A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then combined into a “geometric technique” used to prove several results about countable non-standard models of set theory. In particular, back-and-forth constructions are carried out to establish various generalizations and refinements of Friedman’s theorem on the existence of rank-initial embeddings between countable non-standard models of the fragment \({\mathrm {KP}}^{{\mathcal {P}}}\) + \(\Sigma _1^{{\mathcal {P}}}\)-Separation of \({\mathrm {ZF}}\); and Gaifman’s technique of iterated ultrapowers is employed to show that any countable model of \({\mathrm {GBC}} + \text {``}{\mathrm {Ord}}\text { is weakly compact''}\) can be elementarily rank-end-extended to models with well-behaved automorphisms whose sets of fixed points equal the original model. These theoretical developments are then utilized to prove various results relating self-embeddings, automorphisms, their sets of fixed points, strong rank-cuts, and set theories of different strengths. Two examples: The notion of “strong rank-cut” is characterized (i) in terms of the theory \({\mathrm {GBC}} + \text {``}{\mathrm {Ord}}\text { is weakly compact''}\), and (ii) in terms of fixed-point sets of self-embeddings.

中文翻译：

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集理论的非标准模型的秩初始嵌入

进行了理论发展，以建立关于可数非标准集合理论模型的秩初始嵌入和自同构的基本结果，并敏锐地关注它们的不动点集合。然后将这些结果组合成一种“几何技术”，用于证明关于可数的非标准集合论模型的若干结果。尤其是，在片段\（{\ mathrm {KP}} ^ {{ \ mathcal {P}}} \） + \（\ Sigma _1 ^ {{\ mathcal {P}}} \） - \（{\ mathrm {ZF}} \\）的分隔; 并且采用迭代ultrapowers的Gaifman的技术表明，任何可数模型\（{\ mathrm {GBC}} + \文本{``} {\ mathrm {奥德}} \文本{是弱紧''} \）可以基本上对具有良好自同构的模型进行定级扩展，这些同构的固定点集等于原始模型。然后利用这些理论发展来证明各种有关自嵌入，自同构，它们的不动点集，强秩削减以及不同强度的理论集的结果。两个例子：“强等级降级”的概念以（i）\（{\ mathrm {GBC}} + \ text {``} {\ mathrm {Ord}} \ text {紧凑“”} \） ，和（ii）在固定点集自的嵌入的条款。