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