Lecture 9: Axiomatic Set Theory December 16, 2014 Lecture 9: Axiomatic Set Theory Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Lecture 9: Axiomatic Set Theory I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: Lecture 9: Axiomatic Set Theory I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: I The iterative concept of set. Lecture 9: Axiomatic Set Theory I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). Lecture 9: Axiomatic Set Theory I Justification of the axioms based on the iterative concept of set. Today Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). Lecture 9: Axiomatic Set Theory Today Key points of today's lecture: I The iterative concept of set. I The language of set theory (LOST). I The axioms of Zermelo-Fraenkel set theory (ZFC). I Justification of the axioms based on the iterative concept of set. Lecture 9: Axiomatic Set Theory The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I Lecture 9: Axiomatic Set Theory Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Lecture 9: Axiomatic Set Theory x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form Lecture 9: Axiomatic Set Theory We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: Lecture 9: Axiomatic Set Theory x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write Lecture 9: Axiomatic Set Theory for the negation of x = y and x 2 y, respectively. The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y Lecture 9: Axiomatic Set Theory The language of set theory, I The language of set theory (LOST) consists of the logical symbols and =, and a single binary relation symbol, 2, called membership. Note that the atomic formulas of LOST are of the form x = y and x 2 y: We will write x 6= y and x 2= y for the negation of x = y and x 2 y, respectively. Lecture 9: Axiomatic Set Theory In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II Lecture 9: Axiomatic Set Theory x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters Lecture 9: Axiomatic Set Theory to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: Lecture 9: Axiomatic Set Theory We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. Lecture 9: Axiomatic Set Theory The language of set theory, II In addition to the variables of our formal language, v1; v2;:::, we will also use the letters x; y; z; u; w; x1; x2;:::; y1; y2;:::; w1; w2;::: to stand for arbitrary formal variables. We will allow ourselves to use the standard abbreviations: _; ^; $ and 9 for \or", \and", \if and only if", and \there exists". Lecture 9: Axiomatic Set Theory 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 Lecture 9: Axiomatic Set Theory 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: Lecture 9: Axiomatic Set Theory (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: Lecture 9: Axiomatic Set Theory 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) Lecture 9: Axiomatic Set Theory 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: Lecture 9: Axiomatic Set Theory (Sets that have the same members are identical.) Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): Lecture 9: Axiomatic Set Theory Axioms 0 and 1 0. Axiom of Set Existence: 9v1v1 = v1: (There is a set.) 1. Axiom of Extensionality: 8v18v2(8v3(v3 2 v1 $ v3 2 v2) ! v1 = v2): (Sets that have the same members are identical.) Lecture 9: Axiomatic Set Theory 2. Axiom of Foundation: 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): (Every non-empty set has a member which has no members in common with it.) Axiom 2 Lecture 9: Axiomatic Set Theory 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): (Every non-empty set has a member which has no members in common with it.) Axiom 2 2. Axiom of Foundation: Lecture 9: Axiomatic Set Theory (Every non-empty set has a member which has no members in common with it.) Axiom 2 2. Axiom of Foundation: 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): Lecture 9: Axiomatic Set Theory Axiom 2 2. Axiom of Foundation: 8v1(9v2 v2 2 v1 ! 9v2(v2 2 v1 ^ 8v3 v3 2= v1 _ v3 2= v2)): (Every non-empty set has a member which has no members in common with it.) Lecture 9: Axiomatic Set Theory 3. Axiom Schema of Comprehension: For each LOST formula '(x; z; w1;:::; wn) with the (distinct) free variables among those shown, the following is an axiom: 8w1 · · · 8wn8z9y8x(x 2 y $ (x 2 z ^ ')): (For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) ! Axiom 3 Lecture 9: Axiomatic Set Theory 8w1 · · · 8wn8z9y8x(x 2 y $ (x 2 z ^ ')): (For any sets z and any property P which can be expressed by a formula of LOST, there is a set whose members are those members of z that have property P.) ! Axiom 3 3.