有限域中的不完全性

摘要：

受寻找经典不完备性定理的有限版本问题的启发，我们提出了一些超越${\bf NP\neq co NP}$的方法。这些理论将计算复杂性与某些句子的证明难度正式联系起来，这意味着与句子相关的问题的高计算复杂性意味着该句子在弱理论中不可证明，或者需要长时间的证明。提出这些命题的另一个原因是证明复杂性中的某些结果似乎是这种一般命题的特例，我们想形式化并充分理解这些命题。在本文中，我们回顾了我们以前提出的一些命题，介绍了新的命题，使它们系统化，并证明了它们与以前研究过的其他一些命题之间的新联系。

英文摘要：

Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be special cases of such general statements and we want to formalize and fully understand these statements. In this paper we review some conjectures that we have presented earlier, introduce new conjectures, systematize them and prove new connections between them and some other statements studied before.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。