Godel, Escher, Bach [1]
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1. 哥德尔定理
用比较通俗的英文来说，就是
All consistent axiomatic formulations of number theory include undicidable propositions.
2. 图形和衬底也许会不带有完全相同的信息
There exist formal systems whose negative space (set of non-theorems) is not the positive space (set of theorems) of any formal system.
用更technical的说法
There exist recursively enumerable sets which are not recursive.
这里recursively enumerable指能按照typographical规则生成，而recursive则对应域指衬底也是个图形的图形。
由此得出一个结论
There exist formal systems for which there is no typographical decision procedure.
证明很简单，用反证法。如果所有的形式系统都能有typographical的判定方法，那么逐个测试所有的符号串，从而能生成一个非定理集合，与前面的定理矛盾。
3. 关于那个数列谜题
{Ai} = 1, 3, 7, 12, 18, 26, 35, 45, 56, 69, …
既然讲到了衬底自然要考虑这个，负空间数列为
{Bi} = 2, 4, 5, 6, 8, 9, 10, 11, 13, 14, …
An = An-1 + Bn-1
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