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 | English |
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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
Publisher Copyright:© 2021, The Author(s).
Keywords
- Infinite decreasing chain
- Mitchell order
- Rank-to-rank extender