Theorem(概括原则) For any predicate v(x),there is a set X: X={x|(x)}. Definition (Russell's Paradox) (x)≌“x夫x” R={x|x生x} Q:R∈R? Hengfeng Wei (fweinju.edu.cn Set Theory:Axioms and Operations 2019年11月26日6/38
Theorem (概括原则) For any predicate ψ(x), there is a set X: X = {x | ψ(x)}. Definition (Russell’s Paradox) ψ(x) ≜ “x /∈ x” R = {x | x /∈ x} Q : R ∈ R ? Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 6 / 38
Q:既然朴素集合论存在悖论，你是如何做作业的？ 但装看不到… Hengfeng Wei (hfweiinju.edu.cn)Set Theory:Axioms and Operations 2019年11月26日7/38
Q : 既然朴素集合论存在悖论，你是如何做作业的? Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 7 / 38
YOU HAVE OXT▣ Theorem (Russell's Paradox) xrg rh is not a set. Hengfeng Wei (fweinju.edu.cn Set Theory:Axioms and Operations 2019年11月26日8/38
Theorem (Russell’s Paradox) {x | x /∈ x} is not a set. Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 8 / 38
Axiomatic Set Theory (ZFC) Ernst Zermelo (1871-1953) Abraham Fraenkel (1891-1965) Hengfeng Wei (hfweiinju.edu.cn)Set Theory:Axioms and Operations 2019年11月26日9/38
Axiomatic Set Theory (ZFC) Ernst Zermelo (1871–1953) Abraham Fraenkel (1891–1965) Hengfeng Wei (hfwei@nju.edu.cn) Set Theory: Axioms and Operations 2019 年 11 月 26 日 9 / 38
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. 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