论文标题
在具有不可数的标准系统的非标准算术模型上
On Non-standard Models of Arithmetic with Uncountable Standard Systems
论文作者
论文摘要
在1960年代，达娜·斯科特（Dana Scott）对算术算术模型的标准系统进行了递归理论表征，即在非标准模型中编码的标准自然数集集合。后来，奈特（Knight）和纳德尔（Nadel）证明了斯科特（Scott）的特征也适用于具有基数$ \ aleph_1 $的非标准算术模型。但是，是否可以从上述表征中删除基数的限制仍然是一个长期存在的问题，称为斯科特设定问题。本文介绍了两种非标准算术模型的结构，并提供了非平布性的无数标准系统。第一个导致了上述骑士和纳德尔定理的新证明，第二个证明了具有非平凡的标准系统的模型存在。在某些集合理论假设下，对Scott集问题的部分答案也来自第二个结构。
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.
