不可数标准系统算术的非标准模型
On Non-standard Models of Arithmetic with Uncountable Standard Systems
摘要：
20世纪60年代，Dana Scott给出了可数非标准算术模型的标准系统的递归理论刻画，即，以非标准模型编码的标准自然数集合。后来，奈特和纳德尔证明了斯科特的刻画也适用于基数为$\aleph_1$的非标准算术模型。但问题是，基数的限制是否可以从上述特征中删除，仍然是一个长期存在的问题，称为斯科特集问题。本文给出了两种利用非平凡不可数标准系统构造算术非标准模型的方法。第一个导致一个新的证明上述定理的骑士和纳德尔，第二个证明存在的模型与非平凡的标准系统的基数的连续。第二种构造在一定的集合论假设下也给出了Scott集合问题的部分答案。
英文摘要：
In 1960s, Dana Scott gave a recursion theoretic characterization of standard systems of countable non-standard models of arithmetic, i.e., collections of sets of standard natural numbers coded in non-standard models. Later, Knight and Nadel proved that Scott's characterization also applies to non-standard models of arithmetic with cardinality $\aleph_1$. But the question, whether the limit on cardinality can be removed from the above characterization, remains a long standing question, known as the Scott Set Problem. This article presents two constructions of non-standard models of arithmetic with non-trivial uncountable standard systems. The first one leads to a new proof of the above theorem of Knight and Nadel, and the second proves the existence of models with non-trivial standard systems of cardinality the continuum. A partial answer to the Scott Set Problem under certain set theoretic hypothesis also follows from the second construction.
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