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(κ) ≥ λ, and V(λ) is a subset of N. Without loss of generality, we can demand also that N contains all its sequences of length 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.
[citation needed] A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact.