论文标题
在戈德尔的“弱”假设上
On Godel's "Much Weaker" Assumption
论文作者
论文摘要
如果理论一致，则在戈德尔（Godel）1931年对不完整定理的开创性论文中建造的足够强大和递归列举的理论的戈德式句子是无法证明的。但是，它们可能是可以驳斥的。当该理论是所谓的欧米茄一致性时，这些句子是独立的。 Godel引入的一个概念，比（简单）一致性更强，但“弱”比健全性更强。 Godel竭尽全力详细说明欧米茄一致性比一致性强，但从未显示或似乎忘记说，为什么它比健全弱得多。在本文中，我们研究了这种证明理论的概念，并将其某些特性与一致性和（变体）的特性进行了比较。
Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable. These sentences are independent when the theory is so-called omega-consistent; a notion introduced by Godel, which is stronger than (simple) consistency, but ``much weaker'' than soundness. Godel goes to great lengths to show in detail that omega-consistency is stronger than consistency, but never shows, or seems to forget to say, why it is much weaker than soundness. In this paper, we study this proof-theoretic notion and compare some of its properties with those of consistency and (variants of) soundness.
