形式化弱逻辑中的集合论，寻找具有哥德尔不完备性的最弱逻辑
Formalizing set theory in weak logics, searching for the weakest logic with Gödel's incompleteness property
摘要：
我们证明了一阶逻辑可以转化为一个非常简单的弱逻辑，从而集合论可以在这个弱逻辑中形式化。这个弱逻辑系统等价于具有三个交换补闭包算子的布尔代数的方程理论，即，无对角三维柱形代数（Df_3's）的性质。等价地，集合论可以用3个交换S5模态（即，在多模态逻辑[S5，S5，S5]中）。有许多后果，例如，由自由子空间生成的Df_3不是原子的，[S5，S5，S5]具有哥德尔不完全性.这里报告的结果是书中主要结果的有力改进：Tarski，A.和Givant，S. R.，形式化无变量的集合论，AMS，1987年。
英文摘要：
We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three commuting complemented closure operators, i.e., that of diagonal-free 3-dimensional cylindric algebras (Df_3's). Equivalently, set theory can be formulated in propositional logic with 3 commuting S5 modalities (i.e., in the multi-modal logic [S5,S5,S5]). There are many consequences, e.g., free finitely generated Df_3's are not atomic and [S5,S5,S5] has Gödel's incompleteness property. The results reported here are strong improvements of the main result of the book: Tarski, A. and Givant, S. R., Formalizing Set Theory without variables, AMS, 1987.
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