Extendible cardinal
Encyclopedia
In mathematics
, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principle
s. A cardinal number
κ is called η-extendible if for some λ there is a nontrivial elementary embedding j of
into
where κ is the critical point
of j.
κ is called an extendible cardinal if it is η-extendible for every ordinal number
η.
Vopenka's principle
implies the existence of extendible cardinals. All extendible cardinals are supercompact cardinals.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principle
Reflection principle
In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle depending on exactly what is meant by "resemble"...
s. A cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
κ is called η-extendible if for some λ there is a nontrivial elementary embedding j of
- Vκ+η
into
- Vλ
where κ is the critical point
Critical point (set theory)
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....
of j.
κ is called an extendible cardinal if it is η-extendible for every ordinal number
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
η.
Vopenka's principle
Vopenka's principle
In mathematics, Vopěnka's principle, named after Petr Vopěnka, is a large cardinal axiom. A cardinal κ is called a Vopěnka cardinal if Vopěnka's principle holds in the rank Vκ .....
implies the existence of extendible cardinals. All extendible cardinals are supercompact cardinals.