First-order Language for Sets Cset={E} Parentheses:(, Variables:,,z,.. Connectives:∧，V,一，→，→ Quantifiers:V,3 Equality:= Constants: Functions: Predicates:∈ Everything we consider in Cset is a set. 4口，1①，43，t夏，30Q0 Jun Ma (majunainju.edu.cn) Set Theory:Axioms and Operations 2021年11月25日10/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 10 / 40
Q:What is“e"? Q:What are“sets"? 4口￥0,43，t夏里Q0 Jun 1 (unmjtcdncn Set Theory:Axioms and Operations 2021 11 25 11/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q : What is “∈”? Q : What are “sets”? We don’t define them directly. We only describe their properties in an axiomatic way. Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 11 / 40
Q:What is“e"? Q:What are“sets"? We don't define them directly. We only describe their properties in an axiomatic way. 4口，1①，43，t夏，30Q0 Jun 1E (tacmjmcdncn Set Theory:Axioms and Operations 2021 11 25 11/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q : What is “∈”? Q : What are “sets”? We don’t define them directly. We only describe their properties in an axiomatic way. Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 11 / 40
几何原本 Eacid (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Q0 Jun Ma (majunnjuedu.cn)Set Theory:Axioms and Operations 2021年11月25日12/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (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. Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 12 / 40
Definition () x生A≌(x∈A). 4口，1①，43，t夏，30Q0 Juni jtmcmjtedncn Set Theory:Axioms and Operations 2021 11 25 13/40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition (∈/) x /∈ A , ¬(x ∈ A). Definition (⊆) A ⊆ B , ∀x(x ∈ A =⇒ x ∈ B) Jun Ma (majun@nju.edu.cn) Set Theory: Axioms and Operations 2021 年 11 月 25 日 13 / 40