## Principles of Mathematical LogicDavid Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. This translation is based on the second German edition and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Godel's completeness proof for the predicate calculus has been updated. In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic. |

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### Contents

THE SENTENTIAL CALCULUS | 3 |

Normal Form for Logical Expressions | 11 |

The Disj unctive Normal Form for Logical Expressions | 17 |

9 Systematic Survey of All the Deductions from Given | 23 |

Examples of the Proof of Theorems from the Axioms | 30 |

The Consistency of the System of Axioms | 38 |

THE CALCULUS 0F CLASSES MONADIC PREDICATE CALCULUS | 44 |

Inadequacy of the Foregoing Calculus | 55 |

Consistency and Independence of the System of Axioms | 87 |

Derivation of Consequences from Given Premises | 101 |

12 The Decision Problem | 112 |

1 The Predicate Calculus of Second Order | 125 |

2 Introduction of Predicates of Second Level Logical | 135 |

4 The Logical Paradoxes | 143 |

5 The Predicate Calculus of Order to | 152 |

Applications of the Calculus of Order w | 158 |

### Other editions - View all

Principles of Mathematical Logic David Hilbert,Norman John Berrill,Wilhelm Ackermann Snippet view - 1950 |

Principles of Mathematical Logic David Hilbert,Norman John Berrill,Wilhelm Ackermann Snippet view - 1950 |

Principles of Mathematical Logic David Hilbert,Norman John Berrill,Wilhelm Ackermann Snippet view - 1950 |

### Common terms and phrases

abbreviation argument places assertion axiom system bound variables cate combination of sentences combined calculus conjunctive normal form connectives contains corresponding Ct(P decision problem deﬁned deﬁnition disjunction disjunctive normal form domain of individuals dyadic predicate elementary sentences Entscheidungsproblem equivalent example existential quantiﬁers false sentence ﬁnally ﬁnd ﬁnite number ﬁrst free variables further given holds inﬁnite logical formulas Math mathematical mathematical logic means monadic predicates mulas negation obtain predicate constants predicate F predicate variables predicates of second preﬁx prenex normal form primitive formula proof proved rational numbers real numbers relation replaced restricted predicate calculus Rule a3 Rule of Implication rules of inference satisﬁability second level second-level predicate sentential calculus sentential combination sentential variables Skolem normal form sufﬁcient syllogisms system of axioms theorem Theorem 34 tion transformed true formula true sentence truth value universal and existential universal quantiﬁer universally valid formulas variables occur versally valid