Editing Set theory (section)

== Areas of study ==
Set theory is a major area of research in mathematics with many interrelated subfields:

=== Combinatorial set theory ===
{{Main|Infinitary combinatorics}}

''Combinatorial set theory'' concerns extensions of finite [[combinatorics]] to infinite sets. This includes the study of [[cardinal arithmetic]] and the study of extensions of [[Ramsey's theorem]] such as the [[Erdős–Rado theorem]].

=== Descriptive set theory ===
{{Main|Descriptive set theory}}

''Descriptive set theory'' is the study of subsets of the [[real line]] and, more generally, subsets of [[Polish space]]s. It begins with the study of [[pointclass]]es in the [[Borel hierarchy]] and extends to the study of more complex hierarchies such as the [[projective hierarchy]] and the [[Wadge hierarchy]]. Many properties of [[Borel set]]s can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

The field of [[effective descriptive set theory]] is between set theory and [[recursion theory]]. It includes the study of [[lightface pointclass]]es, and is closely related to [[hyperarithmetical theory]]. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.

A recent area of research concerns [[Borel equivalence relation]]s and more complicated definable [[equivalence relation]]s. This has important applications to the study of [[invariant (mathematics)|invariants]] in many fields of mathematics.

=== Fuzzy set theory ===
{{Main|Fuzzy set theory}}

In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In ''[[fuzzy set theory]]'' this condition was relaxed by [[Lotfi A. Zadeh]] so an object has a ''degree of membership'' in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

=== Inner model theory ===
{{Main|Inner model theory}}

An ''inner model'' of Zermelo–Fraenkel set theory (ZF) is a transitive [[proper class|class]] that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the [[constructible universe]] ''L'' developed by Gödel.
One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model ''V'' of ZF satisfies the [[continuum hypothesis]] or the [[axiom of choice]], the inner model ''L'' constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.

The study of inner models is common in the study of [[axiom of determinacy|determinacy]] and [[large cardinal]]s, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).<ref>{{citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition= Third Millennium | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 | page=642 | url=https://books.google.com/books?id=CZb-CAAAQBAJ&pg=PA642 }}</ref>

=== Large cardinals ===
{{Main|Large cardinal property}}

A ''large cardinal'' is a cardinal number with an extra property. Many such properties are studied, including [[inaccessible cardinal]]s, [[measurable cardinal]]s, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in [[Zermelo–Fraenkel set theory]].

=== Determinacy ===
{{Main|Determinacy}}

''Determinacy'' refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The [[axiom of determinacy]] (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the [[Wadge degree]]s have an elegant structure.

=== Forcing ===
{{Main|Forcing (mathematics)}}

[[Paul Cohen (mathematician)|Paul Cohen]] invented the method of ''[[forcing (mathematics)|forcing]]'' while searching for a [[model theory|model]] of [[ZFC]] in which the [[continuum hypothesis]] fails, or a model of ZF in which the [[axiom of choice]] fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the [[natural number]]s without changing any of the [[cardinal number]]s of the original model. Forcing is also one of two methods for proving [[consistency (mathematical logic)|relative consistency]] by finitistic methods, the other method being [[Boolean-valued model]]s.

=== Cardinal invariants ===
{{Main|Cardinal characteristics of the continuum}}

A ''cardinal invariant'' is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of [[meagre set]]s of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

=== Set-theoretic topology ===
{{Main|Set-theoretic topology}}

''Set-theoretic topology'' studies questions of [[general topology]] that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the [[Moore space (topology)|normal Moore space question]], a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.