奎因哲学词条：哥德尔定理

奎因哲学词条：预言

（《奎因短文集：一本不拘泥于分类学标准的哲学词典》，译者：翟玉章）

译者按：哥德尔定理的哲学价值在于凸显了逻辑和数学之间的区别。逻辑归根到底是自明的，但数学归根到底不是自明的。哥德尔指出有些数学真理无法从公理（无论是否自明）中推导出来，因此是不自明的。数学更像自然科学而不是更像逻辑。至少，在集合论中，人们已经放弃了公理必须是自明的这一理想；公理成了假说，它的可信性，就像自然科学的假说一样，在于对自明的东西的蕴涵。数学本质上也是一项试错的事业。这并没有贬低逻辑，因为试错也不能违背逻辑；逻辑不再是数学的全部，但仍构成它和其他自然科学的底线。

Gödel's Theorem

哥德尔定理

【试错法和演绎法的鲜明对比。悖论对公理自明性的挑战。哥德尔不完全性定理：任何有效的演绎系统都不能覆盖初等数论中的所有真理。哥德尔数。附后、引用作为句法运算。从回顾角度看哥德尔定理。检验方法和完全性证明方法在数论中的等价性。】

本文提到的词条：哥德尔定理、悖论、费马大定理、使用与提及、递归

Mathematics, proverbially, is where proof is. Natural science goes forward by trial and error, and a good job too. But mathematics, one felt, proceeds inexorably from self-evident truths by infallible steps of inference. Euclid's Elements set the style.

众所周知，哪里有数学，哪里就有证明。自然科学是靠着试错法向前发展的，而且发展得很不错。但人们认为数学的方法与此不同；数学的发展过程是一个从自明的真理出发，通过不可错的推理步骤而一往无前的过程。欧几里得的《几何原本》奠定了这样的风格。

GOOGLE: As we all know, wherever there is mathematics, there is proof. Natural science develops by trial and error, and it does so very well. But people think that the method of mathematics is different from this; the development process of mathematics is a process of starting from self-evident truths and moving forward through infallible reasoning steps. Euclid's Elements established this style.

Self-evidence turned out to be too much to ask. Euclid's postulate of parallels, for one, resisted that requirement. Set theory, again, shaken by PARADOXES, settles pragmatically for a serviceable set of postulates without further aspiration to self-evidence.

事实证明，自明性是一个过高的要求。欧几里得的平行公设就达不到自明性的要求。集合论提供了另外一个例子；它在悖论的打击下变得务实了，只满足于一组有用的公设，而不再要求它们具有自明性。

GOOGLE: It turns out that self-evidence is too high a requirement. Euclid's parallel postulate fails to meet the requirement of self-evidence. Set theory provides another example; it has become pragmatic under the attack of PARADOXES, contenting itself with a set of useful postulates without requiring them to be self-evident.

But self-evidence could still reasonably be expected in the foundations of pure arithmetic, or elementary number theory. Self-evidence indeed already prevailed in the little set of axioms for number theory that is customarily credited to Dedekind and customarily named for Peano. There remained a question of adequacy: perhaps there were valid laws, expressible wholly within the notation of elementary number theory, that could not be derived from the Peano axioms. There were. But presumably the axioms could be completed by further additions, self-evident or not. The same was presumed, as a matter of course, for any other branch of mathematics.

但人们仍然合乎情理地期望，在纯算术的基础（即数论）中，公理仍应该是自明的。那时，数论已经被公理化了，而且其中的那组公理（通常归功于戴德金而用皮亚诺的名字命名）确实是自明的。但关于这组公理，存在着一个充分性的问题：也许有一些有效定律，虽然能够用初等数论的语言加以表达，但却不能从皮亚诺公理中推导出来。确实有这样的有效定律。不过也许，公理组添入一些自明或不自明的新公理后就完备了。人们想当然地认为，不但数论如此，其他数学分支也如此。

GOOGLE: But it is still reasonable to expect that the axioms of the foundations of pure arithmetic (i.e., number theory) should still be self-evident. By then, number theory had been axiomatic, and the set of axioms within it (usually attributed to Dedekind and named after Peano) was indeed self-evident. But there is a sufficiency problem regarding this set of axioms: there may be some valid laws that, although they can be expressed in the language of elementary number theory, cannot be derived from Peano's axioms. There are indeed such valid laws. Perhaps, however, the set of axioms would be complete by adding some new axioms, whether self-evident or not. People take it for granted that this is true not only of number theory but also of other branches of mathematics.

No such luck. In his ground-breaking, bond-breaking, road-breaking, epoch-making theorem of 1931 Kurt Gödel proved that a complete deductive system was impossible for even so modest a fragment of mathematics as elementary number theory.

但这只是一厢情愿。库尔特·哥德尔于1931年提出了他的具有划时代意义的开创性定理：即使对于数论——数学中的一个相当小的部分——来说，完全的演绎系统也是不可能的。

GOOGLE: But this is just wishful thinking. Kurt Gödel proposed his epoch-making seminal theorem in 1931: Even for number theory, a fairly small part of mathematics, a complete deductive system is impossible.

Several points cry out for clarification. First, the scope of elementary number theory. The numerals and the notations for plus, times, power, and equality are there, along with the {83} sentence operators 'not', 'and', and 'or' and the quantifiers 'every number x is such that ... ' and 'there is a number x such that ... '. The numbers concerned are just the positive integers and zero. This formal language is adequate for expressing such homely truths as

For all numbers x and zthere is a number y such that

x + y = z or x = y + z

and indeed all the rest of the arithmetical truths that one thinks of as elementary. It can also express FERMAT'S LAST THEOREM.

有几点需要澄清一下。首先是初等数论的范围。其中有数字，有加法、乘法、乘方、等式的符号，还有语句算子“并非”、“而且”和“或者”，还有量词“所有数都是如此这般的x”、“有些数是如此这般的x”。所涉及的数仅限于正整数和零。这个形式语言足以表达算术中常见的真理，例如：

对所有的数x和z来说，都存在一个如此这般的数y：x + y= z 或x = y + z

事实上，它能表达所有人们所认为的初等数学真理。它也能表达费马大定理。

GOOGLE: There are several points that need to be clarified. The first is the scope of elementary number theory. There are numerals in it, and there are symbols for addition, multiplication, exponentiation, and equality, as well as sentence operators "not", "and" and "not", as well as quantifiers "all numbers are x such that", "some number is x such that”. The numbers involved are limited to positive integers and zero. This formal language is sufficient to express common truths in arithmetic, such as:

For all numbers x and z, there exists a number y such that

x + y = z or x = y + z

In fact, it can express all that is thought to be elementary truths in mathematics. It can also express FERMAT'S LAST THEOREM.

What Gödel proved, then, is that no axiom system or other deductive apparatus can cover all the truths expressible even in that modest notation; any valid proof procedure will let some true statements, indeed infinitely many, slip through its net.

因此，哥德尔所证明的是：即使是用这种相当朴素的语言来表达的真理，任何公理系统或其他演绎工具也不能将它们全部涵盖；任何有效的证明方法都会漏掉一些（实际上在数量上有无穷多）真语句。

GOOGLE: So what Gödel proved is that no axiomatic system or other deductive tool can cover all of the truths expressed in this rather plain language; any valid proof procedure will miss some (indeed infinitely many) true sentences.

Further clarification is now called for. Since we have given up on self-evidence, why not just take all those truths as axioms? Is it because there are infinitely many? No, not exactly. Some infinite sets of axioms are admissible, and anyway there is no requirement that our deductive apparatus even consist of axioms and rules of inference. What is required is just that proofs admit of being checked. The rules of proof must be so formulated that a computer could be programmed to scan a purported proof, line by line, and determine whether it qualified as a proof of its bottom line.

这里又要做进一步的澄清了。既然我们已经放弃了对公理的自明性要求，何不把这些不能被证明的真理当作公理呢？是因为它们的数量有无穷多吗？不，不完全是。有些无限公理集是可以接受的，而且无论如何，我们的演绎工具甚至无需由公理和推理规则所组成。证明的规则必须这样来制定：计算机可以通过程序，逐行扫描一个所谓的证明，并判断它是不是一个满足了最低要求的合格证明。

GOOGLE:Here again, further clarification is needed. Now that we have given up the requirement for self-evidence of axioms, why not treat these unprovable truths as axioms? Is it because there are infinitely many of them? No, not exactly. Some infinite sets of axioms are admissible, and in any case our deductive tool need not even consist of axioms and rules of inference. The rules of proof must be formulated in such a way that a computer can be programmed to scan a so-called proof line by line, and determine whether it is a qualified proof that meets the minimum requirements.

Gödel's theorem is akin to the reflexive PARADOXES. Its proof hinges on coaxing the notation of elementary number theory into talking, in effect, about itself. Thus suppose the notation has been regimented in symbolic logic together with the arithmetical signs. We can number the total stock of single characters -- digits, letters, logical signs, parentheses -- in an arbitrary order, using say just two-digit numbers, of which there are enough and to spare. We can then assign numbers to expressions generally -- strings of characters -- by concatenating the pairs of digits for the component characters. Thus if the characters 'x', '+', 'y', '=', and '6' are numbered 13, 10, 27, 21, and 47, then the equation 'x + y = 6' is numbered 1310272147. {84} Such is its Gödel number. Gödel's method was different, but no matter.

哥德尔定理类似于自反悖论。它的证明要领是设法用初等数论的语言构造一个实际上说自己的语句。让我们假定它的标记法和算术符号都已经经过符号逻辑的处理了。我们可以用绰绰有余的数字资源（比如两位数的数字）给所有单独的字符（数位、字母、逻辑符号、括号）人为地编号。然后，每一个表达式都将得到一个编号，方法是将其中所有字符的编号连接起来。举例说明：如果字符“x”、“+”、“y”、“=”和“6”的编号分别是13、10、27、21 和 47，则等式“x + y = 6”的编号为 1310272147。这就是“x + y = 6”的哥德尔数。哥德尔自己的配数法有所不同，但这是无关紧要的细节。

Gödel's theorem is similar to reflexive PARADOXES. Its proof is to try to use the language of elementary number theory to construct a statement that actually speaks of itself. Let us assume that its notation and arithmetic symbols have been processed by symbolic logic. We can artificially number all individual characters (one-digit numerals, letters, logical symbols, parentheses) with more than adequate numerical resources (such as two-digit numerals). Each expression is then given a number by concatenating the numerals of all the characters in it. Here is an example: if the characters "x", "+", "y", "=", and "6" have the numbers 13, 10, 27, 21, and 47 respectively, then the equation "x + y = 6" has the number 1310272147. This is the Gödel number of "x + y = 6". Gödel's method of numbering was different, but this is an inconsequential detail.

We begin to see that Gödel's proof will demand utter clarity over matters of USE VERSUS MENTION. The numeral '6' names the number 6 and is assigned the G6del number 47. If we think of the Gödel numbers as going proxy for the expressions to which they are assigned, then the number 47 may be seen as naming the number 6.

我们开始看到，哥德尔的证明要求在使用与提及上非常清晰。数字“6”是6的名称，而被配以数47。如果我们将哥德尔数看成它们所分配给的表达式的代理者，我们不妨说数47是数6的名称。

We begin to see that Gödel's proof requires complete clarity on USE VERSUS MENTION. The numeral "6" is the name of the number 6 and is paired with the number 47. If we think of Gödel numbers as proxies for the expressions to which they are assigned, we might as well say that the number 47 is the name of the number 6.

Now that all expressions have been assigned their Gödelnumbers, various syntactical operations on expressions can be mirrored by arithmetical operations on the numbers. Appending is a trivial example, on our approach: the appending of expression number 2147, namely '=6', to expression number 131027, namely 'x + y', is mirrored by the arithmetical operation on those numbers that gives 1310272147.

现在，既然所有表达式都被分配了它们的哥德尔数，所以对表达式的句法运算都可以有一个镜像运算，即对于数的运算。在我们的方法中，附后是一个简单的例子：将数2147这个表达式（“=6”这个表达式的镜像）附加到数131027这个表达式（“x+y”这个表达式的镜像）的后面，就得到数1310272147这个表达式（“x + y = 6”这个表达式的镜像）。

Now, since all expressions are assigned their Gödel numbers, syntactic operations on expressions can have mirrored operations on numbers. In our approach, appending a simple example: append the expression 2147 (the mirror image of the expression "=6") to the expression 131027 (the mirror image of the expression "x+y" Mirror), you get the expression 1310272147 (the mirror image of the expression "x + y = 6").

Quotation is a subtler matter. Since it does not appear in the notation of symbolic logic and arithmetic, its parallel in terms of Gödel numbers cannot simply be read off by Gödel numbering; it must be worked out from scratch. Now the quotation of an expression is meant to name that expression, just as a numeral '6' names the number 6. Proxy-wise, then, 47 names 6. Similarly, 47 is named in turn by 5361, if 53 and 61 happen to be the Gödel numbers of the characters '4' and '7'. Accordingly the quotation relation is represented by an arithmetical relation that 5361 bears to 47 and that 47 bears to 6. The general relation can be expressed in the notation of elementary number theory, though not easily. The arithmetical reconstruction of syntactical notions such as these was a substantial part of Gödel's achievement.

引用是一个更微妙的问题。由于引号并没有出现在符号逻辑和算术的记号系统中，因此与它对应的哥德尔数，就不能简单地通过哥德尔编号法获得解决，而只能另行解决。引用一个表达式意味着命名该表达式，就像数字“6”命名了数6一样。从代理的角度看，“6”的代理者，即数47，也命名了数6。同样地，5361命名了数47，如果53和61分别是字符“4”和“7”的哥德尔数的话。因此，引用关系是5361和47之间，以及47和6之间的那种算术关系。一般的引用关系在初等数论中也可以被表达出来，只是不那么容易。诸如此类的句法概念的算术重建，是哥德尔成就的重要组成部分。

GOOGLE: Quotation is a subtler issue. Since a quotation mark does not appear in the notation system of symbolic logic and arithmetic, the Gödel number corresponding to it cannot be solved simply through Gödel numbering, but can only be solved separately. To quote an expression is to name the expression, just as the numeral "6" names the number 6. From a proxy point of view, the proxy of "6", i.e., the number 47, also names the number 6. Likewise, 5361 names 47, if 53 and 61 are the Gödel numbers of the characters "4" and "7" respectively. Therefore, the relation of quotation is the relation between 5361 and 47, 47 and 6, etc. The general relation of quotation can also be expressed in elementary number theory, but it is not so easy. The arithmetic reconstruction of syntactic concepts such as these was an important part of Gödel's achievement.

My two examples -- appending and quotation -- ominously echo my formulation of the Liar Paradox near the end of the entry on PARADOXES. That paradox is instrumental to one of two parts into which the proof of Gödel's theorem may, in retrospect, be resolved. His bombshell detonates when the parts are joined.

我的两个例子——附后和引用——是对我在悖论结尾处对说谎者悖论的表述的不祥呼应。哥德尔定理从回顾的角度看可以分为两个部分，那个悖论有助于理解其中的一个部分。当这两部分会合时，他的重磅炸弹就会引爆。

GOOGLE: My two examples—appending and the quotation—are ominous echoes of my formulation of the Liar Paradox at the end of the entry onPARADOXES. In retrospect, Gödel's theorem can be divided into two parts and that paradox helps to understand one of them. When the two halves meet, his bomb will detonate.

My cited rendering of the Liar Paradox, q. v., can by Gödel {85} numbering and a measure of ingenuity be translated into the notation of elementary number theory in its entirety, except for one word: 'truth'. If we could accommodate 'truth' too, we would have accommodated the Liar Paradox in full, discrediting elementary number theory itself. We must conclude rather that truth for elementary number theory is not definable, via Gödel numbering, in elementary number theory. Spelled out: no formula in the notation of elementary number theory is true of all and only the Gödel numbers of truths of elementary number theory.This, innocuous in itself, is one of two parts of the bombshell.

我所援引的“说谎者悖论”版本，可以借助于哥德尔编号法和一些技巧，被整个地翻译为初等数论的语言，但“真”这个词例外。如果没有这个例外，这个悖论就可以完全翻译为初等数论的语言，从而证伪初等数论。我们必须得到这个结论：初等数论的真理性，在（经过哥德尔编号法处理过的）初等数论中是无法定义的。更具体地：用初等数论的语言所表达的任何公式，都不正好对于所有初等数论真理的哥德尔数是真的。这个结论本身是无害的，但却是重磅炸弹的两个部分之一。

GOOGLE: The version of the "Liar's Paradox" that I have quoted can be translated entirely into the language of elementary number theory with the help of Gödel's numbering and some dexterity, with the exception of the word "true". Without this exception, this paradox can be fully translated into elementary number theory, thus constituting its falsification. We must come to this conclusion: the truth of elementary number theory cannot be defined in elementary number theory (processed by Gödel's numbering). More specifically: any formula expressed in the language of elementary number theory is not true of all and only those Gödel numbers that correspond to the truths in elementary number theory. The conclusion itself is innocuous, but it is one of two parts of the bombshell.

The other part treats of any proper proof procedure, that is again, any whose proofs can be checked, as by a programmed computer. Gödel shows that, given any such proof procedure, a certain formula in the notation of elementary number theory is true of all and only the Gödelnumbers of provable formulas[1].

另一部分讨论任何的适当证明方法。再说一遍，适当证明方法是指其中的每一个证明都可由程序化的计算机加以检验的方法。哥德尔表明：对于任何这样的证明方法，一定存在着某个用初等数论语言表达的公式，它刚好对所有可证明的公式的哥德尔数是真的。

GOOGLE: The other part discusses any proper proof method. Again, a proper proof method is one in which every proof can be checked by a programmed computer. Gödel shows that: for any such proof method, there must be some formula expressed in the language of elementary number theory which is true of all and only the Gödel numbers of provable formulas.

Combining the two parts, we conclude that the provable formulas do not coincide with the truths of elementary number theory. Either they include some falsehoods, God forbid, or they miss some truths.

结合这两部分，我们得出结论：可证明的公式并不与初等数论的真理相重合。它们要么包括一些假语句（但愿不是如此），要么会漏掉一些真理。

GOOGLE: Combining these two parts, we conclude that provable formulas and the truths of elementary number theory do not coincide. They either contain some falsehoods (hopefully not) or they leave out some truths.

Gödel's actual proof took a more direct line. He showed how, given any proper proof procedure, to construct a sentence in the notation of elementary number theory that says of itself, in effect, via Gödel numbering, that it cannot be proved. Either it is false and provable, God forbid, or true and not provable; presumably the latter. One could strengthen the proof procedure by adding this stray truth as a further axiom; but, by a repetition of the argument, there will always be others.

哥德尔的实际证明更为直接。他表明，在任何适当的证明方法下，如何用经过哥德尔编号法处理过的初等数论语言构造一个谈论自己的语句；这个语句的大意是说：它自己无法被证明。显然，这句话要么是假的、可证明的（但愿不是如此），要么是真的、不可证明的（但愿如此）。人们可以将这个不可证明的真理添加为公理，从而强化证明方法；但通过重复上面的论证就会得出结论：总还会有其他不可证明的真理[2]。

GOOGLE: Gödel's actual proof was more direct. He shows how, given any proper method of proof, one could construct a statement about itself in the language of elementary number theory processed by Gödel's numbering; a statement to the effect that it could not be proved. Obviously, this statement is either false and provable (hopefully not) or true and unprovable (hopefully so). One could add this unprovable truth as an axiom, thus strengthening the method of proof; but by repeating the above argument one would conclude that there are always other unprovable truths.

There is a final irony worth noting. It was implausible all along that a general mechanical routine could exist for checking for truth or falsity any and every statement in the notation of elementary number theory. Such a test would settle FERMAT'S {86}LAST THEOREM, along with a bundle of other stubborn conjectures. On the other hand Gödel's result, that there could not ever be a complete proof procedure, came as a bombshell. Now the irony is that the two lacks, the expected one and the astonishing one, could easily have been shown to be equivalent. For, if there were a complete proof procedure, then, as Stephen Kleene has observed, any statement could be tested for truth or falsity as follows. Program your computer to grind out expressions of the sort that qualify as proofs under the given proof procedure. Have it grind out the shortest ones first, in alphabetical order, and then continue into increasing lengths. Eventually, thanks to the completeness, it will print a proof of the statement in question if it is true, and of its negation otherwise. This is a finite test. It would be unfeasibly long for the fastest computer, but the point is theoretical.

最后，还有一个有讽刺意味的情况值得注意。下面这一点一直都是令人难以置信的：初等数论里的每一个陈述，其真假都可以通过一个通用的机械方法加以检验。这样的检验将解决包括费马大定理在内的一系列棘手的猜想。另一方面，哥德尔的结论——不存在完全的证明方法——却是一枚重磅炸弹。这里的讽刺意味就在于：这两种缺乏，虽然一个在意料之中，一个在意料之外，却可以很容易地被表明是等价的。因为，如果有一个完全的证明方法，那么，正如斯蒂芬·克林所指出的那样，任何语句的真假都可以按下面的方法来检验。给你的计算机编程，使之找出给定证明方法之下那些可视为合格证明的表达式。首先，按字母顺序找出那些最短的表达式，然后找出越来越长的表达式。由于证明方法的完全性，计算机终将会打出所讨论的语句（或它的否定句）的真理性的证明。这是一个有限步骤的检验。当然，即使在最快的计算机上，这个检验也会慢得令人无法忍受，我是为了理论的目的才指出这种检验方法的。

GOOGLE: Finally, there is an irony worth noting. It has always been incredible that every statement in elementary number theory can be tested for truth or falsity by a general mechanical method. Such a test would resolve a series of thorny conjectures, including FERMAT'S LAST THEOREM. On the other hand, Gödel's conclusion—that there is no complete method of proof—is a bombshell. The irony here is that these two deficiencies, although one is expected and the other is unexpected, can easily be shown to be equivalent. Because, if there is a complete method of proof, then, as Stephen Kleene points out, the truth or falsity of any statement can be tested as follows. Program your computer to find expressions that qualify as proofs under the given proof method. First, find those shortest expressions in alphabetical order, then find longer and longer expressions. Due to the completeness of the proof method, the computer will eventually produce a proof of the truth of the statement (or its negation) in question. This is a test in finite steps. Of course, this test would be unbearably slow even on the fastest computer, and I point it out for theoretical purpose.

附：奎因在《逻辑方法》中对哥德尔对哥德尔定理的证明过程的概括

Gödel’s way was to argue directly to the incompletability of elementary number theory, by showing how, for any given proof procedure P for elementary number theory, a statement S_P of elementary number theory can be constructed which will be true if and only if it is not provable by the procedure P. Either S_P is provable, in which case it is false and so the general proof procedure P is discredited, or else S_P is true and not provable, in which case the proof procedure P is incomplete.

哥德尔对初等数论不完全性的证明是直接的；他表明，对任何初等数论证明方法P，我们都可以构造出这样一个初等数论的陈述S_P：它是真的，当且仅当它在P中是不可证的。S_P要么是可证的，但却是假的，这说明P被证伪了；要么是真的，但却是不可证的，这说明P是不完全的。

In broadest outlines, Gödel’s construction of S_Pis as follows. It is easy to assign integers in a systematic way to all the finite strings, however long, of signs of a given alphabet. If the alphabet has just nine signs, we can assign them the integers from 1 to 9, and then we can get the integer for any string of signs by concatenating the corresponding digits into a long numeral. If the alphabet has more than nine signs, we can easily adapt this method or choose another.Suppose this done for the notation of elementary number theory. Thus each sentence in that notation has its so called Gödelnumber. Gödelthen shows that, given P, it is possible within the notation of elementary number theory to formulate an open sentence, ‘…x…’, say, which is true of any number x if and only if x is a Gödelnumber of a statement provable by P. If we put for ‘x’ in ‘-(…x…)’ the actual numeral designating some particular number n, clearly the resulting statement, schematically ‘-(…n…)’, will be true if and only if that chosen n is not the Gödelnumber of a statement provable by P, but Gödel shows that n can be so chosen as to turn out to be the Gödelnumber of a statement equivalent to‘-(…n…)’ itself. The statement produced by that sly choice of n is the sought S_P, true if and only if not provable by P.

概括地说，哥德尔对S_P的构造如下。由给定符号表中的符号所组成的任何有限字符串，不论长短，我们都可以很容易地用系统的方法为它指派一个整数。如果字符表只有9个符号，我们可以把从1至9的9个整数指派给它们。在此基础上，我们可以得到任意一串字符的整数，方法是将其中的符号所对应的数字连接起来从而形成一个长数字。如果字符表中的符号超过9个，我们可以很容易地对此法加以调整或选择另一种方法。假定初等数论的语言的编号工作已经完成了。因此，该语言中的每个语句都有其所谓的哥德尔数。哥德尔然后表明，给定P，我们可以用初等数论的语言构造一个开语句，姑且说是“…x…”：它对x是真的，当且仅当x=某个在P中可证的语句的哥德尔数。如果我们把“-(...x...)”中的“x”换成指称某个具体的数n的实际数字，很显然，这样得到的语句（大致是这样的：“-(...n...)”）是真的，当且仅当n≠任何在P中可证的语句的哥德尔数[3]，但哥德尔又表明，n可以选择得正好等于某个与“-(...n...)”等价的语句的哥德尔数。这种精心选择的n所产生的语句就是所寻求的S_P：它是真的当仅当它在 P中不可证。[4]

[1]I slur over Church's Thesis; see RECURSION.我略去了丘奇论题；见递归。

[2]较详细的说明见附在后面的奎因在《逻辑方法》一书中的两段文字。——译者注

[3]我把得出这个“显然的”结论的过程写在下面：

I、所有事物都是如此这般的x: “…x…”是真的，当且仅当x=某个在P中可证的语句的哥德尔数…已证

II、“…n…”是真的，当且仅当n=某个在P中可证的语句的哥德尔数…I，全称例化（n）

III、并非（“…n…”是真的），当且仅当并非（n=某个在P中可证的某个语句的哥德尔数）…与II等价

IV、“…n…”是假的，当且仅当并非（n=某个在P中可证的某个语句的哥德尔数）…与III等价，因为说‘并非（“…n…”是真的）’，就是说‘“…n…”是假的’

V、“-(…n…)”是真的，当且仅当并非（n=某个在P中可证的某个语句的哥德尔数）…与IV等价，因为说‘“…n…”是假的’，就是说‘“-(…n…)”是真的’。

VI、“-(…n…)”是真的，当且仅当n≠任何在P中可证的语句的哥德尔数…与V等价，因为说“并非（n=某个在P中可证的某个语句的哥德尔数）”，就是说“n≠任何在P中可证的语句的哥德尔数”

[4]我把这个结论的详细过程写在下面：（1）如果“-(...n...)”是真的，那么n≠任何在P中可证的语句的哥德尔数。但已知n=“-(...n...)”的哥德尔数（严格说来，是与“-(...n...)”等价的某个语句的哥德尔数，我做了简单化的处理），所以“-(...n...)”的哥德尔数≠任何在P中可证的语句的哥德尔数。进一步地，“-(...n...)”≠任何在P中可证的语句。（2）如果“-(...n...)”是假的，那么“...n...”是真的，所以n=某个在P中可证的语句的哥德尔数。但已知n=“-(...n...)”的哥德尔数，所以“-(...n...)”的哥德尔数=某个在P中可证的语句的哥德尔数。进一步地，“-(...n...)”=某个在P中可证的语句。

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