In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.
If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and
That is, M agrees with V on an initial segment. Then κ is strong means that it is λ-strong for all ordinals λ.
By definitions, strong cardinals lie below supercompact cardinals and above measurable cardinals in the consistency strength hierarchy.
κ is κ-strong if and only if it is measurable. If κ is strong or λ-strong for λ ≥ κ+2, then the ultrafilter U witnessing that κ is measurable will be in Vκ+2 and thus in M. So for any α < κ, we have that there exist an ultrafilter U in j(Vκ) −j(Vα), remembering that j(α) = α. Using the elementary embedding backwards, we get that there is an ultrafilter in Vκ−Vα. So there are arbitrarily large measurable cardinals below κ which is regular, and thus κ is a limit of κ-many measurable cardinals.
Strong cardinals also lie below superstrong cardinals and Woodin cardinals in consistency strength. However, the least strong cardinal is larger than the least superstrong cardinal.
Every strong cardinal is strongly unfoldable and therefore totally indescribable.
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies .
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo. As with all large cardinals, none of these varieties of Mahlo cardinals can be proved to exist by ZFC.
In mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to describe in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott (1961).
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it can be described as a subdivision of all of its subsets into large and small sets such that κ itself is large, ∅ and all singletons {α}, α ∈ κ are small, complements of small sets are large and vice versa. The intersection of fewer than κ large sets is again large.
In set theory, a Woodin cardinal is a cardinal number such that for all functions
In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties.
In mathematics, a cardinal number κ is called huge if there exists an elementary embedding j : V → M from V into a transitive inner model M with critical point κ and
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength.
In mathematics, extendible cardinals are large cardinals introduced by Reinhardt (1974), who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one.
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large". The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
In mathematics, an unfoldable cardinal is a certain kind of large cardinal number.
In set theory, a branch of mathematics, a strongly compact cardinal is a certain kind of large cardinal.
In set theory, the core model is a definable inner model of the universe of all sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set-theoretic assumptions have very special properties, most notably covering properties. Intuitively, the core model is "the largest canonical inner model there is" and is typically associated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase "core model below Φ" refers to the definable inner model that exhibits the special properties under the assumption that there does not exist a cardinal satisfying Φ. The core model program seeks to analyze large cardinal axioms by determining the core models below them.
In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings.
In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal.
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxes encountered in earlier formulations of set theory by recognizing that some classes are too big to be sets. Von Neumann realized that the paradoxes are caused by permitting these big classes to be members of a class. A class that is a member of a class is a set; a class that is not a set is a proper class. Every class is a subclass of V, the class of all sets. The axiom of limitation of size says that a class is a set if and only if it is smaller than V—that is, there is no function mapping it onto V. Usually, this axiom is stated in the equivalent form: A class is a proper class if and only if there is a function that maps it onto V.
In mathematics, a shrewd cardinal is a certain kind of large cardinal number introduced by, extending the definition of indescribable cardinals.
This is a glossary of set theory.
In set theory, Berkeley cardinals are certain large cardinals suggested by Hugh Woodin in a seminar at the University of California, Berkeley in about 1992.
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