与少量的质疑哥德尔不完备性定理的讨论相比，网上有大量质疑康托尔对角线法讨论。我编辑几个可能有代表性的资料：

1. 质疑康托尔对角线法的论坛（1）

2. 质疑康托尔对角线法的论坛（2）

3. 质疑康托尔对角线法的书

4. 质疑康托尔对角线法的最新文章

5. 追本溯源康托尔对角线法的文章

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1. 质疑康托尔对角线法的论坛（1）

https://news.ycombinator.com/item?id=6844885

dmfdmf (2013) :

Cantor is wrong, yes, but you won't prove that to contemporary mathematicians who believe his ideas because this is an epistemological issue and can't be resolve with mathematical arguments.

康托尔是错的，是的，但你无法向相信他的思想的当代数学家证明这一点，因为这是一个认识论的问题，不能用数学论证来解决。

This is really an age-old philosophic dispute that the mystics won decades ago. As David Hilbert described it "No one shall expel us from the Paradise that Cantor has created." It is the "paradise" of deuces wild for the mathematicians and it rests on an equivocation of the meaning of "infinity". Cantor's idea of a "completed infinity" is a self contradiction if you grasp what the concept of infinity actually means and keep it tied to reality. It is the error of treating infinity as a real thing and not an abstraction. A similar error, for similar reasons, is made in the history of the philosophy by the mystics of nihil who wanted to treat nothingness as on par with existence via the Reification of Zero.

这实际上是一个古老的哲学争论，神秘主义者在几十年前就赢了。正如David Hilbert所描述的那样， “没有人可以把我们从康托尔所创造的天堂中驱逐出去”，这是数学家们的“天堂”，它建立在对“无限”意义的含糊其辞上。如果你掌握了“无限”概念的实际含义并将其与现实联系起来，康托尔的“完成的无限”的想法就是一个自我矛盾。这是把无限当作一个真实的东西而不是一个抽象的东西的错误。一个类似的错误，出于类似的原因，在哲学史上被虚无的神秘主义者所犯，他们想通过“零的再化”（Reification of Zero）把虚无与存在等同起来。

2. 质疑康托尔对角线法的论坛（2）

https://groups.google.com/g/sci.math/c/zQoeAX799BY/m/9F1zYo8qWDoJ?pli=1 

Mark Adkins :

Cantor's diagonal proof of the uncountability of certain infinite sets (such as the set of real numbers) is fatally flawed. Cantor's proof begins with what is taken to be a  complete list of real numbers. It then constructs "a real  number not on the list" by a diagonal method which is no doubt familiar to those with a basic knowledge of this issue. Thus, his proof claims to demonstrate that the initial assumption (that the reals are countable) is false. In fact, the analytical flaw in Cantor's proof is in thinking that such a procedure would generate a real number at all. Since the list is assumed complete, there is no real number not on the list, and no such number can be generated.

康托尔对某些无限集（如实数集）的不可数性的对角线证明存在致命的缺陷。康托尔的证明从被认为是一个完全的实数列表开始，然后通过对角线的方法构造出“一个不在列表上的实数”，这对那些对这个问题有基本了解的人来说无疑是熟悉的。因此，他的证明声称证明了最初的假设（实数是可数的）是错误的。事实上，康托尔证明中的分析性缺陷在于认为这样的程序会产生实数。既然假设列表是完全的，就没有不在列表上的实数，也就不可能产生这样的数字。

The flaw is in thinking that a procedure which can be applied to any single element of a complete list (to produce a single digit of the diagonal number), or to any finite number of elements of a complete list (to produce a finite number of digits of the diagonal number), or to any infinite *subset* of a complete list (to produce an infinite diagonal number, i.e. a real number -- albeit one elsewhere on the list) can be applied to a complete list as a whole. Obviously not, since this contradicts the axiomatic assumption that it is complete!

其缺陷在于认为一个程序可以应用于一个完全列表的任何单一元素（产生对角线数的一个数字），或一个完全列表的任何有限数量的元素（产生对角线的有限数字），或一个完全列表的任何无限*子集（产生对角线的无限的数字，即一个实数—尽管是在列表的其他地方）都可以应用于整个完全列表。显然不能，因为这与它是完全的公理假设相矛盾!

And any objective claim to proof of the existence of uncountable sets must begin with an assumption that lists of reals may be complete, otherwise one is simply begging the question. The same procedure which can be applied to an incomplete infinite list (such as a list of the rationals) to produce a real number not on the list, cannot be applied to a complete list of reals as a whole to produce a real number (or anything else). In such an instance it produces neither a real number on the list, nor a real number not on the list, nor does it produce anything else.

而任何关于证明不可数集存在的客观主张，都必须从一个假设开始，那就是实数的列表可能是完全的，否则就只是在乞求问题。同样的程序可以应用于一个不完全的无限列表（比如有理数的列表）来产生一个不在列表上的实数，但不能应用于一个完全的实数列表，作为一个整体来产生一个实数（或任何其他东西）。在这种情况下，它既不能产生列表上的实数，也不能产生不在列表上的实数，更不能产生其他东西。

Paola Cattabriga :

Good point! Till now none has never recognized that this axiomatic assumption of completeness is incorrect. But not the same does the theory itself! by means of its own axioms!

说得好! 直到现在都没有人认识到，这种完全性的公理假设是不正确的。但理论本身却不一样！通过它自己的公理！

The explanation goes directly through Cantor's ‘theorem’.

解释直接通过康托尔的’定理'。

The axiomatic assumption of completeness of the list is expressed by the axioms of ZF by means of the universal quantifiers of the Subsets Axiom Schema

列表完全性的公理假设是由ZF的公理通过子集公理模式的普遍量词来表达的：

AzEyAx(x in y <-> x in z and P(x))

where the existential quantifier states the existence of a set, i.e. GENERATES the fateful set.

其中存在量词说明了一个集合的存在，也就是产生了注定集合。

Till now no objection about the fact that in Cantor theorem the property P which define the incriminate set is in itself a contradiction, a flaw (The axiom of comprehension was erased by this reason and it was replaced by the above one which was considered yielding no contradiction - what irony! the axioms of the Subsets was constructed by Zermelo to prevent any contradiction but finished to legitimate the use of the
property P as a contradiction of the kind (A <-> not A) within Cantor's 'theorem' !).

到现在为止，没有人反对在康托尔定理中定义有罪集合的属性P本身就是一个矛盾，一个缺陷（理解力公理因为这个原因被抹去了，它被上述被认为不会产生矛盾的公理所取代—多么讽刺！子集公理是泽梅洛为了防止任何矛盾而构建的，但却完成了将属性P作为康托尔'定理'中的那种矛盾（A<->非A）的合法使用！）。

But it's easy to show that the axioms of ZF itself firmly objects to this assumption of completeness when the property P is a contradiction.

但很容易表明，当属性P为矛盾时，ZF的公理本身就坚决反对这种完全性的假设。

By the same axiom of the Subsets the complementary set of the incriminate Cantor's set

根据子集的同一公理，罪魁祸首康托尔集的互补集

B = {x in A| x notin g(x)}

(which is only another notation for
Ax(x in B <-> x in A and x notin g(x)) )

can be defined

~B = {x in A| x in g(x)} (*

(which is only another notation for
Ax(x in ~B <-> x in A and not (x notin g(x)))
i.e. the relative complement of B in A).

Hence directly by the axiom of Extensionality
(AxAy[Az(z in x <-> z in y) -> x = y]) we get

(b in ~B <-> b in g(b)) -> ~B = g(b)

since A was 'already' defined (by Zermelo axiom ;-)

(b in ~B <-> b in g(b)) is only an example of the above definition of ~B (*.

We can then derive by modus ponens

~B = g(b)

which is equivalent to

|-ZF not (B = g(b))

(where (B = g(b)) is diagonalization in Cantor's ‘theorem').

In simple words, the axioms of ZF by themselves tell us that to consider P = 'x notin g(x)' along Cantor's 'theorem' is unacceptable, since in ZF the negation of the diagonalization can be derived as a theorem of ZF.

简单地说，ZF的公理本身就告诉我们，沿着康托尔的’定理'考虑P='x不在g(x)'是不可接受的，因为在ZF中，对角线化的否定可以作为ZF的定理来推导。

More explanations can be found in
http://www.serdata.it/cattabriga/

I know you were talking of the Cantor's diagonal method over reals, but Cantor's 'theorem' talk of the UNcountability of 'all' the subsets of a given set and does it by diagonalization (i.e. a self-referring procedure (i.e. an incontrovertible contradiction)) which is at the end the same thing of the diagonal method (…and exactly the same of the Goedel formula in the 'theorem' of UNcompleteness for PA) and to understand/clarify precisely where the « axiomatic assumption of completeness" is wrong you must refer to the universal quantifiers in front at the formula that usually mathematicians utilize for defining sets!

我知道你们说的是康托尔在实数上的对角线法，但康托尔的’定理'说的是一个给定集合的'所有'子集的不可数性，并且是通过对角线化（即一个自指的过程（即一个无可争议的矛盾）来实现的。，这与对角线方法是一样的（......与PA的不完备性'定理'中的Goedel公式完全一样），为了准确理解/澄清 "完备性公理假设 « 的错误之处，你必须参考通常数学家用来定义集合的公式前面的通用量词！

3. 质疑康托尔对角线法的书

https://www.amazon.com/Why-Cantor-Diagonal-Argument-Valid/dp/1720899770

https://img1.wsimg.com/blobby/go/7cb7b799-04e5-49b2-ab17-c5e9889ccb0d/downloads/Why the Cantor Diagonal Argument is Not Valid.pdf?ver=1596896913748

Why the Cantor Diagonal Argument is Not Valid

And there is no such thing as a number written to Infinite Digits

Pravin K. Johri

The Cantor Diagonal Argument (CDA) is the quintessential result in Cantor’s infinite set theory. It is over a hundred years old, but it still remains controversial. The CDA establishes that the unit interval [0, 1] cannot be put into one-to-one correspondence with the set of natural numbers N = {1, 2, 3 ...} and, consequently, the set of real numbers R is uncountable.

康托尔对角线论证（CDA）是康托尔无限集合理论中最典型的结果。它已经有一百多年的历史了，但它仍然存在争议。康托尔对角线论证指出，单位区间[0，1]不能与自然数集N={1，2，3…}一一对应，因此，实数集R是不可数的。

Many people believe the CDA is flawed and spend a lot of effort trying to disprove it. Mückenheim [6] has an extensive list of the counterarguments against the CDA.

Excerpts from Hodges [7]

Cantor’s argument is short and lucid. It has been around now for over a hundred years. Probably every professional mathematician alive today has studied it and found no fallacy in it.

This argument is often the first mathematical argument that people meet in which the conclusion bears no relation to anything in their practical experience or their visual imagination ... all intuition fails us.

许多人认为CDA是有缺陷的，并花了很多精力试图反驳它。Mückenheim[6]列举了大量反对CDA的反驳意见。

Hodges[7]的摘录

康托尔的论点简短而清晰。它现在已经存在了一百多年了。也许今天活着的每一个专业数学家都研究过它，并且没有发现其中的谬误。

这个论证往往是人们遇到的第一个数学论证，其中的结论与他们的实践经验或视觉想象中的任何东西都没有关系……所有的直觉都让我们失望。

4. 质疑康托尔对角线法的最新文章

https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

Transfinity A Source Book

Wolfgang Mückenheim

14 Mar 2023

Transfinity is the realm of numbers larger than every natural number: For every natural number k there are infinitely many natural numbers n > k. For a transfinite number t there is no natural number n ¥ t.

We will first present the theory of actual infinity, mainly sustained by quotes, in chapter I and then transfinite set theory as far as necessary to understand the following chapters. In addition the attitude of the founder of transfinite set theory, Georg Cantor, with respect to sciences and religion (his point of departure) will be illuminated by various quotes of his as well as of his followers in chapter IV. Also the set of applications of set theory will be summarized there. All this is a prerequisite to judge the social and scientific environment and the importance of set theory. Quotes expressing a sceptical attitude against transfinity or addressing questionable points of current mathematics based on it are collected in chapter V. For a brief overview see also Critics of transfinity. The critique is scrutinized in chapter VI, the main part of this source book. It contains over 100 arguments against actual infinity – from doubtful aspects to clear contra- dictions – among others applying the newly devised powerful method of ArithmoGeometry. Finally we will present in chapter VII MatheRealism, a theory that shows that in real mathematics, consisting of monologue, dialogue, and discourse between real thinking-devices, via necessarily physical means, infinite sets cannot exist other than as names. This recognition removes transfinity together with all its problems from mathematics – although the application of mathematics based on MatheRealism would raise a lot of technical problems.

超限数是指大于每个自然数的数的领域： 对于每一个自然数k，都有无限多的自然数n>k。对于一个转无穷数t，没有自然数n ¥ t。

我们将首先在第一章中介绍主要由引文支撑的实际无限的理论，然后在理解以下各章的必要范围内介绍转置理论。此外，第四章将通过引用乔治-康托尔及其追随者的各种话语，阐明他对科学和宗教（他的出发点）的态度。此外，集合理论的一系列应用也将在这里得到总结。所有这些都是判断社会和科学环境以及集合理论重要性的先决条件。在第五章中收集了一些对变异性表示怀疑的引文，或对当前基于变异性的数学有疑问的地方进行了论述。批评的内容在第六章，即本资料书的主要部分进行了仔细的分析。它包含了100多个反对实无穷的论据—从可疑的方面到明确的反驳--其中包括应用新设计的强大的ArithmoGeometry的方法。最后，我们将在第七章中介绍MatheRealism，这一理论表明，在真实的数学中，由真实的思维设备之间的独白、对话和话语组成，通过必然的物理手段，无限集除了作为名称之外，不可能存在。这种认识从数学中消除了变形及其所有问题--尽管基于MatheRealism的数学应用会引起许多技术问题。

5. 追本溯源的文章

https://www.maa.org/sites/default/files/pdf/pubs/AMM-March11_Cantor.pdf

Was Cantor Surprised?

We look at the circumstances and context of Cantor’s famous remark, “I see it, but I don’t believe it.” We argue that, rather than denoting astonishment at his result, the remark pointed to Cantor’s worry about the correctness of his proof.

我们研究了康托尔的名言“我看到了，但我不相信”的情况和背景。我们认为，这句话并不表示对他的结果感到惊讶，而是指康托尔对他的证明的正确性感到担忧。