我对哥德尔证明也有过类似的震撼，甚至一度怀疑彭罗斯说得有道理——他认为人类心智因此必然是不可计算的。……但托克尔·弗朗岑（Törkel Franzén）写的《哥德尔定理：使用与滥用不完全指南》（Gödel's Theorem: An Incomplete Guide to its Use and Abuse）这本书非常擅长讲清楚一点：我们应该得出的结论其实相当有限，比如该证明根本不适用于算术之外的任何东西。
I felt similar amazement at Gödel's proof, and even suspect that Penrose must be on to something when he says the mind must therefore not be computable. . . . But the book Gödel's Theorem: An Incomplete Guide to its Use and Abuse by Törkel Franzén is very good at explaining that the conclusions we should draw are fairly limited, e.g. that the proof doesn't apply to anything other than arithmetic.
Gödel's theorem in particular lends itself to this sort of thing. For more mathematical schadenfreude, I like Franzen's "An Incomplete Guide to its Use and Abuse" http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/...
Hackers will be interested especially in the rather weak attempts to use Gödel to make all sorts of silly inferences about artificial intelligence and the nature of mind.
我不太清楚你为什么给它加引号。不过供参考（FWIW），这两本好书解开了我的大部分疑惑：Peter Smith 的《An Introduction to Gödel's Theorems》和 Torkel Franzén 的《Gödel's Theorem: An Incomplete Guide to Its Use and Abuse》
I'm not sure why you have it in quotes. But FWIW, most of my questions were cleared up by these two wonderful books: Peter Smith's An Introduction to Gödel's Theorems and Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse
只要技术人员愿意读，有大把材料可看。这个帖子里其他人提到了 Nagel 和 Newman，还有 Smullyan。这里再推荐两本：1) Smith 的《Introduction to Godel's Theorems》http://www.logicmatters.net/igt/ 是一本好书，该有的数学细节全有，而且也愿意探讨哲学层面。2) Franzen 的《Gödel's Theorem: An Incomplete Guide to Its Use and Abuse》http://www.ams.org/notices/200703/rev-raatikainen.pdf……
Lots of reading are available to technical people who are willing. People elsewhere in this thread have mentioned Nagel and Newman, and also Smullyan. Here are two more: 1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy. 2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf…
An informative overview which is not referenced in the article is Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. (https://www.goodreads.com/book/show/12830.G_del_s_Theorem).
I don't agree with that definition of "effectively axiomatized". Granted, that's what Wikipedia says here: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... But the reference given doesn't check out. Franzén's 2005 book Gödel's Theorem doesn't say anything about effective axiomatization on page 112. He does however give the following definition in his lecture notes at: https://books.
我对哥德尔证明也有过类似的震撼，甚至一度怀疑彭罗斯说得有道理——他认为人类心智因此必然是不可计算的。……但托克尔·弗朗岑（Törkel Franzén）写的《哥德尔定理：使用与滥用不完全指南》（Gödel's Theorem: An Incomplete Guide to its Use and Abuse）这本书非常擅长讲清楚一点：我们应该得出的结论其实相当有限，比如该证明根本不适用于算术之外的任何东西。
I felt similar amazement at Gödel's proof, and even suspect that Penrose must be on to something when he says the mind must therefore not be computable. . . . But the book Gödel's Theorem: An Incomplete Guide to its Use and Abuse by Törkel Franzén is very good at explaining that the conclusions we should draw are fairly limited, e.g. that the proof doesn't apply to anything other than arithmetic.
哥德尔定理尤其容易招来这种事。如果想看更多数学上的“看笑话/幸灾乐祸”乐趣，我推荐弗朗岑的《使用与滥用不完全指南》http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/... hacker 们会对书中那些试图借哥德尔之名对人工智能和心智本质进行各种荒谬推论的无力尝试特别感兴趣。
Gödel's theorem in particular lends itself to this sort of thing. For more mathematical schadenfreude, I like Franzen's "An Incomplete Guide to its Use and Abuse" http://www.amazon.com/Godels-Theorem-Incomplete-Guide-Abuse/... Hackers will be interested especially in the rather weak attempts to use Gödel to make all sorts of silly inferences about artificial intelligence and the nature of mind.
我不太清楚你为什么给它加引号。不过供参考（FWIW），这两本好书解开了我的大部分疑惑：Peter Smith 的《An Introduction to Gödel's Theorems》和 Torkel Franzén 的《Gödel's Theorem: An Incomplete Guide to Its Use and Abuse》
I'm not sure why you have it in quotes. But FWIW, most of my questions were cleared up by these two wonderful books: Peter Smith's An Introduction to Gödel's Theorems and Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse
只要技术人员愿意读，有大把材料可看。这个帖子里其他人提到了 Nagel 和 Newman，还有 Smullyan。这里再推荐两本：1) Smith 的《Introduction to Godel's Theorems》http://www.logicmatters.net/igt/ 是一本好书，该有的数学细节全有，而且也愿意探讨哲学层面。2) Franzen 的《Gödel's Theorem: An Incomplete Guide to Its Use and Abuse》http://www.ams.org/notices/200703/rev-raatikainen.pdf……
Lots of reading are available to technical people who are willing. People elsewhere in this thread have mentioned Nagel and Newman, and also Smullyan. Here are two more: 1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy. 2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf…
文中没有引用的一本干货满满的综述是托克尔·弗朗岑的《哥德尔定理：使用与滥用不完全指南》（https://www.goodreads.com/book/show/12830.G_del_s_Theorem）。
An informative overview which is not referenced in the article is Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. (https://www.goodreads.com/book/show/12830.G_del_s_Theorem).
我不同意对“有效公理化”（effectively axiomatized）的那种定义。诚然，维基百科这里是这么说的：https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... 但给出的参考来源对不上。弗朗岑2005年的书《哥德尔定理》第112页上压根没提什么有效公理化。不过他在讲义里确实给出了如下定义，见：https://books.
I don't agree with that definition of "effectively axiomatized". Granted, that's what Wikipedia says here: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... But the reference given doesn't check out. Franzén's 2005 book Gödel's Theorem doesn't say anything about effective axiomatization on page 112. He does however give the following definition in his lecture notes at: https://books.