In this paper we study the notion of C(n)-supercompactness introduced by Bagaria in  and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least C(1)-supercompact but also that the least supercompact is C(1)-supercompact and even C(n)-supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.
Bibliographical noteFunding Information:
The third author has been supported by MECD (Spanish Government) Grant no FPU15/00026, MEC project number MTM2017-86777-P and SGR (Catalan Government) project number 2017SGR-270. 1
- 2020 Mathematics Subject Classification 03Exx 03E50 03E57