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(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.

References

• Unfoldable Cardinals and the GCH, Joel David Hamkins. The Journal of Symbolic Logic, Vol. 66, No. 3 (Sep., 2001), pp. 1186-1198 doi:10.2307/2695100
• Strongly unfoldable cardinals made indestructible, Thomas A. Johnstone. J. Symbolic Logic, Volume 73, Issue 4 (2008), 1215-1248. doi:10.2178/jsl/1230396915
• Diamond (on the regulars) can fail at any strongly unfoldable cardinal, Joel David Hamkins (The City University of New York), Mirna Džamonja (University of East Anglia). (Submitted to arxiv (http://arxiv.org/abs/math/0409304) on 17 Sep 2004)