Nonexistence of universal orders in many cardinals
Nonexistence of universal orders in many cardinals
复制标题
许多枢机主教不存在普遍秩序
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
S. Shelah
中科院分区：
文献类型：
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作者：
M. Kojman;S. Shelah
摘要我们的主题是，不是每一个有趣的问题，在集合论是独立的ZFC。我们给出了一个一阶理论T的例子，它具有可数的D（T），它不能有一个普适的模型在n = 1;没有CH;我们在ZFC中证明了一个覆盖定理，假设某个理论存在一个普适的模型;我们再次在ZFC中证明了对于一个大的基数类，没有普适的线性序（例如在每个正则中）。事实上，我们证明了，如果在正则λ处存在泛线性序，并且它的存在不是一个平凡基数算术原因的结果，那么λ“类似于”λ 1-一个已知具有泛序的一致性的基数。对于奇异基数，我们证明了对于许多奇异基数，如果它们不是强极限，那么它们就没有泛线性序。作为不存在泛线性序的结果，我们证明了所有具有严格序性质的理论（例如，有序域和群，布尔代数，p-adic环和域，偏序，PA的模型等）的泛模型的不存在性.
Abstract Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ1—a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).