Godel Proof

Kurt Friedrich Gödel (German, April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Frege.

The completeness question: whether a certain explicitly given axiom system for the first order predicate calculus “…is complete in the sense that from it all logical formulas that are correct for each domain of individuals can be derived…”.

• 希尔伯特：希望将演绎系统完全形式化。将研究对象本身（数学）和关于这个对象的讨论（元数学）区分。
• 《数学原理》：形式逻辑的系统化。试图 证明：（1）所有的数论概念都可以用纯逻辑的概念来定义；（2）所有数论公理都可以从很少几条可以被确认为纯逻辑的基本命题中演绎出来。（一致性问题就从数论转移到了逻辑上）