In 1908 Zermelo published his paper “Investigations in the foundations of set theory”. This contained the axiom system that eventually led to ZFC. Zermelo opens the paper with this rationale:
Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions “number”, “order”, and “function” … At present, however, the very existence of this discipline seems to be threatened by certain contradictions, or “antinomies” [such as the Russell paradox]. …it no longer seems admissible today to assign to an arbitrary logically definable notion a “set”, or “class”, as its “extension”. Cantor’s original definition of a “set” as “a collection, gathered into a whole, of certain well-distinguished objects of our perception or our thought” therefore certainly requires some restriction … Under these circumstances there is at this point nothing left for us to do but to proceed in the opposite direction and, starting from “set theory” as it is historically given, to seek out the principles required for establishing the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valuable in this theory.1
Zermelo’s motivation is pragmatic, unlike the philosophical approach of Russell and Whitehead’s Principia Mathematica.
Axiomatization was “in the air” at this time, with people throwing out various suggestions. Moore (p.151) offers some examples. Julius König proposed two axioms: (1) There are mental processes satisfying the formal laws of logic. (2) The continuum ℝ, treated as the totality of all sequences of natural numbers, does not lead to a contradiction. Schoenflies opted for the Trichotomy of Cardinals, and wanted to hold on to the Principle of Comprehension. Cantor sent a letter to Hilbert with some principles that look a lot like Zermelo’s axioms, but this letter didn’t come to light until decades later.
Zermelo’s article stands out as the first published proposal with a full set of axioms, demonstrating that it could save some of Cantor’s Paradise, and recognizing that the Principle of Comprehension was kaput.
Zermelo’s eschews philosophy:
The further, more philosophical, question about the origin of these principles and the extent to which they are valid will not be discussed here. I have not yet even been able to prove rigorously that my axioms are “consistent”, though this is certainly very essential…
The paper, he says, develops the theory of equivalence in a manner “that avoids the formal use of cardinal numbers.” He promises a second part, dealing with well-ordering, but this never appeared.
After the introduction, Zermelo begins:
- Set theory is concerned with a “domain” 𝔅 of individuals, which we shall call simply “objects” and among which are the “sets”. …
- Certain “fundamental relations” of the form aεb obtain between the objects of the domain 𝔅. …An object b may be called a set if and—with a single exception (Axiom II)—only if it contains another object, a, as an element. [I will use the modern ∈ in place of Zermelo’s ε from now on.]
- [Definition of subset and disjoint]
- A question or assertion 𝔈 is said to be “definite” if the fundamental relations of the domain, by means of the axioms and the universally valid laws of logic, determine without arbitrariness whether it holds or not. Likewise a “propositional function” 𝔈(x), in which the variable term x ranges over all individuals of a class 𝔎, is said to be “definite” if it is definite for each single individual x of the class 𝔎. Thus the question whether a∈b or not is always definite, as is the question whether M⊆N or not.
Note that item (1) allows for so-called urelements or atoms—things like integers. ZFC is a so-called “pure” set theory, without atoms.
Next come seven axioms, interlarded with extensive discussion.
- Extensionality:
- “Every set is determined by its elements.” In other words, if M⊆N and N⊆M, then M=N.
- Elementary Sets:
- The null set exists. Given any elements a and b of the domain, the sets {a} and {a,b} exist.
- Separation:
- To quote Zermelo: “Whenever the propositional function 𝔈(x) is definite for all elements of a set M, M possesses a subset M𝔈 containing as elements precisely those elements x of M for which 𝔈(x) is true.”
Zermelo notes that “sets may never be independently defined by means of this axiom but must always be separated as subsets from sets already given”, and that this prevents the Russell paradox and the like. Indeed, the Russell paradox is turned into a theorem: for any set M there is a subset M0 such that M0∉M. He also notes that “definiteness” precludes some semantic paradoxes, e.g., Richard’s paradox (see post 5).
Zermelo shows that Separation implies the existence of set differences M∖M1 (denoted M−M1) and intersections M∩N (denoted [M,N]), and even ⋂X∈TX for a set of sets (which he denotes 𝔇T, for “Durchschnitt”).
- Power Set:
- For any set T, there is a set whose elements are precisely all of T’s subsets. He denotes the power set of T by 𝔘T (for “Untermengen”).
- Union:
- For any set T, there is a set whose elements are precisely the elements of the elements of T. In modern notation, ⋃X∈TX. Denoted 𝔖T, for “Summe”. He writes M+N for our M∪N.
- Choice:
- Given any set T of mutually disjoint nonempty sets, the union ⋃X∈TX contains a subset S such that S∩X is a singleton for each X∈T.
Zermelo adds, “We can also express this axiom by saying that it is always possible to choose a single element from each element M,N,R,… of T and to combine all the chosen elements, m,n,r,…, into a set” S.
The set of all S’s satisfying this condition (card(S∩X)=1 for all X∈T) Zermelo calls the product of the elements of T, denoted 𝔓T, or just MN for a pair of disjoint sets M and N.
- Infinity:
- There is a set Z containing the null set 0, and for each of its elements a, it also contains {a}.
This leads to the so-called “Zermelo finite ordinals”, 0, {0}, {{0}}, {{{0}}}, etc. Z contains all these, and using Separation, we can assume Z contains exactly these. The Zermelo finite ordinals have two drawbacks: (1) They don’t extend naturally into the infinite ordinals; (2) Each of them, except 0, contains exactly one element. The von Neumann ordinals removed both of these blemishes.
The rest of the paper develops the theory of equivalence from the axioms. I noted that Zermelo allows atoms. On the other hand, he does not have ordered pairs, and thus neither relations nor functions. This lack calls for some gymnastics. When M and N are disjoint, the set of all unordered pairs MN={{m,n}:m∈M, n∈N} substitutes for our M×N.
To define equivalence between sets M and N, he assumes first that M and N are disjoint. Using MN instead of M×N, he can define “bijection between M and N’’. If one exists, then M and N are “immediately equivalent”. Dropping the disjointness condition, he says M and N are “mediately equivalent” if there exists a third set that is disjoint from both and “immediately equivalent” to both. It takes a couple of pages to show that this definition makes sense.
Zermelo proves the Equivalence Theorem, that is the “(Cantor-Dedekind-Schröder)-Bernstein Theorem”. (A couple of decades later, he discovered that Dedekind had basically the same proof.) He gives detailed proofs of the basic facts about equivalence. He defines “M has lower cardinality than N’’ in the usual fashion (M injects into N but not vice versa) but avoids defining “cardinal number”, as he promised in the introduction. The paper crescendoes in a proof of J. König’s inequality, a generalization of Cantor’s 𝔪<2𝔪. Expressed using cardinal numbers, this says that if 𝔪k<𝔫k for all k in some index set K, then ∑k𝔪k<∏k𝔫k. Zermelo, of course, phrases this without mentioning cardinal numbers.
Zermelo spills a fair amount of ink on the question of “definiteness”. He initially claims that a∈b and M⊆N are definite questions, as we’ve seen. When defining 𝔇T, he notes that for any object a, the set Ta={X∈T: a∈X} exists by Separation (because a∈X is definite). But the question whether Ta=T is also definite. So using Separation again, 𝔇T={a∈A: Ta=T}, where A is any element of T. A similar discussion accompanies his definition of “immediately equivalent” showing that it is definite whether a given subset of MN defines a bijection.
Nonetheless, a certain nimbus of indefiniteness surrounds Zermelo’s “definite”. Twenty-one years later, Zermelo published a paper, “On the concept of definiteness in axiomatics”. By this time, people had suggested replacing “definite” with “definable in first-order logic”. Zermelo did not accept this, and his proposal had no influence on ZFC.
[1] Despite this clear statement from Zermelo, Moore argues that “his axiomatization was primarily motivated by a desire to secure his demonstration of the Well-Ordering Theorem and, in particular, to save his Axiom of Choice” (Moore (p.159)). He notes that Zermelo composed the two 1908 papers—the axiomatization, and the second well-ordering proof—together, and that “there are numerous internal links connecting the two papers” (Moore). Zermelo’s biographer takes an intermediate view: “Above all, however, one has to take into consideration how deeply Zermelo’s axiomatic work was entwined with Hilbert’s programme of axiomatization and the influence of the programme’s ‘philosophical turn’ which was triggered by the publication of the paradoxes in 1903” (Ebbinghaus (p.81)).
Diagonal Argument 