论文标题
正式的数学系统，包括结构性归纳原理
Formal Mathematical Systems including a Structural Induction Principle
论文作者
论文摘要
我们为正式数学系统提出了一个统一的理论，其中包括与形式语法密切相关的递归系统，包括谓词微积分以及正式的归纳原理。我们介绍了递归系统，生成了术语列表之间的递归枚举关系，即所考虑的基本对象。递归系统由公理组成，公理是特殊的无量词正喇叭公式和特定的推理规则。它扩展到形式数学系统会导致相对于基础递归系统的公理的形式结构诱导。这种方法在不使用人造和困难的解释技术的情况下提供了一些新的表示定理。在此框架内，我们还将为一类公理化的正式数学系统的Gödel的第一和第二不完整定理提供版本。
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating the recursively enumerable relations between lists of terms, the basic objects under consideration. A recursive system consists of axioms, which are special quantifier-free positive horn formulas, and of specific rules of inference. Its extension to formal mathematical systems leads to a formal structural induction with respect to the axioms of the underlying recursive system. This approach provides some new representation theorems without using artificial and difficult interpretation techniques. Within this frame we will also derive versions of Gödel's First and Second Incompleteness Theorems for a general class of axiomatized formal mathematical systems.
