Zermelo–Fraenkel Set Theory

Zermelo–Fraenkel set theory, abbreviated ZF, is an axiomatic system of set theory in which mathematical objects are represented by sets. It specifies principles governing membership and set formation, avoiding contradictions associated with unrestricted collection-building. ZF does not include the axiom of choice; adding that principle produces ZFC, a standard foundation for mathematics. Omitting choice does not mean asserting its negation. (people.math.ethz.ch)

Historical development and formal language

Ernst Zermelo published an axiomatization of set theory in 1908. In 1922, Abraham Fraenkel and Thoralf Skolem independently extended and clarified this framework, notably through replacement and the formulation of definable properties. Later developments incorporated foundation; Zermelo presented a revised system in 1930. The modern formulation reflects these contributions rather than a single original publication. (people.math.ethz.ch)

ZF is expressed in first-order logic with equality. Its only primitive nonlogical symbol is the membership relation, $x\in y$, read “$x$ is an element of $y$.” Variables range over sets, and elements of sets are themselves sets: the ordinary formulation has no primitive non-set objects, or atoms. Expressions for subsets, unions, and functions are introduced through definitions rather than additional primitive entities. (people.math.ethz.ch)

An axiom states a basic principle, whereas an axiom schema specifies an axiom for each eligible formula. Separation and replacement are schemas, so their familiar names designate infinitely many first-order axioms, not single sentences quantifying over arbitrary properties. (plato.stanford.edu)

The axioms

A common presentation contains the following principles:

• Extensionality: sets with exactly the same elements are equal.
• Empty set: an empty set, $\varnothing$, exists.
• Pairing: for any sets $a,b$, the set $\{a,b\}$ exists.
• Union: for any set $A$, a set $\bigcup A$ contains exactly the elements belonging to members of $A$.
• Power set: every set $A$ has a power set $\mathcal P(A)$, containing all its subsets.
• Infinity: a set exists that contains $\varnothing$ and is closed under $x\mapsto x\cup\{x\}$.
• Separation: each formula, with permitted parameters, defines a subset of an already given set.
• Replacement: if a formula assigns a unique set to every element of a set $A$, its resulting values form a set.
• Foundation, also called regularity: every nonempty set $A$ has a member $b$ with $A\cap b=\varnothing$. (plato.stanford.edu)

Foundation excludes self-membership and finite membership cycles. Presentations differ in their treatment of redundant axioms: empty-set existence can be derived from other principles, and separation follows from replacement together with the surrounding axioms. Consequently, counting the named axioms is less informative than identifying the theory they generate. (cl.cam.ac.uk)

Replacement concerns a definable assignment that need not already exist as a set-sized function. Its output can contain objects outside the original domain. Separation, by contrast, only selects elements from an existing set. This distinction gives replacement an essential role in transfinite constructions. (plato.stanford.edu)

Avoiding paradoxes and distinguishing classes

Unrestricted comprehension would permit a set for every condition. Russell’s paradox arises by applying this principle to

which yields $R\in R$ exactly when $R\notin R$. ZF permits only the restricted separation expression $\{x\in A:x\notin x\}$, where $A$ is already a set. (plato.stanford.edu)

A proper class is a collection that is not a set, such as the collection of all sets. In ordinary ZF, definable classes are convenient descriptions, not additional objects over which its variables range. Class theories such as von Neumann–Bernays–Gödel set theory provide a separate formal treatment of classes. (cl.cam.ac.uk)

Constructing mathematical objects

The natural numbers can be represented by the finite von Neumann ordinals:

Infinity and separation support the construction of their set, $\omega$. More generally, ordinal numbers extend this representation to transfinite order types. (people.math.ethz.ch)

An ordered pair can be encoded by

This supports the construction of Cartesian products, relations, and functions as sets of ordered pairs satisfying suitable conditions. Such encodings demonstrate how richer mathematical language can be interpreted using membership alone; they are representations, not uniquely mandated identifications. (cl.cam.ac.uk)

The cumulative hierarchy organizes sets into ordinal-indexed stages:

for limit ordinals $\lambda$. Replacement supports the limit-stage construction, and foundation implies that every set belongs to some stage. The entire hierarchy is a proper class, not a universal set. (plato.stanford.edu)

Choice, independence, and consistency

Choice states that every set-indexed family of nonempty sets admits a function selecting one element from each member. Over ZF, it is equivalent to the well-ordering principle for arbitrary sets. Assuming ZF is consistent, choice can neither be proved nor refuted from ZF. (plato.stanford.edu)

Kurt Gödel established relative consistency results using the constructible universe; Paul Cohen subsequently developed forcing to establish complementary independence results. Together, their work shows that the continuum hypothesis is independent of ZFC, assuming consistency. Such results concern what follows from specified axioms, not an inconsistency in those axioms. (plato.stanford.edu)

Gödel’s incompleteness theorems impose another limitation: if ZF is consistent, it cannot prove its own standard arithmetically expressed consistency statement. Accordingly, consistency proofs typically establish a relative claim—consistency follows from that of another theory—rather than supplying an unconditional guarantee from within ZF itself. (plato.stanford.edu)