Ernst Zermelo undertook the first comprehensive axiomatization of (Cantor’s) set theory, establishing key axioms as the basis of subsequent mathematics.

Between 1895 and 1897, Georg Cantor published his chief papers on ordinal and cardinal numbers, the culmination of three decades of research on aggregates, collections, or sets. Because of its novel treatment of topics such as transfinite numbers, Cantor’s set theory also was effectively a new theory of the infinite in mathematics as a legitimate and consistently definable entity. However, many critics from logic, mathematics, and several schools of philosophy refused to accept Cantor’s set theory, not only because of several omissions on Cantor’s part (for example, inadequate discussion of his well-ordering principle) but also because of a number of ambiguities and contradictions that apparently could be deduced from set theory. Mathematics;set theory
Set theory;axiomization
Axiomization of set theory
[kw]Zermelo Undertakes Comprehensive Axiomatization of Set Theory (1904-1908)
[kw]Comprehensive Axiomatization of Set Theory, Zermelo Undertakes (1904-1908)
[kw]Axiomatization of Set Theory, Zermelo Undertakes Comprehensive (1904-1908)
[kw]Set Theory, Zermelo Undertakes Comprehensive Axiomatization of (1904-1908)
Mathematics;set theory
Set theory;axiomization
Axiomization of set theory
[g]Germany;1904-1908: Zermelo Undertakes Comprehensive Axiomatization of Set Theory[00970]
[c]Mathematics;1904-1908: Zermelo Undertakes Comprehensive Axiomatization of Set Theory[00970]
Zermelo, Ernst
Cantor, Georg
Hilbert, David
Russell, Bertrand
Fraenkel, Abraham Adolf
Skolem, Thoralf Albert
Gödel, Kurt

One of the earliest paradoxes arose about 1895 directly from one of Cantor’s theorems. According to the theorem, for every set of ordinal numbers (numbers that designate the order and position of numbers in a series: first, second, and so on), there is one ordinal number larger than all ordinal numbers of the set. An apparent contradiction arises when one applies this theorem to consider the set of all ordinal numbers, as Cantor described to David Hilbert. Greater problems resulted from Bertrand Russell’s antinomy of 1902, involving the set of all sets that do not contain themselves as a subset. Russell’s paradox[Russells paradox]
Paradox, Russell’s[Paradox, Russells] Many aspects of these and other paradoxes arose because Cantor’s works gave no clear and consistent definition of the concept of ordinal versus cardinal numbers (integers one, two, and so on). Russell, Hilbert, and Giuseppe Peano, among others, saw the paradoxes as extremely threatening.

Further problems with Cantor’s “naïve” set theory arose concurrently, when Cantor lost confidence that the well-ordering principle might be a necessary universal law of thought and concluded, instead, that it was in need of clarification and proof, a conclusion with which Hilbert strongly agreed. The well-ordering principle basically states that it is possible to select simultaneously or choose—in an abstract yet valid sense—sets with infinite elements without having to specify or carry out this selecting through actual operations or calculations. In his international conference review paper of 1900 on the major unsolved problems then facing mathematicians, Hilbert placed Cantor’s set theory at the top of his famous list of twenty-three problems.

By 1902, Russell had convinced fellow logician Gottlob Frege Frege, Gottlob that numerous other contradictions were inherent in Cantor’s set theory, some generalized criticisms of which appear in Russell’s 1903 The Principles of Mathematics. Principles of Mathematics, The (Russell) In 1903, Hungarian mathematician Jules Konig Konig, Jules presented a paper that challenged Cantor’s claims for being able to order and proceed between different orders or levels of infinity and also attacked the more general claim that any set can be well-ordered. Within a day of Konig’s presentation, a student of Cantor and Hilbert, Ernst Zermelo, prepared a counterdemonstration that vindicated Cantor’s system by proving that Konig had improperly used a previous result from the work of Felix Bernstein Bernstein, Felix that Beppo Levi Levi, Beppo had examined independently and attacked in his paper on Cantor’s axiom of choice. Nevertheless, in the face of questions from mathematicians such as Hilbert, Russell, Émile Borel, Henri-Léon Lebesgue, and W. H. Young, Zermelo (at the urging of Hilbert and Cantor) sought to establish more strongly the overall validity of Cantor’s set theory.

Zermelo had already become well acquainted with many aspects and problems of Cantor’s theory. In 1899, Zermelo discovered Russell’s paradox and informed Hilbert. In lectures he delivered at the University of Göttingen in spring, 1901, Zermelo expressed doubts regarding extant validations by Cantor of some of his set theoretic axioms on cardinality. Assisted by Erhard Schmidt, Schmidt, Erhard Zermelo developed the background to what in 1904 would be his published proof of Cantor’s well-ordering principle, using the controversial axiom of choice. In this much-criticized paper, Zermelo proved that in every set, an ordering principle can be introduced in the form of the relation “a comes before b.” Central to controversies around Zermelo’s proof was his argument that for every subset M, a corresponding element m can be imagined. Many doubted that such formal imaginatory postulation had any real meaning, unless, for example, a concrete procedure or algorithm for carrying out the one-to-one correspondence between members of two sets could be given. For Zermelo, Hilbert, and other “formalists,” however, such principles were irreducible and self-evident.

The other key aspect of Hilbert’s formalism that influenced Zermelo’s efforts to clarify and reorder set theory was that of axiomatics. Hilbert’s Grundlagen der Geometrie (1899; The Foundations of Geometry, 1902) Foundations of Geometry, The (Hilbert) first made clear the benefits and economy derived from using a minimum system of primitive assumptions, combined through explicit axioms to redefine naïve/intuitive areas of mathematics such as geometry. Hilbert was interested in seeing a rigorous proof that the real number system is a consistent set, because in showing the completeness and consistency of Euclidean geometry, Hilbert had assumed the consistency of the real numbers. In 1900-1901, Hilbert attempted a preliminary axiomatization of set theory. Proposing the basic concept of number as a given, Hilbert sought to determine the relation between this and other primitive concepts by introducing axioms such as the operations of arithmetic (+, -, ×, ), the commutative and associative laws, several ordering relations, the “Archimedean” axiom of continuity, and a completeness axiom. Hilbert believed (but did not show in detail) that his systematization of set theory was consistent, complete, and noncontradictory with regard to all the then-known theorems about the real numbers.

In his efforts at the axiomatization of set theory, Zermelo was motivated not solely by the logical paradoxes but also by his desire to continue Hilbert’s efforts. In mid-1908, Zermelo published two papers in Hilbert’s journal Mathematische Annalen in which he set out his own more comprehensive set theory axiomatics. He began, like Hilbert, with a specific domain of mathematical objects, including sets, with the defined relations of membership between objects of his domain as a key set theory relation. Zermelo then stated seven central axioms, which he asserted were mutually independent and consistent. Zermelo’s seven axioms can be simply described as the axioms of extensionality, elementary sets, separation, power sets, union, choice, and infinity.

The axiom of choice validated a major means of proving inductively that a given property holds for all set elements if it holds for one element. Zermelo outlined how many extant axioms of arithmetic could be derived from these axioms, including operations of union, intersection, product, equivalence, and functionality. He devoted much of his discussion to developing a theory of cardinals (transfinite numbers) in terms of specific ordinal numbers, showing that the earlier Frege-Russell definition of cardinal numbers was incompatible with his and Hilbert’s system of arithmetic. Zermelo devoted minimal discussion to the paradoxes of Cantor, Russell, and others, as he believed that his “condition of definiteness of subsets” avoided any contradictory construction. Definiteness, or a definite property, E, of a set S is said to be one for which the seven axioms permit one to determine whether E holds for any element of S.

Zermelo’s axiom system for Cantor’s set theory was not acknowledged at first. Most initial attention focused almost exclusively on its weak points, such as Russell’s and Henri Poincaré’s suspicions that Zermelo’s formal system was inconsistent or employed methods or ideas improper for mathematics. Poincaré, in particular, objected to formal axiomatization as contrary to the constructive intuition and creation of mathematical entities. Russell’s criticism was that set theory, with arithmetic, is a part of logic and can be made consistent only when recast into symbolic logic. Russell and Alfred North Whitehead, in their Principia Mathematica
Principia Mathematica (Whitehead and Russell) (1910-1913), presented their theory of “types” to resolve the paradoxes of set theory. This established a hierarchy of types of sets, which includes the real numbers but also contains the unintuitive axiom of irreducibility. This states that for any property of types higher than order -0, there is a property over the same range of order -0 that can be shown as equivalent to the assertion that impredicative definitions are equivalent to predicative definition. (An impredicative definition is one made only in terms of the totality of which it is a part.)

Despite Zermelo’s concise and clear exposition, his axiomatic set theory was largely ignored in favor of nonaxiomatic set theory until 1921. In that year, Abraham Adolf Fraenkel, in his efforts to demonstrate the independence of the axiom of choice from other axioms, first pointed out some shortcomings in Zermelo’s system as well as their remedies. Fraenkel objected that Zermelo’s axiom of infinity was too weak, that his property of definiteness was too vague, and that his total system was insufficiently categorical to encompass all of ordinal arithmetic and the transfinite numbers. This led Fraenkel to add what is now known as the axiom of replacement, which states that if each element of a set is associated with one and only one set, then the collection of associated sets is a set. In place of Zermelo’s notion of a definite property, Fraenkel proposed the notion of a new function. In 1923, the Swedish mathematician Thoralf Albert Skolem independently arrived at a similar result regarding “definiteness” as a property expressible in first-order logic. Skolem proposed that within any predicate calculus (first-order logic) it is impossible to establish an ultimate categorical system of axioms for the natural numbers by means of a finite set of axioms, a result suggestive of Kurt Gödel’s later theorems. With these modifications, Zermelo’s theory has become known as the Zermelo-Fraenkel system for set theory.

In 1923 and 1925, John von Neumann Von Neumann, John published papers presenting, respectively, a new definition of ordinal number sets and an alternate axiomatic for set theory. Paul Bernays Bernays, Paul in 1937 and Gödel in 1940 published alternatives to the Zermelo-Fraenkel theory of sets. A system of ten other axioms was developed, taking “classes” as the basic variable, for which sets are those classes adequate for arithmetic with natural/real numbers and proper sets are those nonset classes for which contradictions can arise. Although others, such as Willard van Orman Quine in 1958, offered further alternative axiomatizations for set theory, to date the Zermelo-Fraenkel and von Neumann-Bernays-Gödel theories remain the only working options within mathematics.

Despite the avoidance of earlier paradoxes, axiomatic set theory is by no means a closed subject. In 1963, Paul Cohen proved that each of Zermelo’s axioms is logically independent, a result that opened several avenues of continuing inquiries. Although the relative consistency of both set theories can be demonstrated with respect to other mathematical theories, any would-be absolute validation of axiomatic set theory is as yet unavailable. Because Gödel’s inconsistency and indecidability theorems apply to both set theories, such absolute verification may never be forthcoming. Most contemporary efforts are directed toward providing at least a more complete and comprehensive set theoretic description, a direction that was first pioneered by Zermelo. Mathematics;set theory
Set theory;axiomization
Axiomization of set theory

• Cohen, Paul J. Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966. An advanced and technical account of Cohen’s independence proof of the axiom of choice in the context of the Zermelo-Fraenkel set theory.
• Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, Mass.: Harvard University Press, 1979. A rigorous yet nontechnical account of naïve set theory as Zermelo found it.
• Fraenkel, Abraham Adolf, and Yehoshua Bar-Hillel. Foundations of Set Theory. Amsterdam: North-Holland, 1958. Fraenkel’s main work on axiomatic set theory. Contains key material and a discussion of his 1921 and 1922 papers.
• Halmos, Paul R. Naive Set Theory. Princeton, N.J.: D. Van Nostrand, 1960. A general undergraduate introductory-level text that addresses what is essentially Cantor’s original set theory.
• Hamilton, A. G. Numbers, Sets, and Axioms: The Apparatus of Mathematics. Cambridge, Mass.: Harvard University Press, 1983. An introductory survey of naïve set theory and the basics of axiomatic set theory in the context of prior and subsequent mathematics.
• Moore, Gregory H. Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. New York: Springer-Verlag, 1982. An exhaustive historical-conceptual treatment of some background and developmental aspects of Zermelo’s set theory axiomatics.
• Potter, Michael. Set Theory and Its Philosophy: A Critical Introduction. New York: Oxford University Press, 2004. A comprehensive introduction to modern set theory. Includes extensive coverage of cardinal and ordinal arithmetic, equivalents of the axiom of choice, and other axiom candidates.
• Stoll, Robert Roth. Set Theory and Logic. San Francisco: W. H. Freeman, 1960. A logical representation of Cantor’s set theory. Points out sources of the paradoxes and their avoidance by axiomatic set theory.
• Suppes, Patrick. Axiomatic Set Theory. Princeton, N.J.: D. Van Nostrand, 1960. An intermediate-level account of set theory from a logician’s perspective.
• Tiles, Mary. The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise. Mineola, N.Y.: Dover, 2004. Begins with perspectives on the finite universe, classes, and Aristotelian logic and then examines, among many other topics, Cantor’s transfinite paradise, axiomatic set theory, and the constructs and reality of mathematical structure. Appropriate for undergraduate- and graduate-level readers.
• Watson, S., and J. Steprans, eds. Set Theory and Its Applications. New York: Springer-Verlag, 1989. An intermediate-advanced text on set theoretic applications in computer and engineering science.

Levi Recognizes the Axiom of Choice in Set Theory

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Gödel Proves Incompleteness-Inconsistency for Formal Systems

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