科学与数学stopa.io编辑精选
哥德尔发现了什么
What Gödel Discovered
2020 — 2022，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
这篇文章是 stopa.io 上关于哥德尔不完备定理的通俗讲解。作者用类似 Lisp 的代码和逐步构造来展示编码、自指与证明思路，让形式逻辑的抽象论证变得可操作。
为什么经久不衰：它把高度抽象的证明转成程序员能跟读的代码，同时被数学从业者评价为准确、少常见误解，所以多年后仍是值得读的入门讲解。
HN 评论精选
数学从业者称赞这是准确且平衡的通俗讲解，连原始证明都做了易读概述。
我是一名数学家，花了很多时间思考哥德尔定理及相关主题。作者自谦说不是数学家只是程序员，别在意：这是我见过的哥德尔定理最佳通俗阐述之一。所有内容解释得很准确，没有常见于哥德尔定理语境中的愚蠢错误和不实说法，甚至以非常平易近人的方式勾勒了原始证明，在传达所有关键思想和跳过大多数分散注意力的技术细节之间取得了很好的平衡（这些细节重要有趣，但我认为只对数学家来说，哈哈）。整体非常出色。
xyzzyz查看英文原文
I’m a mathematician who spent many hours thinking about Goedel theorem and adjacent topics. The author in a self-deprecating way says that he’s not a mathematician, but just a programmer. Bear no mind to that: this is one of the best popular expositions of Goedel theorem I’ve seen. Everything is very accurately explained, there are no silly mistakes and untruths one often sees mentioned in context of Goedel theorem, and even the original proof is sketched in a very approachable manner, striking a very good balance between getting across all crucial ideas, and skipping most of the distracting technical aspects (which are important and interesting, but I think only to mathematicians, haha). Great job all around.
入门讲解应避免“真”概念，并应强调系统不能证明更强系统的一致性，而非自身一致性。
我想回复过但可能没成功，回复变成了顶层回复，也好。我只想提两个小批评，免得其他读哥德尔入门文章的人掉进同样的陷阱：1. 请不要在入门解释中引入“真”的概念（比如在“数的力量”一节或希尔伯特计划的简短介绍中）。考虑到哥德尔的另一项里程碑结果——完备性定理，它同时适用于许多相同的逻辑系统，可以非常模糊地描述为“所有为真的事情都可证明”，引入真会让人非常困惑。坚持在句法层面谈不完全性和一致性就好，即总能添加新公理的能力，以及不存在句法矛盾。2. 一个系统 S 不能证明自身一致性本身并不有趣，因为我们不会信任一个可能不一致的系统给出的证明。因此，需要一座越来越强的系统之塔来证明 S 和它们自身的一致性，从某种意义上说并不那么有趣。有趣的方向相反：S 不能证明更强系统的一致性，这意味着我们不能有一个用来探索所有其他系统一致性的“最小”系统（当然还有些细微之处……）。
dwohnitmok查看英文原文
I thought I replied to this post, but I guess not and my reply ended up being a top-level reply, which is just as well. I will just mention my main two quibbles to this otherwise excellent post and others like it so that other people who embark on introductory posts to Godel's results don't fall into the same trap. 1. Please don't bring the notion of truth into an introductory explanation (such as in the section "Power of Numbers" or in the short intro to "Hilbert's Program"). It makes it very confusing, especially in light of Godel's other landmark result, i.e Godel's Completeness Theorem which simultaneously applies to many of the same logical systems and can be very very vaguely described as "all true things are provable." Just stick to incompleteness and consistency at a syntactic level instead of appealing to truth, that is respectively the ability to always add new axioms and the lack of syntactic contradictions. 2. The lack of a system S's ability to prove its own consistency is not interesting by itself since we would not trust a proof from a potentially inconsistent system. Hence it is not, in a sense, terribly interesting that we require an infinite tower of increasingly stronger systems to prove the consistency of S and themselves. Rather the interesting direction goes the other way. S cannot prove the consistency of stronger systems, which means we cannot have a "minimal" system we use to explore the consistency of all other systems (again subject to some nuance).
哥德尔不完备与停机问题同源，是自指悖论而非深刻意识局限。
人们表现得好像哥德尔不完备定理全是在说形式系统在“对真的洞察”上有限。但其实它和停机问题是同一个观察：你有一个试图推理其他系统行为的系统（如图灵机或公理集合论），它不可能总是内省自身行为，因为你也可以配置它根据自己对自己行为的结论而采取不同行为。比如我让你预测你会向左还是向右，但我又告诉你我会强迫你朝你预测的反方向走。那么你能准确预测的事情集合就是不完全的。但这不是人类意识的深刻局限，只是任何算命师或时间旅行场景里熟悉的悖论。
Liron查看英文原文
People act like Godel’s Incompleteness Theorem is all about how formal systems are limited in their level of “insight about truth”. But actually it’s just the same observation as the Halting Problem: you have a system that tries to reason about the behavior of other systems (like Turing Machines or axiomatic set theory), but it can’t possibly always introspect about its own behavior because you can also configure it to behave differently based on its own conclusion about its behavior. What if I ask you to predict whether you’re going to go left or right, but I also tell you that I’m going to force you to go the opposite direction as what you predict? Then the set of things you can accurately predict is Incomplete. But that’s not a profound limitation of human consciousness, it’s just the familiar paradox of any fortune teller or time travel scenarios.
Lisp 代码让非专业读者也能清楚理解困难主题。
这真的很出色。作者的 Lisp 代码把事情讲得清楚多了（即使对非数学家和不懂 Lisp 的人也是如此）。很难找到对困难主题解释得如此好的文章，我希望这成为一种趋势。
montebicyclelo查看英文原文
This is really excellent. Amazing how much clearer the author’s Lisp code makes things (even to a non mathematician and non lisper). It’s rare to find explanations of difficult topics that are this well communicated, but I hope it becomes a trend.
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