恩斯特·策梅洛Ernst Zermelo

恩斯特·策梅洛的父母是Ferdinand Zermelo和Maria Augusta Elisabeth Ziegler。他的父亲是一位大学教授，因此策梅洛在一个鼓励学术追求的家庭中长大。他的中学教育是在柏林的Luisenstädtisches 文理中学（Gymnasium）完成的，他于1889年从文理中学毕业。

当时，德国的学生在多所不同大学学习的习惯很普遍，策梅洛正是这样做的。他在三所大学学习，即柏林、哈勒和弗莱堡，所学科目相当广泛，包括数学、物理和哲学。

在这些大学里，他听了费迪南德·格奥尔格·弗罗贝尼乌斯、拉扎勒斯·福克斯、马克斯·普朗克、艾尔哈德·施密特、赫尔曼·阿曼杜斯·施瓦茨和Edmund Husserl的课程。这是一批令人印象深刻的、富有启发性的讲师，策梅洛在完成第一个学位后开始从事数学研究。他于1894年完成博士学位，当时柏林大学授予他学位，学位论文为Untersuchungen zur Variationsrechnung Ⓣ（变分法研究），该论文遵循卡尔·魏尔斯特拉斯的方法处理变分法。在这篇论文中，他[1]：

……将卡尔·魏尔斯特拉斯关于一类曲线上积分极值的方法推广到被积函数依赖于任意高阶导数的情况，同时给出了曲线空间中邻域概念的仔细定义。

获得博士学位后，策梅洛留在柏林大学，被任命为马克斯·普朗克的助手，后者在那里担任理论物理讲席。在这个阶段，策梅洛的工作更多地转向应用数学领域，并在马克斯·普朗克的指导下，他开始撰写关于流体力学的教授资格论文（Habilitation）学位论文。

1897年，策梅洛前往哥廷根，那里也许是当时世界上数学研究的领先中心，他在那里完成了他的任教资格论文，于1899年提交了他的学位论文Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche Ⓣ（球面上涡旋运动的流体动力学研究）。获得学位后，他立即凭借对统计力学以及变分法的贡献被任命为哥廷根的讲师。

策梅洛的研究方向很快发生了重大变化。格奥尔格·康托尔在1878年提出了连续统假设，猜想连续统的每个无限子集要么是可数的（即可以与自然数建立一一对应），要么具有连续统的基数（即可以与实数建立一一对应）。大卫·希尔伯特看到了这一点的重要性，他将连续统假设作为他在1900年巴黎演讲中提出的问题清单中的第一个。大卫·希尔伯特将此视为数学家们在1900年代应该攻克的最基本问题之一，他进一步提出了一种攻克这一猜想的方法。他建议首先应该尝试证明格奥尔格·康托尔的另一个猜想，即任何集合都可以良序化。

此时或许有必要给出良序集的定义。一个集合$S$是良序的，如果它上面定义了一个关系<，满足三个性质：

（i）对于$S$中的任意元素$a, b$，要么$a = b, a < b$，要么$b < a$。

(ii) 对于$S$中的每个$a, b, c$，若$a < b$且$b < c$，则$a < c$。

（iii）$S$的每个非空子集都有一个最小元素。

因此，带有通常序关系的自然数集合是一个良序集，但带有通常序关系的整数集合不是良序集，因为负整数子集没有最小元素。

策梅洛开始研究集合论的问题，特别是采纳了大卫·希尔伯特的想法，朝着解决连续统假设问题的方向前进。1902年，策梅洛发表了他关于集合论的第一部著作，内容是关于超限基数的加法。两年后，即1904年，他成功迈出了大卫·希尔伯特所建议的通往连续统假设的第一步，证明了每个集合都可以良序化。这一结果使策梅洛声名鹊起，也为他赢得了快速晋升，因为1905年12月，他被任命为哥廷根大学的教授。

选择公理是策梅洛证明每个集合都可以良序化的基础；事实上，选择公理等价于良序性质，所以我们现在知道这条公理必须被使用。他对良序性质的证明使用了选择公理，通过超限归纳法来构造集合。尽管策梅洛无疑因他对良序性质的证明而声名鹊起，但此时的集合论处于一种相当不寻常的境地，许多数学家拒绝接受策梅洛所发现的那种证明。关于数学中这些非构造性部分是否是合法的研究领域，存在着强烈的情绪，而策梅洛的想法肯定不被相当多的数学家所接受[1]：-

这个证明震动了数学界，并招致了大量批评——其中大部分是不公正的——策梅洛在Neuer Beweis Ⓣ（新证据）中优雅地回应了这些批评……

正如这段引文所示，策梅洛对这些批评的反应是试图用一种能被更广泛接受的证明来证明良序性质，而他在1908年发表的论文Neuer Beweis Ⓣ（新证据）中成功地做到了这一点。这是一篇他专门针对其工作的批评者而写的论文。一方面，他强调了其良序新证明的形式特征；另一方面，他论证说，他的批评者以及其他数学家在处理无限集合时也使用了选择公理。

策梅洛对公理化集合论做出了其他基础性贡献，这部分是他对该主题的第一个重大贡献受到批评的结果，部分是因为集合论开始在哥廷根成为一个重要的研究课题。集合论悖论最早出现在1903年前后，随着伯特兰·罗素悖论的发表。策梅洛实际上自己发现了一个类似的集合悖论，但没有发表这一结果。相反，这促使他首次尝试将集合论公理化，他于1905年开始这项任务。在产生了一个公理系统之后，他想在发表这项工作之前证明他的公理是一致的，但他未能做到这一点。

1908年，策梅洛尽管未能证明一致性，还是发表了他的公理系统。他给出了七条公理：外延公理、初等集合公理、分离公理、幂集公理、并集公理、选择公理和无穷公理。

策梅洛通常用文字而不是符号来陈述他的公理和定理。事实上，他并不经常使用当时正在使用的诸如∃或∀这样的量词形式语言以及绑定变量，相反，他使用诸如“存在”或“对所有”这样的普通表达。

值得评论的是，陶拉尔夫·斯科伦和亚伯拉罕·弗兰克尔在1922年前后独立地改进了策梅洛的公理系统。由此产生的系统，有十条公理，现在是公理化集合论最常用的系统。它使得集合论的矛盾可以被消除，同时经典集合论中排除悖论的结果可以被推导出来。

1910年，策梅洛被任命为苏黎世大学数学讲席后离开了哥廷根。他的健康状况很差，但因其对集合论的重大贡献而获得5000马克奖金，这帮助了他的处境。该奖金是在大卫·希尔伯特的倡议下颁发的，这无疑是为了让策梅洛休息并恢复健康。

1916年，他的健康状况仍未好转，策梅洛辞去了在苏黎世的讲席，迁往德国黑森林地区，在那里生活了十年。1926年，他被任命为弗赖堡（布赖斯高）的名誉讲席，但因不赞同希特勒的政权，于1935年放弃了这一讲席。第二次世界大战结束时，II Zermelo请求恢复他在弗赖堡的名誉职位，并于1946年确实得以复职。

Ernst Zermelo's parents were Ferdinand Zermelo and Maria Augusta Elisabeth Ziegler. His father was a college professor so Zermelo was brought up in a family where academic pursuits were encouraged. His secondary school education was at the Luisenstädtisches Gymnasium in Berlin and he graduated from the gymnasium in 1889.

At this time it was the custom for students in Germany to study at a number of different universities and indeed that is precisely what Zermelo did. His studies were undertaken at three universities, namely Berlin, Halle and Freiburg, and the subjects he studied were quite wide ranging and included mathematics, physics and philosophy.

At these universities he attended courses by Frobenius, Lazarus Fuchs, Planck, Schmidt, Schwarz and Edmund Husserl. This was an impressive collection of inspiring lecturers and Zermelo began to undertake research in mathematics after completing his first degree. His doctorate was completed in 1894 when the University of Berlin awarded him the degree for a dissertation Untersuchungen zur Variationsrechnung Ⓣ which followed the Weierstrass approach to the calculus of variations. In this thesis he [1]:-

... extended Weierstrass's method for the extrema of integrals over a class of curves to the case of integrands depending on derivatives of arbitrarily high order, at the same time giving a careful definition of the notion of neighbourhood in the space of curves.

After the award of his doctorate, Zermelo remained at the University of Berlin where he was appointed assistant to Planck who held the chair of theoretical physics there. At this stage Zermelo's work was turning more towards areas of applied mathematics and, under Planck's guidance, he began to work for his habilitation thesis studying hydrodynamics.

In 1897 Zermelo went to Göttingen, perhaps the leading centre for mathematical research in the world at that time, where he completed his habilitation, submitting his dissertation Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche Ⓣ in 1899. Immediately following the award of the degree he was appointed as a lecturer at Göttingen on the strength of his contributions to statistical mechanics as well as to the calculus of variations.

The direction of Zermelo's research was soon to take a major change. Cantor had put forward the continuum hypothesis in 1878, conjecturing that every infinite subset of the continuum is either countable (i.e. can be put in 1-1 correspondence with the natural numbers) or has the cardinality of the continuum (i.e. can be put in 1-1 correspondence with the real numbers). The importance of this was seen by Hilbert who made the continuum hypothesis the first in the list of problems which he proposed in his Paris lecture of 1900. Hilbert saw this as one of the most fundamental questions which mathematicians should attack in the 1900s and he went further in proposing a method to attack the conjecture. He suggested that first one should try to prove another of Cantor's conjectures, namely that any set can be well ordered.

Perhaps it would be helpful to give a definition of a well ordered set at this point. A set $S$ is well ordered if it has a relation < defined on it which satisfies three properties:

(i) for any elements $a, b$ in $S$ either $a = b, a < b$ or $b < a$.

(ii) for every $a, b, c$ in $S$ with $a < b$ and $b < c$ then $a < c$.

(iii) every non-empty subset of $S$ has a least element.

The set of natural numbers with the usual ordering is therefore a well ordered set but the set of integers is not well ordered with the usual ordering since the subset of negative integers has no least element.

Zermelo began to work on the problems of set theory, in particular taking up Hilbert's idea to head towards a resolution of the problem of the continuum hypothesis. In 1902 Zermelo published his first work on set theory which was on the addition of transfinite cardinals. Two years later, in 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved that every set can be well ordered. This result brought fame to Zermelo and also earned him a quick promotion for, in December 1905, he was appointed as professor in Göttingen.

The axiom of choice is the basis for Zermelo's proof that every set can be well ordered; in fact the axiom of choice is equivalent to the well ordering property so we now know that this axiom has to be used. His proof of the well ordering property used the axiom of choice to construct sets by transfinite induction. Although Zermelo certainly gained fame for his proof of the well ordering property, set theory at this time was in the rather unusual position that many mathematicians rejected the type of proofs that Zermelo had discovered. There were strong feelings as to whether such non-constructive parts of mathematics were legitimate areas for study and Zermelo's ideas were certainly not accepted by quite a number of mathematicians [1]:-

The proof stirred the mathematical world and produced a great deal of criticism - most of it unjustified - which Zermelo answered elegantly in Neuer Beweis Ⓣ...

As this quote indicates, Zermelo's reaction to these criticisms was to try to prove the well ordering property with a proof that would find more widespread acceptance, and this he succeeded in doing in the paper Neuer Beweis Ⓣ which he published in 1908. It was a paper which he specifically directed at critics of his work. On the one hand he emphasised the formal character of his new proof of the well ordering and on the other hand he argued that his critics, and other mathematicians, also used the axiom of choice when dealing with infinite sets.

Zermelo made other fundamental contributions to axiomatic set theory which were partly a consequence of the criticism of his first major contribution to the subject and partly because set theory began to become an important research topic at Göttingen. The set theory paradoxes first appeared around 1903 with the publication of Russell's paradox. Zermelo had in fact discovered a similar set paradox himself but did not publish the result. Rather it prompted him to make the first attempt to axiomatise set theory and he began this task in 1905. Having produced an axiom system he wanted to prove that his axioms were consistent before publishing the work, but he failed to achieve this.

In 1908 Zermelo published his axiomatic system despite his failure to prove consistency. He gave seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity.

Zermelo usually stated his axioms and theorems in words rather than symbols. In fact he did not often use the formal language for quantifiers such as ∃ or ∀ and binding variables that were then being used, instead, he used ordinary expressions such as "there exists" or "for all".

It is worth commenting that Skolem and Fraenkel independently improved Zermelo's axiom system in around 1922. The resulting system, with ten axioms, is now the most commonly used one for axiomatic set theory. It enables the contradictions of set theory to be eliminated yet the results of classical set theory excluding the paradoxes can be derived.

In 1910 Zermelo left Göttingen when he was appointed to the chair of mathematics at the Zürich University. His health was poor but his position was helped by the award of a prize of 5000 marks for his major contributions to set theory. The prize was awarded on the initiative of Hilbert and certainly it was an attempt to enable Zermelo rest and so to regain his health.

When his health had not improved by 1916 Zermelo resigned his chair in Zürich and moved to the Black Forest in Germany where he lived for ten years. He was appointed to an honorary chair at Freiburg im Breisgau in 1926 but he renounced his chair in 1935 because of his disapproval of Hitler's regime. At the end of the World War II Zermelo requested that he be reinstated to his honorary position in Freiburg and indeed he was reinstated to the post in 1946.