复杂性是不完整性的来源吗？
Is Complexity a Source of Incompleteness?
摘要：
在本文中，我们证明Chaitin的“启发式原理”，{ \it定理的有限指定的理论不能显着更复杂的理论本身}，为一个适当的措施的复杂性。我们证明了该测度在哥德尔数的变化下是不变的。对于这个测度，一个有限指定的、可靠的、相容的理论的定理，强到足以将算术形式化，这是算术上可靠的（如带选择的策梅洛-弗伦克尔集合论或皮亚诺算术），具有有限的复杂性，因此理论的每一个句子，如果比理论复杂得多，都是不可证明的。以前的结果表明，不完全性不是偶然的，而是普遍存在的，在这里得到了概率方面的加强：当$n$趋于无穷大时，长度为$n$的真句子在理论中可证明的概率趋于零，而长度为$n$的句子为真的概率是严格正的。
英文摘要：
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is invariant under the change of the Gödel numbering. For this measure, the theorems of a finitely-specified, sound, consistent theory strong enough to formalize arithmetic which is arithmetically sound (like Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded complexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. Previous results showing that incompleteness is not accidental, but ubiquitous are here reinforced in probabilistic terms: the probability that a true sentence of length $n$ is provable in the theory tends to zero when $n$ tends to infinity, while the probability that a sentence of length $n$ is true is strictly positive.
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。
