ⓘ Unfoldable cardinal
In mathematics, an unfoldable cardinal is a certain kind of large cardinal number.
Formally, a cardinal number κ is λunfoldable if and only if for every transitive model M of cardinality κ of ZFCminuspower set such that κ is in M and M contains all its sequences of length less than κ, there is a nontrivial 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 λunfoldable for all ordinals λ.
A cardinal number κ is strongly λunfoldable if and only if for every transitive model M of cardinality κ of ZFCminuspower set such that κ is in M and M contains all its sequences of length less than κ, there is a nontrivial elementary embedding j of M into a transitive model "N" with the critical point of j being κ, j κ ≥ λ, and Vλ 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 λunfoldable 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 κstrongly unfoldable, and κunfoldable, if and only if it is weakly compact. A κ+ωunfoldable cardinal is totally indescribable and preceded by a stationary set of totally indescribable cardinals.
 Unfold may refer to: Unfoldable cardinal in mathematics Unfold higher  order function in computer science a family of anamorphism functions Unfoldment
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 relation unfoldable cardinal An unfoldable cardinal a cardinal κ such that for every ordinal λ and every transitive model M of cardinality κ of ZFC  minus  power

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