We introduce the $Σ_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $Σ_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end-extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.
本文探讨了描述集合论与内模型论之间的联系。根据描述集合论，我们将取一个可数的、可定义的实数集A。然后我们将证明A等于M的实数，其中M是来自内模型论的正则模型。用专业术语来说，M是一只“老鼠”。因此，我们说A是一个鼠标集。对于我们正在处理的集合A的类型的一个具体例子，令OD（n）是在模型L（R）的ω-1水平上可由序数参数定义的实集合。在本文中，我们将证明，对于所有的n，OD（n）是一个鼠标集。我们的工作推广了D.A. Martin，J. R. Steel和H.伍丁在这方面，有几个令人感兴趣的问题仍然悬而未决。
Using Indices of Points on an Elliptic Curve to Construct A Diophantine Model of $\Z$ and Define $\Z$ Using One Universal Quantifier in Very Large Subrings of Number Fields, Including $\Q$
