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 "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.

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.3

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 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 and preceded by a stationary set of totally indescribable cardinals.

References

• Hamkins, Joel David (2001). "Unfoldable cardinals and the GCH". The Journal of Symbolic Logic. 66 (3): 1186–1198. arXiv:math/9909029. doi:10.2307/2695100. JSTOR 2695100. S2CID 6269487.
• Johnstone, Thomas A. (2008). "Strongly unfoldable cardinals made indestructible". Journal of Symbolic Logic. 73 (4): 1215–1248. doi:10.2178/jsl/1230396915. S2CID 30534686.
• Džamonja, Mirna; Hamkins, Joel David (2006). "Diamond (on the regulars) can fail at any strongly unfoldable cardinal". Annals of Pure and Applied Logic. 144 (1–3): 83–95. arXiv:math/0409304. doi:10.1016/j.apal.2006.05.001. MR 2279655.