通向哥德尔之路

The Road to Gettier

哥德尔不完全性定理是思想史近乎必然的结果,而哥德尔在证明第一不完全性定理时构造的哥德尔句表达了一个可以实际诉说的简易陈述。首先,集合论悖论的存在表明,不受限制的概括公理模式是不一致的,而这足以得出不完全性定理的非构造性证明。其次,每个关于属于关系的集合论悖论都可以转化为关于满足关系的悖论,再通过满足到真的归约转化为语义悖论,这恰好是哥德尔句的形式:"对自身不可证"对自身不可证。

Gdel’s incompleteness theorem is almost the inevitable result of a historic line of thought, and shows that the Gdel statement, the one Gdel constructed when proving the first incompleteness theorem, makes a fairly intelligible statement that can actually be stated. First, the existence of set theory paradoxes shows the inconsistency of unrestricted comprehension axiom schema, which suffices a non- constructive proof of incompleteness theorem. Second, any set theory paradox regarding membership can be changed into one of satisfaction, and then changed into a semantic paradox through the reduction of satisfaction to truth, which is precisely the form of the Gdel statement: 'Unprovable of itself' is unprovable of itself.