## Elements of IntuitionismThis is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an informal but thorough introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics has been completely revised for this second edition. Brouwer's proof of the Bar Theorem has been reworked, the account of valuation systems simplified, and the treatment of generalized Beth Trees and the completeness of intuitionistic first-order logic rewritten. Readers are assumed to have some knowledge of classical formal logic and a general awareness of the history of intuitionism. From reviews of the first edition: This excellent book can be recommended to the student of mathematics or philosophy wishing to get a comprehensive and reliable introduction to modern intuitionism. Zentralblatt fur Mathematik The relevance of this book extends well beyond the confines of the philosophy of mathematics, right to the centre of debates in the philosophy of language. Philosophical Books ...a valuable and original contribution to the philosophy of mathematics. Journal of Symbolic Logic |

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### Contents

Introductory remarks | 1 |

Elementary intuitionistic mathematics | 22 |

Choice sequences and spreads | 40 |

The formalization of intuitionistic logic | 88 |

The semantics of intuitionistic logic | 119 |

Some further topics | 211 |

Concluding philosophical remarks | 250 |

311 | |

325 | |

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### Common terms and phrases

A V B Aa,fc applied arithmetic assert assume atomic formula Axiom of Choice axiom schema Bar Induction basic sequent Beth trees canonical proof choice sequences classical logic closed formula condition consider constructive functions Continuity Principle counter-example decidable defined definition derived determined domain dual sequence dual tree effective elements equivalent extensional Fa,fc fact Fan Theorem finite sequences follows free variables given Harrop formula hence Heyting lattice holds induction hypothesis infinite initial segment internal interpretation intuitionistic logic intuitionistic mathematics intuitive Kripke trees lawless sequences lawlike Lemma logical constants mathematical statements means natural numbers node notion obtain operation path platonistic predicate logic predicate-letter premiss primitive recursive proof tree-trunk proof-tree provable prove quantifiers real number recognize restriction rules of inference satisfies semantics sentence sentence-letter sentential logic species suppose theory topological truth truth-value valid valuation system verified vertex Vx A(x