Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.

2013年11月11日 · Gödel’s incompleteness theorems are among the most important results in modern logic. These discoveries revolutionized the understanding of mathematics and logic, and had dramatic implications for the philosophy of mathematics.

哥德尔不完全性定理是二十世纪最具影响力的数学真理，他使人类对世界的认识变得“面目全非”， 也使人类对理性的认识上升到一个新的高度。 哥德尔不完全性定理告诉人们，人类寻求知识的努力永远都不会到达终点，我们始终都有获得新知识的挑战， 人类社会总有“事与愿违”的事情需要我们去克服，没有这种挑战，人类就会停滞。 [3] 任意一个包含一阶谓词逻辑与初等数论的 形式系统，都存在一个命题，它在这个系统中既不能被证明为真，也不能被证明为否。 如果系统S含有 …

在 数理逻辑 中， 哥德尔不完备定理 是 库尔特·哥德尔 于1931年 证明 并发表的两条 定理。 第一条定理指出： 任何 自洽 的 形式系統，只要蕴涵 皮亚诺算术公理，就可以在其中构造在 体系 中不能被 证明 的真 命题，因此通过 推理 演绎 不能得到所有真命题（即体系是不 完备 的）。 这是 形式逻辑 中的定理，容易被错误表述。 有许多命题听起来很像是哥德尔不完备定理，但事实上并不是。 具体实例见 对哥德尔定理的误解。 把第一条定理的证明过程在体系内部形式化后，哥德尔 …

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

1999年1月25日 · What Godel's theorem says is that there are properly posed questions involving only the arithmetic of integers that Oracle cannot answer. In other words, there are statements...

2013年7月28日 · The problem with Gödel's incompleteness is that it is so open for exploitations and problems once you don't do it completely right. You can prove and disprove the existence of god using this theorem, as well the correctness of religion and its incorrectness against the correctness of science.

The completeness theorem essentially asserts that true statements are the result of deductions (there is another theorem, the soundness theorem, that asserts the converse that all deductions lead to true statements). The statement of the theorem is that if ˚satis es a language, , then ˚is deducible from . Theorem 2.4. (a) If j= ˚then ‘˚

2025年2月25日 · Incompleteness theorem, in foundations of mathematics, either of two theorems proved by the Austrian-born American logician Kurt Gödel. In 1931 Gödel published his first incompleteness theorem, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On Formally

This book presents Gödel’s incompleteness theorems and the other limitative results which are most significant for the philosophy of mathematics. Results are stated in the form most relevant for use in the philosophy of mathematics. An appendix considers their implications for Hilbert’s Program for the foundations of mathematics.