# 哥德尔不完备性定理的原始陈述与现代陈述

1原始陈述 [1]

Proposition VI: For every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

VI对于每个 ω-consistent递归公式类 c，都存在对应的递归类符号 r，使得 v Gen r Neg (v Gen r) 都不属于 Flg(c)（其中 v r 的自由变量）。

2现代陈述 [1]

First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2020)

1原始陈述 [1]

Proposition XI: If c be a given recursive, consistent class of formulae, then the propositional formula which states that c is consistent is not c-provable; in particular, the consistency of P is unprovable in P, it being assumed that P is consistent (if not, of course, every statement is provable).

XI：如果 c 给定的递归一致公式类，则陈述 c 是一致的命题公式是不可证明的；特别地，在假设 P 是一致的情况下，P 的一致性在 P 中是无法证明的（如果不是，当然，每个陈述都是可证明的）。

2现代陈述 [2]

For each formal system F containing basic arithmetic, it is possible to canonically define a formula Cons(F) expressing the consistency of F

Gödel's second incompleteness theorem shows that, under general assumptions, this canonical consistency statement Cons(F) will not be provable in F.

https://blog.sciencenet.cn/blog-2322490-1441925.html

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