关于哥德尔1931的评论
Observations concerning Gödel's 1931
摘要：
本文论证了哥德尔1931年专著中定理VI的无效性，指出由定义（8.1）导出的命题（15）和（16）在其证明中在PA中是假的。这是通过两个步骤实现的。首先，通过添加新的关系Wid（x）来定义与著名的哥德尔谓词Bew（x）互补的谓词，并且相应地建立新的逻辑连接，引理（6）。其次，由定义（8.1）和引理（6）导出（15）和（16）的否定式。这等于说（15）和（16）是错误的，系统无法接受。因此，这两个著名的案例1。[17] Gen r不是k-PROVABLE，2。否定（17 Gen r）不是k-可证明的，不能被画出来，因此定理VI是无效的。
英文摘要：
This article demonstrates the invalidity of Theorem VI of Godel's monograph of 1931, showing that propositions (15) and (16), derived from definition (8.1), in its proof, are false in PA. This is achieved in two steps. First, the predicate complementary to the well-known Godel's predicate Bew(x) is ndefined by adding a new relation Wid(x), and new logical connections are accordingly established, Lemma (6). Second, the negations of (15) and (16) are derived by definition (8.1) and Lemma (6). It amounts to saying that (15) and (16) are false and unacceptable for the system. On the account of that, the two well-known cases 1. 17 Gen r is not k-PROVABLE, 2. Neg(17 Gen r) is not k-PROVABLE, can not be drawn, and Theorem VI is therefore invalid.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
