Set Theory
Set theory studies collections of objects and serves as the standard foundation of modern mathematics, in which essentially every mathematical object can be represented as a set and every theorem derived from a short list of axioms.
Definition
Set theory is the mathematical study of sets, well-defined collections of objects, together with the membership relation, developed axiomatically so as to provide a uniform foundation for mathematics and to analyze notions of infinity.
Scope
This area covers the axiomatic development of sets (chiefly Zermelo-Fraenkel set theory with the axiom of choice), the theory of ordinal and cardinal numbers and their arithmetic, the constructible universe and inner models, the method of forcing for proving independence results, and the hierarchy of large cardinal axioms that extend the standard axioms. It spans both the foundational role of set theory and its development as an autonomous mathematical discipline.
Sub-topics
Core questions
- Which axioms suffice to develop ordinary mathematics, and what are their consequences?
- How are the sizes of infinite sets compared and computed?
- Which statements are independent of the standard axioms, and how is independence established?
- What stronger axioms of infinity exist, and how do they extend the provable consequences of set theory?
Key theories
- Zermelo-Fraenkel set theory with choice (ZFC)
- A first-order axiom system whose axioms (extensionality, pairing, union, power set, infinity, separation, replacement, foundation, and choice) provide the standard foundation in which mathematics is formalized.
- Independence of the continuum hypothesis
- Goedel showed the continuum hypothesis is consistent with ZFC via the constructible universe and Cohen showed its negation is also consistent via forcing, so the hypothesis is independent of the standard axioms.
- Theory of ordinals and cardinals
- Ordinals generalize counting into the transfinite as canonical well-ordered sets, while cardinals measure size; together they organize the cumulative hierarchy and transfinite recursion.
Clinical relevance
Set theory provides the common foundational language of mathematics: its axioms underlie the construction of the number systems, its theory of infinity shapes analysis and topology, and its independence results clarify the limits of what the standard axioms can settle.
History
Set theory began with Cantor's nineteenth-century discovery that infinite sets come in different sizes. Paradoxes such as Russell's prompted the axiomatic systems of Zermelo and Fraenkel in the early twentieth century. Goedel's constructible universe (1938) and Cohen's invention of forcing (1963) resolved the consistency and independence of the continuum hypothesis and the axiom of choice, and the subsequent study of large cardinals and determinacy turned set theory into a deep autonomous field.
Key figures
- Georg Cantor
- Ernst Zermelo
- Abraham Fraenkel
- Kurt Goedel
- Paul Cohen
Related topics
Seminal works
- jech2003
- kunen2011
- cohen1963
Frequently asked questions
- Why is set theory considered a foundation of mathematics?
- Almost every mathematical object such as numbers, functions, and spaces can be encoded as a set, and the usual theorems can be derived from the ZFC axioms, so set theory provides a single formal system in which mathematics can be carried out.
- What does it mean for the continuum hypothesis to be independent?
- It means neither the continuum hypothesis nor its negation can be proved from the ZFC axioms, so the axioms leave the size of the continuum undetermined; this was established by combining Goedel's and Cohen's results.