In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers.
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A.
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph.
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that is a regular cardinal if and only if every unbounded subset has cardinality . Infinite well-ordered cardinals that are not regular are called singular cardinals. Finite cardinal numbers are typically not called regular or singular.
In mathematics, the beth numbers are a certain sequence of infinite cardinal numbers, conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers, but there may be numbers indexed by that are not indexed by .
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.
In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable.
In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory (ZFC).
In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.
In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of cardinality κ in which all levels have size less than κ and all branches have height less than κ. They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935).
In mathematics, infinitary combinatorics, or combinatorial set theory, is an extension of ideas in combinatorics to infinite sets. Some of the things studied include continuous graphs and trees, extensions of Ramsey's theorem, and Martin's axiom. Recent developments concern combinatorics of the continuum and combinatorics on successors of singular cardinals.
In set theory, a branch of mathematical logic, Martin's maximum, introduced by Foreman, Magidor & Shelah (1988) and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent.
In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph.
In axiomatic set theory, the Rasiowa–Sikorski lemma is one of the most fundamental facts used in the technique of forcing. In the area of forcing, a subset E of a poset is called dense in P if for any p ∈ P there is e ∈ E with e ≤ p. If D is a family of dense subsets of P, then a filter F in P is called D-generic if
This is a glossary of set theory.
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two countable dense unbounded linear orders are order-isomorphic. It is named after Georg Cantor, and can be proved by the back-and-forth method sometimes attributed to Cantor, but Cantor's original proof only used the "going forth" half of this method.
