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 that are less than κ, and implies .
In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.
In set theory, a Woodin cardinal is a cardinal number λ 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 mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by Erdős & Tarski (1961); weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory.
In set theory, a branch of mathematics, a rank-into-rank 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 axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal is called Shelah iff for every , there exists a transitive class and an elementary embedding with critical point ; and .
In mathematics, a remarkable cardinal is a certain kind of large cardinal number.
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, a subcompact cardinal is a certain kind of large cardinal number.
In mathematical set theory, a Reinhardt cardinal is a large cardinal κ in a model of ZF, Zermelo–Fraenkel set theory without the axiom of choice. They were suggested by William Nelson Reinhardt.
Herbert Kenneth Kunen is an emeritus professor of mathematics at the University of Wisconsin–Madison who works in set theory and its applications to various areas of mathematics, such as set-theoretic topology and measure theory. He also works on non-associative algebraic systems, such as loops, and uses computer software, such as the Otter theorem prover, to derive theorems in these areas.
In set theory, 0† is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0† does not exist" is consistent. ZFC + "0† exists" is not known to be inconsistent. In other words, it is believed to be independent. It is usually formulated as follows:
In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter is the most basic case of an extender.
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen (1971), shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.
This is a glossary of set theory.
