# 质疑康托尔对角线法讨论的文本编辑

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.

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.

2. 质疑康托尔对角线法的论坛（2

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.

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!

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.

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!

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

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

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.

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' !).

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.

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

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.

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!

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.

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.

Hodges[7]的摘录

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.

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.

https://m.sciencenet.cn/blog-2322490-1393577.html

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