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 \(\omega _1\).

众所周知，由于不再有根据，米歇尔阶的行为在逐级扩展器的级别上发生了实质性的变化。尽管人们认为在等级到等级扩展器下面的米切尔阶的可能的部分阶结构很容易理解，但在病情不明的情况下，对这种结构的了解却很少。本文的目的是通过研究Mitchell指令的不合理程度来迈出了解这种情况的第一步。我们的主要结果是（i）在存在等级到等级的扩展程序的情况下，存在任何可数长度的扩展程序的传递Mitchell阶递减序列，并且（ii）没有这样的长度\（\ omega _1 \ ）。