定义法比较无限集合元素数目的相对多少

The method by the definition of sets to compare the relative number of elements of infinite sets

                                               Li Hongyi

摘要  在数学中，从定义出发的推导是可靠的。 因此，本文首创的从定义出发来推断无限集合元素数目相对大小的方法(定义法)具有高度的可靠性和精确性，远比传统的一一对应法更为可靠，而且不会产生任何悖论。本文还用该法消除了广被垢病的伽利略，无限旅馆，整体等于部分等悖论。

关键词 无限集合；定义法；一一对应；伽利略悖论；无限旅馆悖论；整体等于部分悖论

ABSTRACT In mathematics, the derivation from the definition is reliable. Therefore, the method proposed in this paper to compare the relative number of elements of infinite sets from the definition of sets is highly reliable and accurate, far more reliable than the traditional one-to-one correspondence method, and will not cause any paradox. This paper also eliminates the paradoxes of Galileo, the infinite hotel, the whole equals the part and so on.

Key words infinite set; Definition method; One-to-one correspondence; Galileo's paradox;  Infinite Hotel Paradox; the whole equals parts paradox

数学家是聪明的，数学本身也历来以严格著称，在学术界享有崇高的声誉。

但在与无限相关的领域，却完全是另外一番景象。这里不但充斥着任何一门严肃的科学绝对不允许的诸如伽利略悖论之类的各种自相矛盾，且人们对此似乎毫无办法，不得不“适应”悖论，与悖论共存，甚至用悖论来做学问。

这种局面至少已经延续几百年，引起不少人士的批评。例如，文献[1]认为集合论不过是一种宗教信条，缺乏科学意义；文献[2]认为公理化集合论的逻辑是不连贯的,无限量词没有意义;文献[3-4]讨论了集合论的一致性问题，文献[5]则试图重建一个无悖论的集合论。

在悖论丛生的集合论，如何判断无限集合的大小(size)，不但是集合论的一个基本问题，也是各种悖论的根本来源。

对于有限集合，只要比较集合的元素数目或在集合之间建立一一对应就可以精确地比较两个集合的大小。但对无限集合，人们误以为难以比较其元素数目，所以康托将一一对应推广到无限集合，从而形成了所谓的基数理论:若两个无限集合之间能够建立一一对应，则称其基数相等，否则就不相等，并用基数的大小来衡量无限集合的大小。

然而，有限集合与无限集合有本质的不同。对于有限集合行之有效的一一对应，能否推广到无限集合，是需要证明的。

可惜的是，从来没有人能够证明这一点，相反，反例倒是大量的。

一个最简单的反例就是有理数集合和自然数集合之间的一一对应。

应该承认的是，康托确实巧妙地在两者之间建立了一一对应，任何一本集合论教材对此都有似乎可以引以为豪的详细叙述。

然而，有理数集合和自然数集合的大小真的是一样的吗？

答案显然是否定的：有理数集合中包含了自然数集合，两个集合的大小怎么可能是一样大的？

在数学中，根据定义出发的推导是可靠的。任何脱离定义或与定义冲突，甚至随意引入未加证明的假设的推导，要么是不可靠的，要么是错的，都没有科学的意义。

关于集合的推导也是如此。

集合的元素数目是对集合的元素进行计数得到的结果。由于无限集合的元素数目是无限多的，计数无法完成，所以我们不可能给出任何一个无限集合元素数目的计数结果。但这并不意味着集合元素的相对多少是无法比较的。事实上，数学是从定义出发的，例如，当我们在定义集合A的真子集的时候，由于真子集不过是A的部分元素组成的集合，任何一个尚有正常思维能力的人都知道，它的元素数目必然是比A少的。

由此可见，我们可以根据无限集合的定义可靠地得到它们的相对数值。这是因为，当我们用某一个或多个集合的元素来定义另一个集合时，这些集合的元素数目之间的相对关系就已经确定了，所以可以直接根据定义确定它们元素数目之间的相对关系。

例如，有理数集合可以定义为:

有理数集合={0}U{正整数}U{负整数}U{分母不等于1的正分数}U{分母不等于1的负分数}

由于上述等式右边的各集合的交集为空，所以有理数集合的元素数目是上述各集合元素数目之和。这就严格证明了有理数集合大于自然数集合。

由此可见，用未严格证明的一一对应并不能比较无限集合的大小。

由于所有集合都是定义出来的，所以只要定义明确，它们之间元素数目的相对关系都是可以确定的。

为了讨论方便，将用上述得到的无限集合元素数目相对数值的方法称为定义法。

由于一一对应最多也只能比较无限集合的相对大小，且其结果还不可靠，而定义法是直接从定义出发的，因此具有高度的精确性和可靠性，是所谓一一对应法所不可比拟的。

由于定义法高度可靠，所以用定义法来讨论问题，不可能出现任何不可靠的结果。这在上面的讨论中已经看到了。

以下用定义法来消解伽利略悖论。

伽利略悖论影响巨大，可以说是集合论的基本悖论，任何一个集合论教师在叙述集合论历史的时候，都不大可能不提该悖论。

该悖论延续至今已近400年，在笔者之前，从来没有人真正解决过，希尔伯特和康托对此也毫无办法。

对该悖论的误读还导致了所谓无限集合可以与其真子集一一对应这一根本性错误。

所谓伽利略悖论，意思是说，偶数(或平方数)只是自然数的一部分：

1，2，3，4，5，6.......                            （1）

因此偶数的数目理所当然是应该比自然数少的。

但实际上又可以通过将每个自然数乘以2，在两者之间建立如下一一对应：

 1→2，2→4，3→6，......                      （2）

即每个自然数对应且只对应了一个偶数，似乎又表明偶数的数目与自然数一样多，于是就产生了悖论。

 虽然在伽利略时代，并没有集合的概念，但仍然可以从集合论的角度来分析他的思想。

用定义法来解决伽利略悖论时，由于定义法是从集合的定义出发的，所以首先必须给出各相关集合的明确定义。

首先设自然数集合为N={1,2,3……}，悖论中提到的只是自然数一部分的偶数即（1）式中的偶数实际上就是N的偶数真子集，其定义为：        

E={x|x mod 2=0,x∈N}={2,4,6,……}                     （3）                     

由于每两个相邻的N的元素只能定义一个E集合的元素，所以正如伽利略所说的那样，E的元素数目比N少，且实际上只是N的一半。

这就是根据可靠的定义法得到的N和E之间的元素数目的关系，

 而（2）所示的能与N一一对应的偶数集合的定义则为:

E'={y|y=2x,x∈N}={2,4,6,……}，                      （4）      

从定义法的角度，由于N的每一个元素都定义且只定义一个E’的元素，所以E’与N的元素数目精确一致，确实形成了严格的一一对应。

有了以上的定义和分折，很容易消解所谓伽利略悖论。

伽利略首次敏锐地发现了E'和E的元素数目不同，但由于两者都可表示为{2，4，6…}，故他像几乎所有的数学家那样，误以为它们是同一个集合，同一个集合元素数目怎么会不同呢？所以才认为产生了悖论。

但从定义法的角度，集合是由定义决定的。既然集合E'和E的定义不同，如果没有充分的理由，怎么可以认为是同一个集合？

何况两个集合的元素数目都不同，根本不可能是同一个集合。

因此，只要认识到这两个集合并不是同一个集合，所谓的悖论也就不再存在:不同的集合元素数目当然可以不同，何来悖论之说？

事实上，尽管E和E'都可表示成{2,4,6....}，但对无限集合，不能仅仅因为元素似乎是一样的，就认为它们一定是同一个集合。这是因为，只有元素完全相同的两个集合才是同一个集合。E'与E的元素数目不同，这是铁的事实，元素数目都不同，元素又怎么可能完全相同？所以，E'与E是两个不同的偶数集合，这是再明显不过的事实了。既然不是同一个集合，悖论也就消解了。

而且，由于E'和N的元素数目是严格相等的，而N的任何真子集不过是N的部分元素组成的，其元素数目理所当然是少于N的，与N具有相同元素数目的E’不可能是N的真子集。

由此可见，误把E'当做N的真子集E是伽利略，希尔伯特和康托以及绝大多数数学家所犯的共同错误，也是所谓伽利略悖论四百来年得不到消解的原因所在。

康托显然在这个错误的道路上走得更远: 他不但错误地把E'当成了N的真子集，而且干脆把这个错误扩大化和“合法”化了:认为任何无限集合都可以与其真子集一一对应，从而诗史般地又形成了“部分等于整体”这一惊天地，泣鬼神的悖论。

事实上，N与其真子集E是不可能一一对应的: E只能与N中的一半元素形成单射，因此此单射并不是满射。所以，因为错误解读伽利略悖论而导致的所谓“无限集合都可以与其真子集一一对应”的论断也是完全错误的。

实际上，不但偶数集合不是唯一的，就是都可以表示成{1,2,3....}的自然数集合，也不是唯一的。

例如，假定甲乙两台机器永不停歇地生产零件，甲的速度是乙的两倍，设用自然数分别对甲乙两台机器生产的零件进行编号，则无限时间后，得到两个不同的自然数集合。设用n表示机器乙生产的零件数，则机器甲生产的零件数可用2n表示。因为n→∞时，Lim[(2n)/n]=2，所以与甲对应的自然数集合的元素数目是与乙对应集合的两倍。

再例如，假定一个学校由两个无限班级A，B组成，其中，B的学生数目是A的两倍，则两个班级的班学号可以分别用两个不同的自然数集合表示，而校学号又可用另一个自然数集合表示，这三个自然数集合都是不一样的。

甚至，假定有无限多人民币，分别用分、元、亿元、万亿元为单位的自然数集合{1,2,3,……}，这些自然数集合可能是同一个集合吗？

由此可见，在没有任何严格可靠的证明的情况下，就断言自然数集合是唯一的，实在是太想当然，太不严谨了。

作为自然数集合的一部分，偶数集或奇数集当然也不是唯一的。

进一步的研究，例如，具体哪些元素属于E’但不属于E，笔者以后会进一步讨论。

集合论中还存在其他的悖论，用定义法都不难消解。例如，设N₀={0}UN，显然N₀比N多了一个元素0，根据定义法，相应的元素数目当然也比N多了1个，并不相等，所以不会出现根据一一对应而产生的错误的无限旅馆，∞+1=∞等悖论。

其实，所有的悖论最后都可以归结为某些概念的混淆，或某些未加证明的假设的引入所致。例如，所谓伽利略悖论就是因为混淆了E’和E所致，本质上不过是一个低级错误。因此，只要足够细心地区分各种实际上并不完全相同的概念，不引入任何未加证明且并不显然成立的假设，任何悖论都是不应该存在的。

集合论界要整顿学风，虚心向数学界其他领域学习，不要再脱离集合的定义讨论集合，也不要随意引入未加证明的假设，更不要因为暂时没法解决的问题或为了维护权威而把集合论界打造成不允许讨论和质疑、只靠信仰来维系的宗教甚至邪教（cult）组织^[1]。否则的话，哪怕再明显、再荒唐的错误，例如自然数和有理数一样多，也会被宗教氛围掩盖，成为不可置疑的“真理”。然而，纸是包不住火的，错误总要暴露。并不是每一个数学家都心甘情愿被愚蠢的教义所蒙蔽，甚至不顾一切地为之辩护，最终成为历史的笑话。

建议教育部门暂停在中学和本科教育中与无限集合有关理论尤其是一一对应和基数理论的灌输式教育，以防误人子弟。

 参考文献

[1] N J Wildberger，Set Theory: Should You Believe?

http://web.maths.unsw.edu.au/~norma

[2] Fletcher, P. (1998). What’s Wrong with Set Theory?. In: Truth, Proof and Infinity.]Synthese Library, vol 276. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3616-9_2

[3]John Mayberry.On the consistency problem for set theory: An essay on the Cantorian foundations of classical mathematics (I),British Journal for the Philosophy of Science 28 (1):1-34 (1977)  

[4] John Mayberry，The Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Mathematics (II)，The British Journal for the Philosophy of Science，Vol. 28, No. 2 (Jun., 1977), pp. 137-170 (34 pages)

[5] Li Hongyi , Rebuilding Set Theory, https://vixra.org/abs/2210.0144

Mathematicians are intelligent, and mathematics itself has always been known for its rigor and has a high reputation in academia.

But in the realm of infinity, it's a different story. It is not only full of paradoxes such as Galileo's paradoxe, which no serious science would allow, but people seem to have no way of dealing with them, and have to "adapt" to them, live with them, and even use them for learning.

This situation has been going on for at least a few hundred years, and it has aroused criticism from many people. For example, literature [1] argues that set theory is no more than a religious creed and lacks scientific significance; Literature [2] holds that the logic of axiomatic set theory is incoherent and infinite quantifiers have no meaning; Literature [3-4] discusses the consistency of set theory, while literature [5] attempts to reconstruct a set theory without paradox.

In the paradoxical set theory, how to judge the size of infinite sets is not only a basic problem in set theory, but also the fundamental source of all kinds of paradoxes.

For finite sets, the sizes of two sets can be compared precisely by comparing the number of elements in the sets or establishing one-to-one correspondence between the sets. However, for infinite sets, people thought that it was difficult to compare the number of elements, so Cantor extended the one-to-one correspondence to infinite sets, thus forming the so-called cardinal theory: if two infinite sets can establish a one-to-one correspondence, their cardinal numbers are said to be equal, otherwise their are not equal, and the size of the infinite set is measured by the cardinal number.

However, finite sets are fundamentally different from infinite sets. It is necessary to prove whether the effective one-to-one correspondence for finite sets can be generalized to infinite sets.

Unfortunately, no one has ever been able to prove this, and contrary examples abound.

One of the simplest counterexamples is the one-to-one correspondence between the set of rational numbers and the set of natural numbers.

It should be admitted that Cantor did a masterful job of establishing a one-to-one correspondence between the two, which any set theory textbook would seem to be proud of.

However, is the set of rational numbers and the set of natural numbers really the same size?

The answer is obviously no: the set of rational numbers contains the set of natural numbers, how can two sets be the same size?

In mathematics, derivations by definition are reliable. Any derivation that deviates from or conflicts with the definition, or even arbitrarily introduces unproven assumptions, is either unreliable or wrong, and has no scientific significance.

The same is true of the derivation of sets.

The number of elements in a set is the result of counting the elements of the set. Since the number of elements in an infinite set is infinite, the  counting can not be finished, so we cannot give the result for any infinite set. But this does not mean that the relative number of elements in sets cannot be compared. In fact, mathematics starts with definitions, for example, when we define a proper subset of  set A, since the proper subset is just a set of some of the elements of A, and anyone with normal thinking ability knows that it must have fewer elements than A.

From this, we can reliably obtain their relative values according to the definition of infinite sets. This is because, when we define another set in terms of the elements of one or more sets, the relative relation between the number of elements of these sets is already determined, so the relative relation between the number of elements of these sets can be determined directly from the definition.

For example, the set of rational numbers can be defined as:

Set of rational numbers ={0}U{positive integer}U{negative integer}U{positive fraction with denominator not equal to 1}U{negative fraction with denominator not equal to 1}

Since the intersection of the sets to the right of the above equation is empty, the number of elements of the set of rational numbers is the sum of the number of elements of the above sets. This strictly proves that the number of elements of rational set is greater than that of natural numbers.

Thus, it is not possible to compare the sizes of infinite sets with unproven one-to-one correspondence.

Since all sets are defined, so long as the definition is clear, the relative number of elements between them can be determined.

For the sake of discussion, the method to compare the relative number of elements of the infinite sets by their definition is called the definition method.

Because one-to-one correspondence can only compare the relative size of infinite sets at most, and its results are not reliable, and the definition method is directly from the definition, so it has a high degree of accuracy and reliability, which is incomparable to the so-called one-to-one correspondence method.

Since the definition method is highly reliable, it is impossible to produce any unreliable results when use definition method. This has been seen in the discussion above.

The following discussion is used to resolve Galileo's paradox.

Galileo's paradox is so influential that it can be said to be the fundamental paradox of set theory that it is unlikely that any teacher of set theory will recount the history of set theory without mentioning the paradox.

This paradox has been going on for nearly 400 years, and no one had really solved it before this writer, and Hilbert and Cantor had no idea what to do about it.

A misreading of this paradox also leads to the fundamental error that the so-called infinite set can correspond to its proper subset one by one.

Galileo's paradox means that even numbers (or square numbers) are only part of the natural numbers:

1,2,3,4,5,6.......                                         (1)

Therefore, the number of even numbers should be less than the natural numbers.

But you can actually establish the following one-to-one correspondence between the two by multiplying each natural number by 2:

1→2, 2→4, 3→6,......                                 (2)

That is, each natural number corresponds to and only corresponds to one even number, which seems to indicate that there are as many even numbers as there are natural numbers, thus creating a paradox.

Although there was no concept of set in Galileo's time, it is still possible to analyze his ideas from the perspective of set theory.

When using the definition method to solve Galileo's paradox, since the method starts from the definition of sets, we must first give the clear definition of each related set.

Let the set of natural numbers be N={1,2,3...... }, the paradox refers to an even number that is only part of the natural numbers, i.e., the even number in eq. (1) is actually an even proper subset of N, defined as:

E={x|x mod 2=0,x∈N}={2,4,6,...... }                                (3)

Since every two adjacent elements of N can define only one element of the set E, as Galileo said, E has fewer elements than N, and is actually only half elements of N.

This is the relationship of the number of elements betwee sets N and E according to the reliable definition method.

The definition of the set of even numbers shown in (2) that can correspond to N one by one is:

E’ = {y | x, y = 2 x ∈ a.} = {2,4,6...... },                            (4)

From the point of view of the definition method, since each element of N is defined only one element of E’, the number of elements of E’ and N are exactly the same, indeed forming a strict one-to-one correspondence.

With the above definitions and breaks, it is easy to dispel the so-called Galileo paradox.

Galileo was the first to make the astute observation that E’ and E have different numbers of elements, but since both can be expressed as {2,4,6...... }, Therefore, as almost all mathematicians do, they mistakenly believe that they are the same set, how can the number of elements of the same set be different? That's why we think there's a paradox.

But from the definition point of view, the set is determined by the definition. Since the sets E’ and E are defined differently, how can they be considered to be the same set without good reason?

Besides, the two sets have different numbers of elements, so they can't be the same set at all.

Therefore, as long as the two sets are not the same set, the so-called paradox no longer exists: of course, the number of elements of different sets can be different, how can there be a paradox?

In fact, although E and E’ can both be expressed as {2,4,6....}, but for infinite sets, just because the elements seem to be the same doesn't mean they must be the same set. This is because only two sets with exactly the same elements are the same set. E’ and E have different numbers of elements, which is a fact, how can the elements be the same if the number of elements is different? Therefore, it is obvious that E’ and E are two different sets of even numbers. Since it is not the same set, the paradox is resolved.

Moreover, since the number of elements of E’ and N is strictly equal, and any proper subset of N is composed of only some elements of N, and of course has fewer elements than N, E’ having the same number of elements as N cannot be a proper subset of N.

Thus, the mistake of E’ as proper subset of N was made by Galileo, Hilbert, and Cantor, as well as by most mathematicians, and is the reason why the so-called Galileo paradox has not been resolved for about 400 years.

Cantor obviously went further on this wrong path: not only did he mistake E’ as proper subset of N, but he simply enlarged and "legitimized" this mistake: he assumed that any infinite set could correspond to its proper subset one by one, thus forming the astonishing paradox of "the part equals the whole."

In fact, there is no one-to-one correspondence between N and its proper subset E : E can only form an injective form with half of the elements in N, so this injective form is not surjective. Therefore, the assertion that "any infinite set can correspond to its proper subset" due to a misinterpretation of Galileo's paradox is also completely wrong.

In fact, not only is the set of even numbers not unique, but also the set of natural numbers is not unique although the set of natural numbers can all be represented as {1,2,3.......}.

For example, if two machines A and B are never stopping to produce parts, and A is twice as fast as B, let the natural numbers be used to number the parts produced by two machines A and B respectively, then after infinite time, two different sets of natural numbers are obtained. Let n represent the number of parts produced by machine B, then the number of parts produced by machine A can be represented by 2n. Because Lim[(2n)/n]=2 when n→∞, the set of natural numbers corresponding to A has twice as many elements as the set corresponding to B.

For another example, if a school consists of two infinite classes A and B, in which B has twice as many students as A, then the class numbers of the two classes can be represented by two different sets of natural numbers, and the school numbers can be represented by another set of natural numbers, which are all different.

Even, assuming there are infinite number of RMB, can the set of natural numbers in units of cent, yuan, billion yuan and trillion yuan {1,2,3,.....} be the same natural numbers set?

Thus, without any strict and reliable proof, to assert that the set of natural numbers is unique is too much to take for granted and too unrigorous.

As part of the set of natural numbers, the even or odd set is certainly not unique.

Further research, for example, which elements belong to E’ but not to E, will be discussed in the future.

There are other paradoxes in set theory that are not difficult to resolve by definition. For example, let N’={0}UN, obviously N’ has one more element 0 than N, according to the definition method, the corresponding number of elements is of course 1 more than N, , so there will be no infinite hotel, ∞+1=∞ and other paradoxes obtained from one-to-one correspondence.

In fact, all paradoxes can ultimately be attributed to the confusion of some concept, or the introduction of some unproven hypothesis. For example, the so-called Galileo paradox, caused by the confusion of E’ and E, is essentially a simple error. Therefore, as long as enough care is taken to distinguish between various concepts that are not actually identical, and no unproven and not obviously valid assumptions are introduced, no paradox should exist.

The community of set theory should rectify its style of study and learn from other fields of mathematics with an open mind. It should no longer discuss sets apart from the definition of sets, nor introduce unproven hypotheses at will, nor build the community of set theory into a religious or even cult organization that does not allow discussion and questioning and only depends on faith because of problems that cannot be solved for the time being or in order to maintain authority ^[1]. Otherwise, even the most obvious and absurd errors, such as the equal number of natural and rational numbers, will be covered up by the religious atmosphere and become unquestionable "truths." However, the paper can not cover fire, mistakes must be exposed. Not every mathematician is willing to be blinded by foolish doctrines, or even defend them at all costs, and eventually become the joke of history.

It is suggested that the education department should suspend the infuse of infinite set theory, especially the theory of one-to-one corresponding and cardinal number, in middle school and undergraduate education, so as not to mislead children.