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.{{cite arXiv|eprint=math/9611209 |last1=Villaveces |first1=Andres |title=Chains of End Elementary Extensions of Models of Set Theory |date=1996 }}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.{{citation needed|date=July 2023}}
In L, any unfoldable cardinal is strongly unfoldable; thus unfoldable and strongly unfoldable have the same consistency strength.{{citation needed|date=July 2023}}
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.{{citation needed|date=July 2023}}
References
{{refbegin}}
- {{cite journal |authorlink=Joel David Hamkins |first=Joel David |last=Hamkins |s2cid=6269487 |title=Unfoldable cardinals and the GCH |journal=The Journal of Symbolic Logic |year=2001 |volume=66 |issue=3 |pages=1186–1198 |doi=10.2307/2695100|jstor=2695100 |arxiv=math/9909029 }}
- {{cite journal |journal=Journal of Symbolic Logic |title=Strongly unfoldable cardinals made indestructible|year=2008|last1=Johnstone|first1=Thomas A.|volume=73|issue=4|pages=1215–1248 |doi=10.2178/jsl/1230396915|s2cid=30534686 }}
- {{cite journal
| last1 = Džamonja | first1 = Mirna
| last2 = Hamkins | first2 = Joel David | author2-link = Joel David Hamkins
| arxiv = math/0409304
| doi = 10.1016/j.apal.2006.05.001
| issue = 1-3
| journal = Annals of Pure and Applied Logic
| mr = 2279655
| pages = 83–95
| title = Diamond (on the regulars) can fail at any strongly unfoldable cardinal
| volume = 144
| year = 2006}}
{{refend}}