集合论的每一个可数模型都嵌入到它自己的可构造宇宙中
Every countable model of set theory embeds into its own constructible universe
摘要：
本文的主要定理是集合论M的每一个可数模型，包括每一个良基模型，都同构于它自己的可构造宇宙的一个子模型。换句话说，有一个嵌入式$j:M\to L^M$，它是无量词断言的基础。证明使用泛有向图组合学，包括可数随机有向图的非循环版本，我称之为可数随机Q-分次有向图，以及作为不可数Fraisse极限而产生的更高的类似物，导致hypnagogic有向图，一个集齐次的，类泛的，超实数分次的非循环类有向图，与超实数紧密相连。证明了$L^M$包含一个秩为$Ord^M$的泛无圈有向图的子模型。证明的方法还建立了集合论的可数模型是线性预排序的嵌入性：对于任何两个可数模型的集合论，其中一个是同构的子模型的其他。实际上，它们是通过顺序类型的嵌入性精确地预先良好排序的$ω_1+1$。具体来说，可数良基模型根据其序数的高度按可嵌入性排序;每个较短的模型嵌入到每个较高的模型中;集合论$M$的每个模型对于秩至多为$Ord^M$的所有可数良基二元关系是通用的;并且集合论的每个病态模型对于所有可数非循环二元关系是通用的。最后，加强Ressayre的一个经典定理，同样的证明方法表明，如果$M$是PA的任何非标准模型，那么集合论的每个可数模型--特别是ZFC的每个模型--同构于$M$的遗传有限集$HF^M$的一个子模型。事实上，$HF^M$对所有可数非循环二元关系都是普适的。
英文摘要：
The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random Q-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $Ord^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other. Indeed, they are pre-well-ordered by embedability in order-type exactly $ω_1+1$. Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $Ord^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory---in particular, every model of ZFC---is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.
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