# 与ChatGPT关于哥德尔不完备性定理的对话（3）

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。

- “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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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 ?

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.

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.

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.

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.

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.

https://blog.sciencenet.cn/blog-2322490-1381052.html

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