# Boole's New Proof of Gödel's Incompleteness Theorem

### Review of Gödel's Proof

$$u: \neg \exists a, provable(a) \cap sub(x, x, a)$$

$$G: \neg \exists a, provable(a) \cap sub(u, u, a)$$

$$G: \neg \exists a, provable(a) \cap a = G$$

$$G: \neg \exists a, provable(G)$$

$$G: \neg provable(G)$$

### Boolo's Proof

Boolo也在试图用形式符号表达函数, 他表达出了一个函数$C(n, x)$: 数$x$可被长为$n$的公式所"named(表示)".

(看到这里有没有觉得很眼熟..)

$$F: \exists n, n = 10 k \cap A(n, x)$$

### Remarks

Gödel的第三步是巧妙的利用对角线法代换, 而Boolos并未使用对角线法. 他在它的那篇论文最后一段简洁地比较了两种方法的异同:

Both our proof and the standard one make use of Gödel numbering. Moreover, the unprovable truth in our proof and in the standard one can both be seen as obtained by the substitution of a name for a number in a certain crucial formula . There is, however, an important distinction between the two proofs. In the usual proof, the number whose name is substituted is the code for the formula into which it is substituted; in ours it is the unique number of which the formula is true.

In view of this distinction, it seems justified to say that our proof, unlike the usual one, does not involve diagonalization.

### References

Boolos的论文: "A new proof of the Gödel Incompleteness Theorem"(1989), Notices of the American Mathematical Society 36: 388–90; 676.