The Hilbert programme was essentially an extension of the work on which he had started in the 1890s.
希尔伯特的计划，本质上是他19世纪90年代开始的工作的拓展。
It did not attempt to answer the question which Frege and Russell had tackled, that of what mathematics really was.
它并不急于解答弗雷格和罗素的问题，也就是数学到底是什么。
In that respect it was less philosophical, less ambitious.
一方面，这样就不那么哲学化，不那么让人吃力。
On the other hand, it was more far-reaching in that it asked profound and difficult questions about the systems such as Russell produced.
另一方面，罗素的那个艰难的困境实际上很难解决。
In fact Hilbert posed the question as to what were, in principle, the limitations of a scheme such as that of Principia Mathematica.
希尔伯特提出的问题主要是，在原则上，《数学原理》的限制是什么。
Was there a way of finding out what could, and what could not, be proved within such a theory?
有没有一种方法，来判断什么可以被这套理论证明，而什么不可以。
Hilbert's approach was called the formalist approach, because it treated mathematics as if a game, a matter of form.
希尔伯特的方法，叫作形式主义，它把数学看成一套形式规则。
The allowable steps of proof were to be considered like the allowable moves in a game of chess, with the axioms as the starting position of the game.
允许的证明步骤，就好比国际象棋中允许的走法，而公理就好比是开局时的摆法。
In this analogy, 'playing chess' corresponded to 'doing mathematics', but statements about chess (such as 'two knights cannot force checkmate') would correspond to statements about the scope of mathematics.
在这个类比中，"下国际象棋"就相当于"做数学"，只是把国际象棋的命题（比如『两个马将不死对方』）换成数学命题。
And it was with such statements that the Hilbert programme was concerned.
希尔伯特计划，就是考虑这样的命题。
At that 1928 congress, Hilbert made his questions quite precise.
在1928年的大会上，希尔伯特明确提出了他的问题。
First, was mathematics complete, in the technical sense that every statement (such as 'every integer is the sum of four squares') could either be proved, or disproved.
第一，数学是完备的吗？是不是每个命题（比如『任意自然数都是四个平方数的和』）都能证明或证伪。
Second, was mathematics consistent, in the sense that the statement '2 + 2 = 5' could never be arrived at by a sequence of valid steps of proof.
第二，数学是相容的吗？也就是说，用符合逻辑的步骤和顺序，永远不会推出矛盾的命题，比如2+2 = 5。
And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true.
第三，数学是可判定的吗？他的意思是，是否存在一个机械式的方法，可以应用于任何命题，然后自动给出该命题的真假。
In 1928, none of these questions was answered.
在1928年，这些问题都不能得到解答。
But it was Hilbert's opinion that the answer would be 'yes' in each case.
但希尔伯特的观点是，每个回答都将会是“是”。
In 1900 Hilbert had declared 'that every definite mathematical problem must necessarily be susceptible of an exact settlement … in mathematics there is no ignorabimus';
早在1900年，希尔伯特宣布"所有数学问题都是有解的……没有数学照耀不到的角落"。
and when he retired in 1930 he went further:
当他1930年退休时，他研究得更深入了：
In an effort to give an example of an unsolvable problem, the philosopher Comte once said that science would never succeed in ascertaining the secret of the chemical composition of the bodies of the universe.
举一个不可解问题的例子来说，哲学家孔德曾经认为，科学永远无法给出宇宙的化学成分。
A few years later this problem was solved.…
但没过几年，这个问题就被解决了……
The true reason, according to my thinking, why Comte could not find an unsolvable problem lies in the fact that there is no such thing as an unsolvable problem.
在我看来，孔德找到一个不可解的问题的真正原因在于，这种不可解的问题压根就不存在。
It was a view more positive than the Positivists.
这个观点，比实证主义者还要激进。
But at the very same meeting, a young Czech mathematician, Kurt Gdel, announced results which dealt it a serious blow.
但就在这同一个大会上，一个年轻的捷克数学家，柯特·哥德尔的宣布，给了他当头一击。
Gdel was able to show that arithmetic must be incomplete: that there existed assertions which could neither be proved nor disproved.
哥德尔能够证明，算术一定是不完备的：存在既不能证明，也不能证伪的命题。
He started with Peano's axioms for the integers, but enlarged through a simple theory of types, so that the system was able to represent sets of integers, sets of sets of integers, and so on.
他从皮亚诺的整数公理开始，经过集合层次理论的拓展，使这个系统可以代表整数的集合、整数的集合的集合，等等。
However, his argument would apply to any formal mathematical system rich enough to include the theory of numbers, and the details of the axioms were not crucial.
总之，他的论点可以应用到任何涵盖了算术公理的形式系统，与其公理本身的内容无关。