A Version of the Second Incompleteness Theorem For Axiom Systems that Recognize Addition But Not Multiplication as a Total Function
A Version of the Second Incompleteness Theorem For Axiom Systems that Recognize Addition But Not Multiplication as a Total Function
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公理系统第二不完备定理的一个版本，该系统将加法而非乘法视为全函数
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发表时间：
2004
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通讯作者：
D. Willard
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作者：
D. Willard
设A（x，y，z）和M（x，y，z）分别表示表示x + y = z和x ≠ y = z的谓词。假设一个公理系统α把加法和乘法都看作是全函数当且仅当它能证明：A（x，y，z）AND A（x，y，z）（1）我们将介绍第二不完全性定理的一些新的变形，这些公理系统把加法看作是“全”函数，但把乘法看作是一个3向关系。第二不完全性定理的这些推广是有趣的，因为我们先前的工作[30，32，34]已经探索了第二不完全性定理的几种类型的边界情况例外，当人们稍微进一步削弱我们的主要定理的假设时，就会发生这种情况。
Let A(x, y, z) and M(x, y, z) denote predicates indicating x + y = z and x ∗ y = z respectively. Let us say an axiom system α recognizes Addition and Multiplication both as Total Functions iff it can prove: ∀x∀y∃z A(x, y, z) AND ∀x∀y∃z M(x, y, z) (1) We will introduce some new variations of the Second Incompleteness Theorem for axiom systems which recognize Addition as a “total” function but which treat Multiplication as only a 3-way relation. These generalizations of the Second Incompleteness Theorem are interesting because our prior work [30, 32, 34] has explored several types of boundary-case exceptions to the Second Incompleteness Theorem that occur when one weakens the the hypothesis for our main theorems only slightly further.