Unfoldable cardinal
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 ≥ 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 "N" with the critical point of j being κ, j ≥ λ, and V is a subset of N. Without loss of generality, we can demand also that N 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.
Relations between large cardinal properties
Assuming V = L, the least unfoldable cardinal is greater than the least indescribable cardinal.p.14 Assuming a Ramsey cardinal exists, it is less than the least Ramsey cardinal.p.3A 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 unfoldable and strongly unfoldable 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 indescribable cardinal|indescribable] and preceded by a stationary set of totally indescribable cardinals.