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Forcing in proof theory

机译:证明论

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摘要

Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.
机译:保罗·科恩(Paul Cohen)的强迫方法,以及索尔·克里普克(Saul Kripke)的模态和直觉逻辑的相关语义,对从集合论和模型论到构造逻辑和分类逻辑的许多数学逻辑产生了深远的影响。在这里,我认为强迫在传统的希尔伯特式证明理论中也占有一席之地,其目的是将普通数学的某些部分形式化为受限制的公理理论,并以建构或句法的方式研究这些理论。我将讨论在这方面有用的强制方面,以及一些示例应用程序。后者包括获得古典和直觉主义理论的保存结果,以建设性理论解释经典理论以及建构模型理论论证的方法。

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