论文标题
直觉数学和逻辑
Intuitionistic Mathematics and Logic
论文作者
论文摘要
数学直觉主义的第一个种子在一个世纪以前在欧洲发芽，以Borel，Baire，Lebesque，Poincaré，Kronecker等的建设性趋势。开花是一个人，卢森·埃格伯斯·扬·布鲁瓦（Luitzen Egbertus Jan Brouwer）的作品，他从1909年至1951年在阿姆斯特丹大学教授数学。通过证明对拓扑不变性和连续映射的固定点的强大理论，布鲁维尔很快就足够的数学声誉来支持他的革命性革命性的数学构想。这些想法影响了希尔伯特和戈德尔，并确立了直觉的逻辑和数学作为值得独立研究的主题。 我们的目的是描述布鲁维尔直觉主义的发展，从拒绝被排除的中间法规到他有争议的连续体理论，对逻辑和数学产生了根本的后果。我们借用Kleene的正式公理系统（将Kolmogorov，Glivenko，Heyting和Peano的早期尝试纳入了直觉的逻辑和算术）作为相应古典理论的子理论，并勾勒出他对Gödel数量的使用数量的使用，以实现gödel数量，以实现直觉主义的句子，包括教堂的特征，包括一部分的特征。最后，我们介绍了克莱恩（Kleene）和维斯利（Vesley）对布鲁维尔（Brouwer）连续体的公理处理，并具有确定其一致性的功能性解释。
The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincaré, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan Brouwer, who taught mathematics at the University of Amsterdam from 1909 until 1951. By proving powerful theorems on topological invariants and fixed points of continuous mappings, Brouwer quickly build a mathematical reputation strong enough to support his revolutionary ideas about the nature of mathematical activity. These ideas influenced Hilbert and Gödel and established intuitionistic logic and mathematics as subjects worthy of independent study. Our aim is to describe the development of Brouwer's intuitionism, from his rejection of the classical law of excluded middle to his controversial theory of the continuum, with fundamental consequences for logic and mathematics. We borrow Kleene's formal axiomatic systems (incorporating earlier attempts by Kolmogorov, Glivenko, Heyting and Peano) for intuitionistic logic and arithmetic as subtheories of the corresponding classical theories, and sketch his use of gödel numbers of recursive functions to realize sentences of intuitionistic arithmetic including a form of Church's Thesis. Finally, we present Kleene and Vesley's axiomatic treatment of Brouwer's continuum, with the function-realizability interpretation which establishes its consistency.
