Zermelo-Fraenkel Axioms

David Baños Abril

12 de July de 2026 7 min read

Set theory was beginning to emerge as a fundamental pillar of mathematics. However, many believed it first needed to attain a sufficient level of rigor. A detailed justification requires a clear and explicit definition of the theory's foundations: an *axiomatization*. This is where the Zermelo-Fraenkel axioms come into play.

Axiomatization

The classical method of axiomatization sought to state the simple and intuitive principles on which a theory could be founded. In the Euclidean tradition, axioms were regarded as self-evident truths that required no proof. There should be no doubt whatsoever about these principles: they were presupposed as the starting point for all subsequent reasoning.

Late nineteenth-century mathematics began to understand the idea of an axiom differently. Axioms did not have to embody intuitive principles or self-evident truths of human thought. They came to establish an unambiguously defined symbolic structure within which proofs could be developed. The objects to which that structure was applied could vary according to the interpretation, provided that they satisfied the relations required by the axioms.

From this new perspective, axioms do not necessarily describe previously known objects. They implicitly determine how the objects in any domain satisfying the theory must relate to one another. We will discuss this formal conception of axiomatization further in the coming articles.

The axiomatization of sets

As we may recall from previous articles, Cantor sought nothing less than to construct the concept of natural number and the basic arithmetic operations from sets and the notions of ordinality and cardinality. Given that many mathematicians regarded Arithmetic as the mother of mathematics, presenting sets as the foundation of Arithmetic was virtually equivalent to grounding all mathematics upon them.

However, the paradoxes had shown that sets could not be formed freely from any property whatsoever. A reform of Set Theory from its very foundations had become urgent. This was the objective pursued by Zermelo.

Set Theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions of “number,” “order,” and “function,” taking them in their simple and pristine form, and thereby to develop the logical foundations of all Arithmetic and Analysis; it therefore constitutes an indispensable component of mathematical science.

Ernst Zermelo. Untersuchungen über die Grundlagen der Mengenlehre. I. 1908.

Zermelo’s axioms

In 1908, Zermelo presented his list of axioms, which would later become known as Zermelo’s Axiomatic Set Theory and would provide the basis for the subsequent Zermelo–Fraenkel axioms.

The aim was not to define directly what a set is by means of a general property. Zermelo sought to specify which set-forming operations were legitimate and under what conditions the existence of a new set could be asserted.

Domain of objects

Zermelo’s original formulation presupposed a universe or domain of objects. Among these objects were sets, characterized by their participation in the membership relation $\in$. Primitive elements, or urelements, could also be admitted: objects that could belong to sets but were not themselves sets.

The domain is the field over which the axioms are interpreted. The axioms are not properties that each object must satisfy individually, but global conditions that the entire structure formed by the domain and the membership relation must satisfy. If the axioms contradicted one another, no domain capable of satisfying them would exist.

Contemporary Set Theory is usually formulated as a pure theory: all of its objects are sets, and every element of a set is itself a set. Nevertheless, to understand Zermelo’s system historically, it is worth remembering that his original domain allowed a distinction between sets and elementary objects.

Axiom of Extensionality

The first of Zermelo’s axioms, the Axiom of Extensionality, states that two sets possessing exactly the same elements are one and the same set.

$$\forall A\forall B\left[\forall x\,(x\in A\leftrightarrow x\in B)\rightarrow A=B\right]$$

A set is determined exclusively by its elements. It does not matter how it was defined, described, or constructed: if two sets contain the same objects, they are equal.

This axiom establishes a criterion of identity for sets, but it does not by itself assert that any property determines a set. The rejection of the unrestricted Principle of Comprehension is expressed especially in the Axiom of Separation, which limits the formation of sets by means of properties.

The distinction is important. Two different properties may select exactly the same elements and therefore determine the same set. The identity of sets is extensional: it depends on their members, not on the way those members are described.

Axiom of Elementary Sets

The Axiom of Extensionality provides a criterion for comparing sets, but it does not yet guarantee that any sets exist. Zermelo’s second axiom, known as the Axiom of Elementary Sets, asserted the existence of certain basic sets.

First, there exists a set to which no element belongs: the empty set, which we denote by the symbol $\emptyset$.

$$\forall x\,(x\notin\emptyset)$$

Moreover, for any object $a$ there exists the singleton set $\{a\}$, whose sole element is $a$. For any objects $a$ and $b$, there also exists the pair set $\{a,b\}$, consisting of exactly those two objects:

$$\{a\}\qquad \{a,b\}$$

In modern formulations, these claims are usually separated into the existence of the empty set and the Pairing Axiom. The singleton set can be obtained by setting $a=b$:

$$\{a,a\}=\{a\}$$

By repeatedly applying this operation, we can form arbitrarily long chains of sets:

$$\emptyset,\quad \{\emptyset\},\quad \{\{\emptyset\}\},\quad \{\{\{\emptyset\}\}\},\ldots$$

This allows us to construct as many finite sets as we wish. It does not yet, however, guarantee the existence of a single set containing all these stages. For that, the Axiom of Infinity will be required.

Axiom of Separation

For Zermelo, a property was well-defined when the fundamental relations of the domain, the axioms, and the laws of logic made it possible to determine without arbitrariness whether or not it held for each object under consideration.

This notion of a well-defined property was still imprecise. Zermelo did not specify exactly which logical expressions could be used. The issue would be debated over the following years and would ultimately be resolved by formulating Set Theory in a formal logical language.

The Axiom of Separation states that a well-defined property makes it possible to select the elements of a set $A$ that satisfy it. The result is a new set $B$, a subset of $A$:

$$B=\{x\in A\mid \varphi(x)\}$$

More explicitly:

$$\forall x\left[x\in B\leftrightarrow\bigl(x\in A\land\varphi(x)\bigr)\right]$$

The reader may notice the apparent similarity between this axiom and the Principle of Comprehension, since both allow sets to be formed by means of properties. The difference is fundamental: the Axiom of Separation only permits elements to be selected from within a previously given set.

We cannot directly form “the set of all things satisfying $\varphi$.” We can only form the subset of those elements of $A$ that satisfy $\varphi$. This restriction prevents Russell’s Paradox from being reproduced.

For example, given a set $A$, we may define:

$$R_A=\{x\in A\mid x\notin x\}$$

The construction does not lead to a contradiction. If $R_A$ belonged to $A$, it would follow that $R_A\notin A$; we therefore conclude simply that $R_A\notin A$. What cannot be assumed is the existence of a universal set containing absolutely every set.

Power Set Axiom

Every set $A$ determines a new set $\mathcal{P}(A)$, called the power set, consisting of all and only the subsets of $A$:

$$\mathcal{P}(A)=\{B\mid B\subseteq A\}$$

For example, if $A=\{a,b\}$, then:

$$\mathcal{P}(A)=\{\emptyset,\{a\},\{b\},\{a,b\}\}$$

Readers who have followed this series will quickly recognize the importance of this axiom for projecting Set Theory onto numerical domains. By Cantor’s Theorem, the power set always has a strictly greater cardinality than the original set:

$$|A|<|\mathcal{P}(A)|$$

When $A=\mathbb{N}$, the power set has the cardinality of the continuum:

$$|\mathcal{P}(\mathbb{N})|=2^{\aleph_0}$$

The assertion that this cardinality is precisely $\aleph_1$ is equivalent to the Continuum Hypothesis and does not follow from the Power Set Axiom alone.

Axiom of Union

Given a set $A$ whose elements are themselves sets, there exists the union set $\bigcup A$, consisting of all the elements that belong to at least one of the sets contained in $A$:

$$\bigcup A=\{x\mid \exists B\,(B\in A\land x\in B)\}$$

For example, if:

$$A=\bigl\{\{a,b\},\{b,c\}\bigr\}$$

then:

$$\bigcup A=\{a,b,c\}$$

Axiom of Choice

As we saw in the previous article, the Axiom of Choice states that, given a family $\mathcal{A}$ of nonempty sets, there exists a function $f$ that selects one element from each of them:

$$\forall A\in\mathcal{A}\qquad f(A)\in A$$

In Zermelo’s original formulation, the axiom was expressed in terms of a family of pairwise disjoint nonempty sets. It then asserted the existence of a set containing exactly one representative from each member of the family.

The formulation in terms of choice functions is more general and does not require the sets to be disjoint. The two versions are equivalent under suitable transformations.

Zermelo had used this axiom to prove the Well-Ordering Theorem. Its inclusion in the 1908 axiomatization made explicit one of the most controversial principles of the new Set Theory.

Axiom of Infinity

The preceding axioms allow finite sets of arbitrarily large size to be constructed, but they do not guarantee the existence of a single infinite set. The Axiom of Infinity states that there exists a set $I$ that contains the empty set and that, together with each element $x$, also contains its set-theoretic successor $x\cup\{x\}$:

$$\emptyset\in I\land\forall x\in I\bigl(x\cup\{x\}\in I\bigr)$$

Starting from the empty set, this operation generates:

$$\emptyset,\quad \{\emptyset\},\quad \{\emptyset,\{\emptyset\}\},\quad \ldots$$

These sets can be identified respectively with the natural numbers $0$, $1$, $2$, and so on. By means of the Axiom of Separation, one can isolate within $I$ the smallest set that contains $\emptyset$ and is closed under the successor operation. This set is denoted by $\omega$ and represents the set of natural numbers.

The combination of the Axiom of Infinity with the Power Set Axiom makes it possible to form $\mathcal{P}(\omega)$, whose cardinality is $2^{\aleph_0}$, strictly greater than that of the natural numbers. Nevertheless, as already noted, identifying $2^{\aleph_0}$ with $\aleph_1$ requires accepting the Continuum Hypothesis.

The introduction of a completed infinite set was one of the most controversial aspects of Zermelo’s axioms. Throughout this series, we have seen the striking conclusions that can be obtained by applying certain forms of reasoning to infinity. It is unsurprising that some mathematicians rejected the acceptance of actual infinity as a given object.

The Axiom of Replacement

A few years after the publication of the axioms, Abraham Fraenkel and Thoralf Skolem independently observed that Zermelo’s principles were insufficient to guarantee the existence of certain sets commonly used in the theory of ordinals and cardinals.

The problem was not that the axioms were contradictory, but that they were too weak to capture certain transfinite operations.

Set of power sets

The Power Set Axiom can be applied repeatedly. Given a set $Z$, we can form:

$$Z,\quad \mathcal{P}(Z),\quad \mathcal{P}(\mathcal{P}(Z)),\quad \mathcal{P}(\mathcal{P}(\mathcal{P}(Z))),\ldots$$

Let $\mathcal{P}^n(Z)$ denote the result of applying the power-set operation $n$ times:

$$\mathcal{P}^0(Z)=Z\qquad \mathcal{P}^{n+1}(Z)=\mathcal{P}(\mathcal{P}^n(Z))$$

For each natural number $n$, Zermelo’s axioms allow us to form $\mathcal{P}^n(Z)$ individually. It does not follow, however, that there exists a set that simultaneously collects all these stages:

$$\{\mathcal{P}^n(Z)\mid n\in\omega\}$$

The difficulty is subtle. We can construct each element of the sequence separately, but we need an additional principle to assert that the results of all those constructions themselves form a set.

Once the sequence has been collected, the Axiom of Union would allow us to form:

$$\bigcup_{n\in\omega}\mathcal{P}^n(Z)$$

This set represents a limit stage that cannot be reached by applying the Power Set Axiom a finite number of times.

Axiom of Replacement

The preceding operation consists in associating with each natural number $n$ a unique set $\mathcal{P}^n(Z)$. The Axiom of Replacement states that if we apply a well-defined functional relation to every element of a set, the results also form a set.

If a property $\varphi(x,y)$ associates each element $x$ of a set $A$ with a unique object $y$, then there exists a set $B$ consisting exactly of those values:

$$\forall x\in A\,\exists!y\,\varphi(x,y)\rightarrow\exists B\,\forall y\left[y\in B\leftrightarrow\exists x\in A\,\varphi(x,y)\right]$$

More intuitively, if $f$ is a function defined on a set $A$, its image is also a set:

$$f[A]=\{f(x)\mid x\in A\}$$

The axiom does not permit each element to be replaced arbitrarily by anything whatsoever without any condition. There must be a functional rule that determines a unique result for each element of the initial set.

Fraenkel and Skolem proposed this principle in 1922. Its true importance would become especially apparent with the development of the modern theory of ordinals.

Von Neumann ordinals

John von Neumann made one of the most fruitful contributions to twentieth-century Set Theory. He took up Cantor’s concept of ordinal and proposed representing each ordinal by the set of all its predecessors.

The first ordinals are defined as follows:

$$0=\emptyset$$

$$1=\{0\}=\{\emptyset\}$$

$$2=\{0,1\}=\{\emptyset,\{\emptyset\}\}$$

$$3=\{0,1,2\}$$

In general, each ordinal $\alpha$ is the set of all preceding ordinals:

$$\alpha=\{\beta\mid \beta<\alpha\}$$

If a set $A$ is well-ordered by a relation $\prec$, we can recursively associate with each element $a$ the set of the images corresponding to all its predecessors:

$$f(a)=\{f(u)\mid u\prec a\}$$

The result is an ordinal representing the order type of $A$. The Axiom of Replacement makes it possible to collect into a set the images $f(u)$ of all predecessors and to develop these transfinite constructions rigorously.

The Axiom of Choice is not required to define von Neumann ordinals. It enters when we wish to prove that an arbitrary set can be well-ordered and therefore placed in correspondence with some ordinal.

The Axiom of Foundation

Zermelo’s original system also did not prevent the existence of infinite descending membership chains:

$$\cdots\in A_3\in A_2\in A_1$$

Nor did it explicitly exclude cycles such as $A\in A$. These possibilities were incompatible with the intuitive picture of sets as objects progressively constructed from earlier sets.

The Axiom of Foundation, developed from work by Mirimanoff, von Neumann, and other authors, states that every nonempty set $A$ contains some element $x$ that shares no element with $A$:

$$\forall A\left[A\neq\emptyset\rightarrow\exists x\in A\,(x\cap A=\emptyset)\right]$$

This axiom excludes infinite descending membership chains and, in particular, prevents a set from belonging to itself. It thus contributes to the cumulative conception of Set Theory: sets are formed in stages from sets constructed earlier.

The Axiom of Foundation was not part of Zermelo’s 1908 system, but it would eventually be incorporated into the standard formulation of the Zermelo–Fraenkel axioms.

Conclusion

Today, the system formed from Zermelo’s principles and enriched primarily by the Axiom of Replacement and the Axiom of Foundation is known as the Zermelo–Fraenkel axioms, or ZF axioms. Separation and Replacement are not in fact single axioms, but axiom schemas: there is a distinct instance for every admissible formula in the language.

The Axiom of Choice is not included in the designation ZF. When it is added, the resulting system is called ZFC, where the letter $C$ comes from the English term Choice.

For a long time, mathematicians debated whether the Axiom of Choice could be proved from the other axioms or whether its negation necessarily led to a contradiction. In 1922, Fraenkel obtained a precursor result using systems with urelements, but it did not yet constitute a proof of independence from pure ZF.

The question would be resolved through the combined work of Kurt Gödel and Paul Cohen. Gödel proved that if ZF is consistent, then so is ZF together with the Axiom of Choice. Cohen later proved that if ZF is consistent, then so is ZF together with the negation of the Axiom of Choice.

Therefore, assuming ZF is consistent, the Axiom of Choice can neither be proved nor refuted from the remaining axioms. ZF and ZFC admit different models and allow different theorems to be proved.

The Zermelo–Fraenkel axioms were repeatedly debated, extended, and reformulated. In 1930, Zermelo presented a new investigation into set domains and the cumulative structure of the set-theoretic universe. The system used today, however, is not simply Zermelo’s version of 1908 or 1930, but the result of contributions by Zermelo, Fraenkel, Skolem, von Neumann, and other mathematicians.

Axiomatization transformed Set Theory. The question was no longer to decide intuitively which collections could be regarded as sets, but to study which objects and operations were licensed by an explicit formal structure. At the same time, it opened the possibility of investigating the axiomatic systems themselves: their consistency, their independence, and the different models capable of satisfying them.