论哥德尔不完全性定理证明中隐藏的悖论

• 作者： 张洪 ¹
• 作者单位：
  1.  机构_srph3ih2km
• 通讯作者： 张洪 
• 提交时间：2024-06-01

摘要: 哥德尔不完全性定理是现代逻辑中的重大发现，一直备受科学界、哲学界瞩目。然而自哥德尔不完全性定理提出至今，其证明的科学性、哲学意义一直备受质疑，特别是维特根斯坦把其看作是某种逻辑悖论。作者对哥德尔不完全性定理的证明提出了几点质疑，找出了证明中存在的具体悖论形式，指出其证明方法与罗素悖论的高度一致性， 认为并不存在“真而不可证”的命题，这有力地支持了维特根斯坦的观点。所谓的哥德尔公式是一个逻辑无效的循环公式，它在谓词W的有效定义域之外。因此，本文的结论是：哥德尔不完全性定理的证明建立在一个无效的公式之上，证明的矛盾源于哥德尔公式自身，不能归结为系统的不完全性，因而证明是错误的。

Abstract: The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regarded it as a certain logical paradox. In the article, the author raises several questions about the proof of the Gödel Incompleteness Theorem, finds out the specific paradox form in the proof, points out the high consistency of its proof method and Russell’s Paradox, and asserts that there is no true but unprovable proposition, which strongly supports Wittgenstein’s view. The so-called Gödel formula is a logically invalid circular formula, which is outside the effective definition domain of predicate W. Therefore, the conclusion of this paper is that the proof of the Gödel Incompleteness Theorem is based on an invalid formula, and the contradiction of the proof originates from the Gödel formula itself. Thus it cannot be attributed to the incompleteness of the system, so the proof is wrong.

• 来自： 张洪
• 分类： 哲学 >> 逻辑学
• 说明： 针对当年伟大的哲学大师维特根斯坦对哥德尔不完全性定理的批评与质疑， 作者找出了哥德尔定理证明中存在的具体悖论形式，这有力支持了维特根斯坦的观点。
• 稿件状态： 未投稿
• 引用： PSSXiv:202406.00225 (或此版本 PSSXiv:202406.00225V1)
  DOI:10.12451/202406.00225
  CSTR:32012.36.PSSXiv.202406.00225
• 推荐引用方式： 张洪.论哥德尔不完全性定理证明中隐藏的悖论.哲学社会科学预印本平台:https://zsyyb.cn/abs/202406.00225.[PSSXiv:202406.00225V1] (点此复制)