First-order Language for Sets Cset =fE Parentheses:(,) Variables:,,,. Connectives:∧，V,一，→，分 Quantifiers:廿，3 Equality:= Constants: Functions: Predicates:∈ Everything we consider in Cset is a set. 4口·￥①，43，t夏，里Q0 Hengfeng Wei (hfweinju.edu.cn)Set Theory:Axioms and Operations 2019年11月26日10/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First-order Language for Sets LSet = {∈} Parentheses: (,) Variables: x, y, z, · · · Connectives: ∧, ∨, ¬, →, ↔ Quantifiers: ∀, ∃ Equality: = Constants: Functions: Predicates: ∈ Everything we consider in LSet is a set. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 10 / 38
Q:What is“e"? Q:What are“sets"? 4口￥0,43，t里，里0Q0 Hengeng Wei iweinjncd.cn Set Theory:Axioms and Operations 2019 11 26 11/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q : What is “∈”? Q : What are “sets”? We don’t define them directly. We only describe their properties in an axiomatic way. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 11 / 38
Q:Vhat is“∈”？ Q:What are“sets"? We don't define them directly. We only describe their properties in an axiomatic way. 4口￥0,43，t里，里0Q0 Hengong We Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 11/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q : What is “∈”? Q : What are “sets”? We don’t define them directly. We only describe their properties in an axiomatic way. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 11 / 38
几何原本 Eudlid's Elements (1)To draw a straight line from any point to any point. (2)To extend a finite straight line continuously in a straight line. (3)To describe a circle with any center and radius. (4)That all right angles are equal to one another. (5)The parallel postulate. 4口·￥①，43，t夏，里0Q0 Hengfeng Wei (bfweiinju.edu.cn)Set Theory:Axioms and Operations 2019年11月26日12/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) To draw a straight line from any point to any point. (2) To extend a finite straight line continuously in a straight line. (3) To describe a circle with any center and radius. (4) That all right angles are equal to one another. (5) The parallel postulate. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 12 / 38
Definition () x生A≌(x∈A). 4口，1①，43，t夏，30Q0 Hengong Wei Chiweinjnedm.cn Set Theory:Axioms and Operations 2019 11 26 13/38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (∈/) x /∈ A ≜ ¬(x ∈ A). Definition (⊆) A ⊆ B ≜ ∀x(x ∈ A =⇒ x ∈ B) Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 13 / 38