Definition (Dedekind-infinite Dedekind-finite (Dedekind,1888)) A set A is Dedekind-infinite if there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. another way This is a theorem in our theory of infinity. Hengfeng Wei (hfweixinju.edu.cn) 1-11 Set Theory (IV):Infinity 2019年12月17日11/49
Definition (Dedekind-infinite & Dedekind-finite (Dedekind, 1888)) A set A is Dedekind-infinite if there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. This is a theorem in our theory of infinity. Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 11 / 49
Ve have not defined“finite”and“infinite"l Hengfeng Wei (hfweiinju.edu.cn)1-11 Set Theory (IV):Infinity 2019年12月17日12/49
We have not defined “finite” and “infinite”! Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 12 / 49
Comparing Sets 子 < Function Hengfeng Wei (hfweignju.edu.cn) 1-11 Set Theory (TV):Infinity 2019年12月17日13/49
Comparing Sets Function Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 13 / 49
Hengfeng Wei (hfweisinju.edu.cn1-11 Set Theory (TV):Infinity 2019年12月17日14/49
Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 14 / 49
Definition (A =B(A B)(1878)) A and B are equipotent if there exists a bijection from A to B. (two abstractions) Abstract from elements:{1,2,3 vs.fa,b,ch Abstract from order::{1,2,3,…}vs.{1,3,5,·,2,4,6,…} Hengfeng Wei (hfweiinju.edu.cn)1-11 Set Theory (IV):Infinity 2019年12月17日15/49
Definition (|A| = |B| (A ≈ B) (1878)) A and B are equipotent if there exists a bijection from A to B. A (two abstractions) Abstract from elements: {1, 2, 3} vs. {a, b, c} Abstract from order: {1, 2, 3, · · · } vs. {1, 3, 5, · · · , 2, 4, 6, · · · } Hengfeng Wei (hfwei@nju.edu.cn) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 15 / 49