Gap-2 morass-definable η1-orderings

Pub Date : 2022-04-12 DOI:10.1002/malq.201800002

Bob A. Dumas

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Abstract

We prove that in the Cohen extension adding ℵ₃ generic reals to a model of $ZFC + CH$ containing a simplified (ω₁, 2)-morass, gap-2 morass-definable η₁-orderings with cardinality ℵ₃ are order-isomorphic. Hence it is consistent that $2^{ℵ_{0}} = ℵ_{3}$ and that morass-definable η₁-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $R$ over ω that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinite back-and-forth construction of order-type ω₁ to an order-preserving bijection between objects of cardinality ℵ₃.

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Gap-2泥沼可定义η - 1排序

在Cohen扩展中，我们证明了在包含一个简化的(ω 1,2)-morass的ZFC + CH $\mathsf {ZFC}+\mathsf {CH}$的模型中添加了λ 3的一般实数是序同构的。因此，2 ~ 0 = ~ 3$ 2^{\aleph _0}=\aleph _3$，连续统的基数上的沼泽可定义η - 1序是序同构的。我们证明了R $\mathbb {R}$ / ω的超幂是gap-2沼泽可定义的。该构造使用简化的间隙-2泥沼，以及泥沼映射和泥沼嵌入的交换性，将阶型ω1的超限来回构造扩展到基数为ω 3的对象之间的保序双射。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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