Unfoldable cardinal
Encyclopedia
In mathematics
, an unfoldable cardinal is a certain kind of large cardinal number.
Formally, a cardinal number
κ is λ-unfoldable if and only if for every transitive model
M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point
of j being κ and j(κ) ≥ λ.
A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals
λ.
A cardinal number
κ is strongly λ-unfoldable if and only if for every transitive model
M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point
of j being κ, j(κ) ≥ λ, and V(λ) is a subset of j(M). Without loss of generality, we can demand also that j(M) contains all its sequences of length λ.
Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ.
These properties are essentially weaker versions of strong
and supercompact cardinals, consistent with V = L
. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the proper forcing axiom
.
A Ramsey cardinal
is unfoldable, and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V, however.
In L, any unfoldable cardinal is strongly unfoldable; thus unfoldables and strongly unfoldables have the same consistency strength.
A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact
. A κ+ω-unfoldable cardinal is totally indescribable
and preceded by a stationary set of totally indescribable 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...
, an unfoldable cardinal is a certain kind of large cardinal number.
Formally, 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 λ-unfoldable if and only if for every transitive model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with 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 being κ and j(κ) ≥ λ.
A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals
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...
λ.
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 strongly λ-unfoldable if and only if for every transitive model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with 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 being κ, j(κ) ≥ λ, and V(λ) is a subset of j(M). Without loss of generality, we can demand also that j(M) contains all its sequences of length λ.
Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ.
These properties are essentially weaker versions of strong
Strong cardinal
In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.- Formal definition :...
and supercompact cardinals, consistent with V = L
Axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the proper forcing axiom
Proper Forcing Axiom
In the mathematical field of set theory, the proper forcing axiom is a significant strengthening of Martin's axiom, where forcings with the countable chain condition are replaced by proper forcings.- Statement :...
.
A Ramsey cardinal
Ramsey cardinal
In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey.With [κ]<ω denoting the set of all finite subsets of κ, a cardinal number κ such that for every function...
is unfoldable, and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V, however.
In L, any unfoldable cardinal is strongly unfoldable; thus unfoldables and strongly unfoldables have the same consistency strength.
A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact
Weakly compact
Weakly compact can refer to:*weakly compact cardinal*compact set in the weak topology...
. A κ+ω-unfoldable cardinal is totally indescribable
Totally indescribable cardinal
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...
and preceded by a stationary set of totally indescribable cardinals.