corollary cut elimination cut-free deﬁnable deﬁned deﬁnition denote derivation disjunction equality equivalent expression Feferman ﬁnd ﬁnite set ﬁrst ﬁrst-order logic formal free variables Frege Frege systems function f function symbols Hence hierarchy implies incompleteness theorem induction hypothesis inductive deﬁnitions inference inﬁnitary inﬁnite interpretation intuitionistic intuitionistic logic iterated Journal of Symbolic language Lemma logic program lower bounds modal modiﬁed modus ponens natural numbers North-Holland notation notion Nuprl obtain operator ordinal ordinal analysis Peano arithmetic polynomial predicate primitive recursive functions proof system proof theory propositional logic propositional proof provability logic provably recursive prove Pudlak quantiﬁer-free quantiﬁers realizability relation symbols resolution rules satisﬁes semantics sentences sequent calculus set theory speciﬁc structure subformula substitution Suppose Symbolic Logic tautology transﬁnite true type theory uniﬁer valid