Definition
For a given cardinal number and a stationary set , is the statement that there is a sequence such that
- every Aδ is a cofinal subset of δ
- for every unbounded subset , there is a so that
is usually written as just .
In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ostaszewski. [1]
For a given cardinal number and a stationary set , is the statement that there is a sequence such that
is usually written as just .
It is clear that ◊ ⇒ ♣, and it was shown in 1975 that ♣ + CH ⇒ ◊; however, Saharon Shelah gave a proof in 1980 that there exists a model of ♣ in which CH does not hold, so ♣ and ◊ are not equivalent (since ◊ ⇒ CH). [2]
In mathematics, specifically set theory, 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.
Freiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidson but the mathematics behind it goes back to Wacław Sierpiński.
In set theory, a Woodin cardinal is a cardinal number such that for all functions , there exists a cardinal with and an elementary embedding from the Von Neumann universe into a transitive inner model with critical point and .
In set theory, a supercompact cardinal is a type of large cardinal independently introduced by Solovay and Reinhardt. They display a variety of reflection properties.
In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by Jensen & Kunen (1969). In the following definitions, will always be a regular uncountable cardinal number.
In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s.
In mathematics, and particularly in axiomatic set theory, the diamond principle◊ is a combinatorial principle introduced by Ronald Jensen in Jensen (1972) that holds in the constructible universe and that implies the continuum hypothesis. Jensen extracted the diamond principle from his proof that the axiom of constructibility implies the existence of a Suslin tree.
In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded relative to the limit ordinal. The name club is a contraction of "closed and unbounded".
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
In mathematics, particularly in set theory, Fodor's lemma states the following:
In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.
In mathematics, particularly in set theory, if is a regular uncountable cardinal then the filter of all sets containing a club subset of is a -complete filter closed under diagonal intersection called the club filter.
In differential geometry, a discipline within mathematics, a distribution on a manifold is an assignment of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle .
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 the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models are too numerous to have any hope of a reasonable classification. A first step of this program was showing that if a theory is not stable then its models are too numerous to classify.
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 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.
