Abstract
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.
| Original language | American English |
|---|---|
| Pages (from-to) | 771-781 |
| Number of pages | 11 |
| Journal | Archive for Mathematical Logic |
| Volume | 60 |
| Issue number | 6 |
| DOIs | |
| State | Published - Aug 2021 |
Bibliographical note
Funding 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.
Publisher Copyright:
© 2021, The Author(s).
Keywords
- Infinite decreasing chain
- Mitchell order
- Rank-to-rank extender