It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length ω1.
Bibliographical noteFunding Information:
This research was initiated whilst the second-listed author visited the first-listed author at the Hebrew University of Jerusalem in February 2019. She wishes to thank the Hebrew University of Jerusalem for its hospitality. The first-listed author was partially supported by the Israel Science Foundation Grant 1832/19. The second-listed author gratefully acknowledges funding from L’ORÉAL Austria, in collaboration with the Austrian UNESCO Commission and in cooperation with the Austrian Academy of Sciences - Fellowship Determinacy and Large Cardinals. The authors would like to thank Grigor Sargsyan and the referee for many valuable comments and suggestions, which greatly improved the presentation of the paper.
© 2021, The Author(s).
- Infinite decreasing chain
- Mitchell order
- Rank-to-rank extender