iterable cardinal
In mathematics, an iterable cardinal is a type of large cardinal introduced by {{harvs|txt|last=Gitman|year=2011}}, and {{harvs|txt|last1=Sharpe|last2=Welch|year=2011}}, and further studied by {{harvs|txt|last=Gitman|last2=Welch|year=2011}}. Sharpe and Welch defined a cardinal κ to be iterable if every subset of κ is contained in a weak κ-model M for which there exists an M-ultrafilter on κ which allows for wellfounded iterations by ultrapowers of arbitrary length.
Gitman gave a finer notion, where a cardinal κ is defined to be α-iterable
if ultrapower iterations only of length α are required to wellfounded. (By standard arguments iterability is equivalent to ω1-iterability.)
References
{{refbegin}}
- {{citation|mr=2830435
|last=Gitman|first= Victoria
|title=Ramsey-like cardinals I
|journal=Journal of Symbolic Logic|volume= 76 |year=2011|issue= 2|pages= 519–540|doi=10.2178/jsl/1305810762 |arxiv=0801.4723|s2cid=16501630 }}
- {{citation|mr=2830435
|last1=Gitman|first1= Victoria|last2= Welch|first2= P. D.
|title=Ramsey-like cardinals II
|journal=Journal of Symbolic Logic|volume= 76 |year=2011|issue= 2|pages= 541–560|doi=10.2178/jsl/1305810763 |arxiv=1104.4448|s2cid=2808737 }}
- {{citation|mr=2817562
|last1=Sharpe|first1= Ian|last2= Welch|first2= P. D.
|title=Greatly Erdős Cardinals with some generalizations to the Chang and Ramsey properties|journal=Annals of Pure and Applied Logic|volume= 162|year=2011|issue= 2|pages= 863–902|doi=10.1016/j.apal.2011.04.002|doi-access=free}}
{{refend}}
External links
- [https://web.archive.org/web/20160522100807/http://boolesrings.org/victoriagitman/files/2013/06/ramseydiagram.jpg Diagram of iterable cardinals]
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