到目前为止在学术界任何质疑哥德尔不完备性定理证明的话题都是禁忌，所以我希望借与Chapgpt的对话，引起学术界对此议题的关注。

Chapgpt的回答在某种程度上代表了学术界的主流观点，比如，Chapgpt说，罗素悖论被视为对数学基础的直接威胁，因为它破坏了集合作为所有数学基础的想法；但说谎者悖论却不是这样的。

如果我没弄错的话，在D. Hilbert and W. Ackermann 的书Principles of Theoretical Logic的第三章，“判定问题（Entscheidungsproblem）”被指定为数理逻辑的主要问题(简介Hilbert and Ackermann’s Principles of Theoretical Logic)。哥德尔对此回答是，“说谎者悖论”是数理逻辑中的“不可判定命题”，由此否定性回答了Hilbert的“判定问题”。

我想辩论的议题是：

- “说谎者悖论”是数理逻辑的“不可判定命题”吗？如果不是，哥德尔的证明就误导了人们对“判定问题”的认知，这难道这不是对数学基础的直接威胁吗？

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分享我今天用英语问Chapgpt的这个问题：

Yu：

When Russell's paradox appeared in set theory, it was considered a crisis, and people worked hard to eliminate it. Then, when the Liar's Paradox appeared in Gö- del's proof, why didn't people have the same vigilance and instead took the paradox proposition as a true arithmetic statement ? 

当罗素悖论出现在集合论中时，它被认为是一个危机，人们努力工作以消除它。那么，当说谎者悖论出现在哥德尔的证明中时，为什么人们没有同样的警惕性，而是把悖论命题当作一个真正的算术语句？

ChatGpt：

There are several reasons why the response to Russell's paradox in set theory was different from the response to the Liar's paradox in Gödel's proof.

在集合论中对罗素悖论的反应与在哥德尔证明中对说谎者悖论的反应不同，这有几个原因。

First, Russell's paradox was seen as a direct threat to the foundations of mathematics, as it undermined the idea of sets as a basis for all of mathematics. The paradox showed that naive set theory, which was the prevailing view at the time, led to contradictions and inconsistencies. This spurred mathematicians to develop new, more rigorous foundations for mathematics, such as Zermelo-Fraenkel set theory.

首先，罗素悖论被视为对数学基础的直接威胁，因为它破坏了集合作为所有数学基础的想法。悖论表明，当时的主流观点—朴素集合论导致了矛盾和不一致之处。这促使数学家们开发新的、更严格的数学基础，如策梅洛-弗拉恩克尔集合理论。

On the other hand, the Liar's paradox was seen as a challenge to logic and semantics, rather than a direct threat to the foundations of mathematics. It did not immediately lead to contradictions or inconsistencies in mathematics, but rather raised questions about the limits of formal systems and the nature of truth. As a result, the response to the Liar's paradox was more focused on exploring its implications for logic and semantics, rather than on immediately trying to eliminate it.

另一方面，说谎者悖论被认为是对逻辑学和语义学的挑战，而不是对数学基础的直接威胁。它并没有立即导致数学中的矛盾或不一致，而是提出了关于形式系统的限制和真理的性质的问题。因此，对说谎者悖论的回应更侧重于探索它对逻辑学和语义学的影响，而不是立即试图消除它。

Furthermore, the Liar's paradox is a more subtle and complex paradox than Russell's paradox. It involves self-reference and the concept of truth, which are inherently difficult to reason about. While there were attempts to solve the Liar's paradox, such as the development of paraconsistent logics, these solutions were not widely accepted or adopted in the same way that Zermelo-Fraenkel set theory was after Russell's paradox.

此外，说谎者悖论是一个比罗素悖论更微妙和复杂的悖论。它涉及到自指和真理的概念，而这些概念本身是很难推理的。虽然有人试图解决说谎者悖论，如发展准一致性逻辑，但这些解决方案并没有像罗素悖论之后的策梅洛-弗拉恩克尔集合理论那样被广泛接受或采纳。

In summary, the response to Russell's paradox in set theory was more urgent and immediate, as it threatened the very foundations of mathematics. The response to the Liar's paradox in Gödel's proof was more focused on exploring its implications for logic and semantics, and the paradox itself was more subtle and difficult to solve.

总之，对集合论中的罗素悖论的反应更为迫切和直接，因为它威胁到数学的基础。对哥德尔证明中的说谎者悖论的反应更侧重于探索其对逻辑和语义的影响，而且悖论本身更微妙，更难解决。