论文标题
在高度等效的非同构可计算模型上
On highly equivalent non-isomorphic countable models of arithmetic and set theory
论文作者
论文摘要
众所周知，一阶Peano Axioms PA具有非同形可计数模型的连续性。这个问题，与同构的这种可计数模型有多近，似乎还没有得到研究。与可数模型的同构的亲密关系的度量是它们之间可以建立的来回序列的长度。我们表明，对于每个可数的序数alpha，都有可计数的PA的非同构模型，它们之间的长度为alpha。这意味着此类模型的Scott高度（或等级）大于$α$。我们还证明了ZFC模型的结果相同。
It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to isomorphism of countable models is the length of back-and-forth sequences that can be established between them. We show that for every countable ordinal alpha there are countable non-isomorphic models of PA with a back-and-forth sequence of length alpha between them. This implies that the Scott height (or rank) of such models is bigger than $α$. We also prove the same result for models of ZFC.
