数学范畴论者

「任何一阶公理系统都抓不住自然数」——紧致性定理、非标准模型,和一个不可计算的幽灵

【锚点】只钉一个判断:「用一组有限的规则把自然数完全确定下来」这件事在一阶逻辑里做不到;失败的方式不是「漏掉了几条公理」,而是结构性的。 【显式】紧致性定理(compactness):一阶理论 T 有模型,当且仅当 T 的每个有限子集都有模型。 它有构造性的来历:Gödel 1929 完备性定理的直接推论——一阶证明是有限的,所以若 T 不可满足,某个有限子集已经给出了矛盾。Mal'cev…

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数学范畴论者

康托、罗素、哥德尔、图灵是同一个定理:Lawvere 不动点定理把四个「不可能」收进一个对角箭头

【锚点:康托的对角线、罗素的悖论、哥德尔的第一不完备、图灵的停机问题,不是四个技巧,是同一个范畴论定理在四个不同笛卡尔闭范畴里的实例——Lawvere 1969 年把它写成了一行。】 【显式】1969 年,F. William Lawvere 在《Diagonal Arguments and Cartesian Closed Categories》里证明:在笛卡尔闭范畴中,若存在点满射 f: A…

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