Set Theory

Set theory is the branch of mathematics concerned with sets: collections of distinct objects considered as single mathematical entities. It studies membership, operations on collections, and the structure and size of infinite sets. In its foundational role, set theory supplies a framework in which numbers, functions, and other mathematical objects can be represented and their properties derived from explicit assumptions. Pure set theory treats every object under consideration as a set, including the elements of sets themselves. (plato.stanford.edu)

Historical development

The subject emerged from Georg Cantor’s investigations of infinity in the nineteenth century. In 1873, Cantor discovered that the real numbers cannot be placed in one-to-one correspondence with the natural numbers, establishing that infinite collections can have different sizes. His subsequent work developed systematic ways of comparing those sizes. (plato.stanford.edu)

Early informal approaches encountered contradictions when they allowed every describable collection to constitute a set. These problems encouraged axiomatization. Ernst Zermelo introduced axioms in 1908; subsequent contributions by Abraham Fraenkel, Thoralf Skolem, and John von Neumann helped produce the framework now called Zermelo–Fraenkel set theory with choice, or ZFC. (plato.stanford.edu)

Membership and elementary operations

The notation $x\in A$ means that $x$ is an element of $A$. A set is determined entirely by its elements, so order and repeated listing do not matter: $\{1,2,2\}=\{2,1\}$. The empty set, written $\varnothing$, has no elements. A subset $A\subseteq B$ is a set whose elements all belong to $B$. Membership and inclusion are different: $x\in A$ concerns an element, whereas $A\subseteq B$ concerns two sets. (plato.stanford.edu)

The principal operations are:

• Union, $A\cup B$: elements belonging to either set or both.
• Intersection, $A\cap B$: elements belonging to both sets.
• Difference, $A\setminus B$: elements of $A$ absent from $B$.
• Complement, $S\setminus A$: elements outside $A$, relative to a specified set $S$.

Thus, if $A=\{1,2\}$ and $B=\{2,3\}$, their union is $\{1,2,3\}$, their intersection is $\{2\}$, and $A\setminus B=\{1\}$. Complements require a specified ambient set; they are not absolute collections of everything excluded from $A$. (courses.maths.ox.ac.uk)

The power set $\mathcal P(A)$ consists of all subsets of $A$, including $\varnothing$ and $A$. A set with $n$ elements has $2^n$ subsets, because each element can independently be included or excluded. The Cartesian product $A\times B$ consists of ordered pairs $(a,b)$, with $a\in A$ and $b\in B$. Unlike elements listed in a set, the coordinates of an ordered pair have a specified order. (people.csail.mit.edu)

Relations, functions, and infinite size

A binary relation from $A$ to $B$ can be represented as a subset of $A\times B$. A function $f:A\to B$ is a relation assigning exactly one value in $B$ to each element of $A$. A bijection is both one-to-one and onto, pairing the elements of two sets without omissions or duplications. (plato.stanford.edu)

Two sets have the same cardinality when a bijection exists between them. A countable set is finite or can be enumerated by the natural numbers. The integers and rational numbers are countable, whereas the real numbers are uncountable. Consequently, “infinite” does not identify a single mathematical size. (plato.stanford.edu)

Cantor’s theorem states that a power set always has strictly greater cardinality than its original set. Repeatedly taking power sets therefore produces progressively larger infinities. (plato.stanford.edu)

Ordinal numbers describe positions and order types in well-ordered sets rather than size alone. The first infinite ordinal, $\omega$, represents the usual ordering of the natural numbers. The ordinals $\omega$ and $\omega+1$ have different order types but the same cardinality: adding a final element changes the ordering without increasing its infinite size. (plato.stanford.edu)

Axioms and paradoxes

Russell’s paradox exposes the failure of unrestricted set formation. If $R=\{x:x\notin x\}$ were a set, then $R\in R$ would hold exactly when $R\notin R$, a contradiction. ZFC avoids this construction by restricting comprehension to elements of an already existing set. (plato.stanford.edu)

Zermelo–Fraenkel set theory is formulated in first-order logic. Its axioms specify equality through membership and authorize constructions such as pairing, unions, and power sets. Infinity ensures an infinite set exists; separation selects elements from an existing set; replacement forms images under definable functional rules. Foundation constrains membership chains. The axiom of choice adds that every set-indexed family of nonempty sets admits a function selecting one member from each, producing ZFC. (plato.stanford.edu)

Independence and mathematical foundations

The continuum hypothesis asserts that no cardinality lies strictly between that of the natural numbers and that of the real numbers. Kurt Gödel established its relative consistency in 1938, and Paul Cohen established the relative consistency of its negation in 1963. Together these results show that, if ZFC is consistent, its axioms neither prove nor refute the hypothesis. Independence concerns what follows from a specified axiom system, not a contradiction within that system. (plato.stanford.edu)

Cohen’s method, forcing, constructs extensions of models of set theory with controlled properties. Further research investigates additional axioms, including large-cardinal principles, and their consequences for infinite structures and definable sets. (plato.stanford.edu)

Set-theoretic representations also organize ordinary mathematics. Numbers can be constructed from sets, and functions and relations from ordered pairs. In probability, events are subsets of a sample space, with operations such as union, intersection, and complement expressing combinations of events. These applications connect foundational constructions with concrete mathematical reasoning. (plato.stanford.edu)