The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of solvability is commonly identified with the decision problem. Given this identification, there is not the slightest doubt that Hilbert’s conviction was falsified by Gödel’s proof and by the negative results for the decision problem. On the other hand, Gödel’s theorems do offer a solution, albeit a negative one, in the form of an impossibility proof. In this sense, Hilbert’s optimism may still be justified. Here I argue that Gödel’s theorems opened the door to proof theory and to the remarkably successful development of generalized as well as relativized realizations of Hilbert’s programs. Thus, the fall of absolute certainty came hand in hand with the rise of partially secure and reliable foundations of mathematical knowledge. Not all was lost and much was gained.

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从可解性到形式可判定性：重温希尔伯特的“非怀疑型”

本文的主题是希尔伯特的可解性公理，即希尔伯特通过有限数量的运算，对每个数学问题的可解性的信念。可解决性的问题通常与决策问题相关。有了这种识别，丝毫不会怀疑希尔伯特的信念被戈德尔的证明和决策问题的否定证伪了。另一方面，哥德尔定理以不可能证明的形式确实提供了一个解决方案，尽管它是一个否定的解决方案。从这个意义上讲，希尔伯特的乐观主义可能仍然是合理的。在这里，我认为，哥德尔定理为证明理论以及希尔伯特程序的广义和相对实现的显着成功发展打开了大门。从而，绝对确定性的下降与部分安全可靠的数学知识基础的兴起同时发生。并不是所有人都失去了，收获了很多。