Abstract

Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. I conclude with an examination of three possible stances towards extensions via ideal elements.

中文翻译：

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数学中的域扩展和理想元素†

摘要

每当给定的数学域被扩充以包括新元素时，数学中的域扩展就会发生。Manders 认为，领域扩展的重要案例的优势被存在闭包和模型完成的模型理论概念所捕获。我认为，在通过理想元素进行域扩展的特定情况下，曼德斯提出的解释是不够的。然后，我基于 Dedekind 的Habilitationsrede开发并形式化了一种不同的域扩展方法，Manders 的帐户与之进行了比较。最后，我通过理想元素检查了三种可能的扩展立场。