This paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a variant of the Mitchell order called the internal relation.

中文翻译：

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逐级嵌入和 STEEL 的猜想

本文建立了 Steel [7] 关于从累积层次结构到自身的基本嵌入结构的猜想。Steel 的问题与这些嵌入的 Mitchell 顺序有关，在 [5] 和 [7] 中进行了研究。虽然这个秩序被认为是没有根据的，但斯蒂尔推测它有一些大的有根据的子秩序，这就是我们所建立的。证明依赖于对更广泛的扩展嵌入类和称为内部关系的米切尔阶的变体的简单和一般分析。