未来集合论简史
A Brief History of Future Set Theory
摘要：
数学家们在生成集合时仍然使用朴素集合论，而不会产生任何矛盾。因此，他们的工作方法可以被认为是一个具有100多年经验的一致推理系统。我的猜想是，这种方法工作得很好，因为数学家只使用那些谓词来形成集合，这些集合产生封闭的遗传一致谓词扩展。对于他们在构造某个（特殊）集合（自底向上）的过程中使用的每个开放公式，我们总是可以找到一个“几乎封闭”的公式（即只有自由变元“x”的无参数公式），它产生与之前在自底向上过程中构造的相同的作为谓词扩展的某个（特殊）集合。因此，在理解方案中使用带有自由参数的谓词不会引起任何困难，并且可以通过元数学考虑来“提升”。关键词：朴素集合论，奎因，新基础，NF，泛集合，理解模式，谓词扩展，集合论哲学，策梅洛，弗伦克尔，ZF，补集。CT是类理论框架：逻辑+“=”+丘奇图式+外延公理。
英文摘要：
Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My conjecture is that this method works well because mathematicians use only those predicates to form sets, which yield closed hereditary consistent predicate extensions. And for every open formula they use in the process of constructing of a certain (special) set (bottom up), we can always find an "almost-closed" formula (i.e. a parameter-free formula with only the free variable "x") which yields the same certain (special) set as predicate extension as constructed in the bottom up process before. Therefore the use of predicates with free parameters in the Comprehension Scheme does not cause any difficulties and can be "lifted" by meta-mathematical considerations. KEYWORDS: naive set theory, Quine, new foundation, NF, universal sets, comprehension schema, predicate extension, philosophy of set theory, Zermelo, Fraenkel, ZF, complement. CT is the Class-Theoretical frame: logic + "=" + Church Schema + Extensionality Axiom.
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