Editing 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 [[inner model|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 (set theory)|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 [[ordinal number|ordinals]] λ.

A [[cardinal number]] κ is '''strongly λ-unfoldable''' if and only if for every [[inner model|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 (set theory)|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 cardinal|strong]] and [[supercompact cardinal|supercompact]] cardinals, consistent with [[Axiom of constructibility|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.<ref name="Villaveces96">{{cite arXiv|eprint=math/9611209 |last1=Villaveces |first1=Andres |title=Chains of End Elementary Extensions of Models of Set Theory |date=1996 }}</ref><sup>p.14</sup> Assuming a Ramsey cardinal exists, it is less than the least Ramsey cardinal.<ref name="Villaveces96" /><sup>p.3</sup>

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 cardinal|weakly compact]]. A κ+ω-unfoldable cardinal is [[totally indescribable cardinal|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}}

==Citations==
{{reflist}}

[[Category:Large cardinals]]


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