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[edit]
Assuming V = L, the least unfoldable cardinal is greater than the least indescribable cardinal.[1]p.14 Assuming a Ramsey cardinal exists, it is less than the least Ramsey cardinal.[1]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]
In L, any unfoldable cardinal is strongly unfoldable; thus unfoldable and strongly unfoldable have the same consistency strength.[citation needed]
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]
References[edit]
- 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.
- Joel David Hamkins; Džamonja, Mirna (2004). "Diamond (On the regulars) can fail at any strongly unfoldable cardinal". arXiv:math/0409304. Bibcode:2004math......9304H.
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Citations[edit]
- ^ a b Villaveces, Andres (1996). "Chains of End Elementary Extensions of Models of Set Theory". arXiv:math/9611209.
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