《论数学的基础》一书是路德维希-维特根斯坦在1937-1944年期间写的关于数学哲学的笔记。

一，维特根斯坦对哥德尔不完备性证明的批评

其中特别有争议的是，维特根斯坦对哥德尔不完备性定理证明的不寻常评论，谈到“真”与“可证明”是相对论域而言的，“可证明与不可证明”的关系。

维特根斯坦在书中写道：

• I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system’. Must I not say that this proposition on the one hand is true, and on the other hand unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.”

• Just as we can ask, “ ‘Provable’ in what system?,” so we must also ask, “ ‘True’ in what system?” “True in Russell’s system” means, as was said, proved in Russell’s system, and “false” in Russell’s system means the opposite has been proved in Russell’s system.—Now, what does your “suppose it is false” mean? In the Russell sense it means, “suppose the opposite is been proved in Russell’s system”; if that is your assumption you will now presumably give up the interpretation that it is unprovable. And by “this interpretation” I understand the translation into this English sentence.—If you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up. If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell’s system. (What is called “losing” in chess may constitute winning in another game.)

二，哥德尔对维特根斯坦的回应

当哥德尔在维也纳时的旧相识门格尔向他指出，维特根斯坦身后出版的《论数学的基础》中有几节提到哥德尔时，哥德尔更多积郁的不满流露出来了

门格尔写道：

70年代早期我开始写一本回忆石里克小组的书。为了完整，我找了一下维特根斯坦发表的关于哥德尔的想法。在后来于1956年出来的《论数学的基础》中有一点。除了第五篇有点含糊的评论之外，第一篇的附录一…有对这个问题的讨论——然而，对哥德尔的工作并没有充分欣赏。事实上，维特根斯坦离谱地说不可判定性证明的唯一用途是“logische kunststiicken”(逻辑小窍门或戏法)。 门格尔将那几节指给哥德尔看后，哥德尔给门格尔回复道:

- 就我关于不可判定命题的定理而言，你引用的段落很清楚地表明维特根斯坦没有理解它（要么就是他假装不懂）。他将其看作是某种逻辑悖论，而事实上恰好相反，这是数学绝对无可争议的部分（有穷主义数论或组合数学）中的数学定理。顺便说一句，你引用的那一整段在我看来都是无稽之谈。例如什么“数学家对于矛盾的莫名恐惧”。

参考文献：

【1】https://en.wikipedia.org/wiki/Remarks_on_the_Foundations_of_Mathematics

【2】https://zhuanlan.zhihu.com/p/283714582