## The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions"A valuable collection both for original source material as well as historical formulations of current problems." — Mathematics of ComputationAn anthology of fundamental papers on undecidability and unsolvability by major figures in the field , this classic reference is ideally suited as a text for graduate and undergraduate courses in logic, philosophy, and foundations of mathematics. It is also appropriate for self-study.The text opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers by Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Supplementary items include corrections, emendations, and added commentaries by Godel, Church, and Kleene for this volume's original publication, along with a helpful commentary by the editor. |

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### Contents

On Formally Undecidable Propositions of | 4 |

On Undecidable Propositions of Formal | 39 |

On Intuitionistic Arithmetic and Number Theory | 75 |

On the Length of Proofs | 82 |

An Unsolvable Problem of Elementary Number Theory | 88 |

A Note on the Entscheidungsproblem | 108 |

Systems of Logic Based on Ordinals | 154 |

J B ROSSER | 222 |

Extensions of Some Theorems of Giidel and Church | 230 |

General Recursive Functions of Natural Numbers | 236 |

Recursive Predicates and Quantifiers | 254 |

Finite Combinatory Processes Formulation I | 288 |

Recursively Enumerable Sets of Positive Integers | 304 |

Absolutely Unsolvable Problems and Relatively | 338 |

407 | |