Gap-2 morass-definable η1-orderings

IF 0.4 4区数学 Q4 LOGIC Mathematical Logic Quarterly 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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来源期刊

Mathematical Logic Quarterly 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.

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