## 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. |

### Contents

Introductory remarks | 1 |

Elementary intuitionistic mathematics | 26 |

Choice sequences and spreads | 40 |

4 | 68 |

5 | 75 |

4 | 85 |

5 | 106 |

The semantics of intuitionistic logic | 119 |

Some further topics | 211 |

Concluding philosophical remarks | 250 |

Bibliography | 311 |

325 | |

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### Common terms and phrases

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