Axiom of Choice, Maximal Independent Sets, Argumentation and Dialogue Games Christof Spanring Department of Computer Science, University of Liverpool, UK Institute of Information Systems, Vienna University of Technology, Austria PhDs Tea Talk Liverpool, October 3, 2014 Outline 1 Introduction Games and Motivations Infinity and Questions 2 Backgrounds Zermelo-Fraenkel Set Theory and related Axioms Games Again, Infinite Style 3 The Stuff Abstract Argumentation An Equivalence Proof Christof Spanring, PhDs Tea Talk Choice and Argumentation 2 / 16 A minor example Example (Games played on Argument Graphs) Can you defend an argument a beyond doubt, i.e. defeat any attackers without running into conflict with your own argument base? Who has a winning strategy, you as the proponent or your oponent? a b c ... Christof Spanring, PhDs Tea Talk Choice and Argumentation 3 / 16 A minor example Example (Games played on Argument Graphs) Can you defend an argument a beyond doubt, i.e. defeat any attackers without running into conflict with your own argument base? Who has a winning strategy, you as the proponent or your oponent? a b c d ... Christof Spanring, PhDs Tea Talk Choice and Argumentation 3 / 16 A minor example Example (Games played on Argument Graphs) Can you defend an argument a beyond doubt, i.e. defeat any attackers without running into conflict with your own argument base? Who has a winning strategy, you as the proponent or your oponent? a b c d ... Christof Spanring, PhDs Tea Talk Choice and Argumentation 3 / 16 Question p How many rational numbers q are there? Question How many decimal numbers are there? Question Is there a set of all sets? The Why? of Infinities I Question How many prime numbers are there? Christof Spanring, PhDs Tea Talk Choice and Argumentation 4 / 16 Question How many decimal numbers are there? Question Is there a set of all sets? The Why? of Infinities I Question How many prime numbers are there? Question p How many rational numbers q are there? Christof Spanring, PhDs Tea Talk Choice and Argumentation 4 / 16 Question Is there a set of all sets? The Why? of Infinities I Question How many prime numbers are there? Question p How many rational numbers q are there? Question How many decimal numbers are there? Christof Spanring, PhDs Tea Talk Choice and Argumentation 4 / 16 The Why? of Infinities I Question How many prime numbers are there? Question p How many rational numbers q are there? Question How many decimal numbers are there? Question Is there a set of all sets? Christof Spanring, PhDs Tea Talk Choice and Argumentation 4 / 16 The Why? of Infinities II Example (jQj = jNj) Example (jNj < jRj) There are only as many rational There are more real than natural as natural numbers. numbers. 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 1 1 1 1 1 1 1 1 2 1 3 1 4 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 2 2 2 2 2 2 1 2 2 2 3 2 4 1 2 3 4 ... i = 0: i i i ; i ; ··· 3 3 3 3 3 3;1 3;2 3 3 3 4 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 4 4 4 4 4 4 1 4 2 4 3 4 4 ... ... ... ... ... ... Christof Spanring, PhDs Tea Talk Choice and Argumentation 5 / 16 The Why? of Infinities II Example (jQj = jNj) Example (jNj < jRj) There are only as many rational There are more real than natural as natural numbers. numbers. 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 1 1 1 1 1 1 1 1 2 1 3 1 4 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 2 2 2 2 2 2 1 2 2 2 3 2 4 1 2 3 4 ... i = 0: i i i ; i ; ··· 3 3 3 3 3 3;1 3;2 3 3 3 4 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 4 4 4 4 4 4 1 4 2 4 3 4 4 ... ... ... ... ... ... Christof Spanring, PhDs Tea Talk Choice and Argumentation 5 / 16 The Why? of Infinities II Example (jQj = jNj) Example (jNj < jRj) There are only as many rational There are more real than natural as natural numbers. numbers. 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 1 1 1 1 1 1 1 1 2 1 3 1 4 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 2 2 2 2 2 2 1 2 2 2 3 2 4 1 2 3 4 ... i = 0: i i i ; i ; ··· 3 3 3 3 3 3;1 3;2 3 3 3 4 1 2 3 4 ... i = 0: i ; i ; i ; i ; ··· 4 4 4 4 4 4 1 4 2 4 3 4 4 ... ... ... ... ... ... Christof Spanring, PhDs Tea Talk Choice and Argumentation 5 / 16 Outline 1 Introduction Games and Motivations Infinity and Questions 2 Backgrounds Zermelo-Fraenkel Set Theory and related Axioms Games Again, Infinite Style 3 The Stuff Abstract Argumentation An Equivalence Proof Christof Spanring, PhDs Tea Talk Choice and Argumentation 6 / 16 Set Theory Definition Zermelo-Fraenkel Set Theory (ZFC-Axioms) 1 Extensionality 8x8y (8z (z 2 x , z 2 y) ) x = y) 2 Foundation 8x (9a (a 2 x) ) 9y (y 2 x ^ :9z (z 2 y ^ z 2 x))) 3 Specification 8z8v18v2 · · · 8vn9y8x (x 2 y , (x 2 z ^ ')) 4 Pairing 8x8y9z (x 2 z ^ y 2 z) 5 Union 8x9z8y8v ((v 2 y ^ y 2 x) ) v 2 z) 6 Replacement 8x8v18v2 · · · 8vn (8y (y 2 x ) 9!z') ) 9w8y (y 2 x ) 9!z(y 2 w ^ ')) 7 Infinity 9x (; 2 x ^ 8y (y 2 x ) (y [ fyg) 2 x)) 8 Power Set 8x9y8z (z ⊆ x ) z 2 y) 9 Choice 8x (; 62 x ) 9f : x ! S x; 8a 2 x (f (a) 2 a)) Christof Spanring, PhDs Tea Talk Choice and Argumentation 7 / 16 Set Theory Definition Zermelo-Fraenkel Set Theory (ZFC-Axioms) 1 Extensionality 8x8y (8z (z 2 x , z 2 y) ) x = y) 2 Foundation 8x (9a (a 2 x) ) 9y (y 2 x ^ :9z (z 2 y ^ z 2 x))) 3 Specification 8z8v18v2 · · · 8vn9y8x (x 2 y , (x 2 z ^ ')) 4 Pairing 8x8y9z (x 2 z ^ y 2 z) 5 Union 8x9z8y8v ((v 2 y ^ y 2 x) ) v 2 z) 6 Replacement 8x8v18v2 · · · 8vn (8y (y 2 x ) 9!z') ) 9w8y (y 2 x ) 9!z(y 2 w ^ ')) 7 Infinity 9x (; 2 x ^ 8y (y 2 x ) (y [ fyg) 2 x)) 8 Power Set 8x9y8z (z ⊆ x ) z 2 y) 9 Choice 8x (; 62 x ) 9f : x ! S x; 8a 2 x (f (a) 2 a)) Christof Spanring, PhDs Tea Talk Choice and Argumentation 7 / 16 Set Theory Definition Zermelo-Fraenkel Set Theory (ZFC-Axioms) 1 Extensionality 8x8y (8z (z 2 x , z 2 y) ) x = y) 2 Foundation 8x (9a (a 2 x) ) 9y (y 2 x ^ :9z (z 2 y ^ z 2 x))) 3 Specification 8z8v18v2 · · · 8vn9y8x (x 2 y , (x 2 z ^ ')) 4 Pairing 8x8y9z (x 2 z ^ y 2 z) 5 Union 8x9z8y8v ((v 2 y ^ y 2 x) ) v 2 z) 6 Replacement 8x8v18v2 · · · 8vn (8y (y 2 x ) 9!z') ) 9w8y (y 2 x ) 9!z(y 2 w ^ ')) 7 Infinity 9x (; 2 x ^ 8y (y 2 x ) (y [ fyg) 2 x)) 8 Power Set 8x9y8z (z ⊆ x ) z 2 y) 9 Choice 8x (; 62 x ) 9f : x ! S x; 8a 2 x (f (a) 2 a)) Christof Spanring, PhDs Tea Talk Choice and Argumentation 7 / 16 Set Theory Definition Zermelo-Fraenkel Set Theory (ZFC-Axioms) 1 Extensionality 8x8y (8z (z 2 x , z 2 y) ) x = y) 2 Foundation 8x (9a (a 2 x) ) 9y (y 2 x ^ :9z (z 2 y ^ z 2 x))) 3 Specification 8z8v18v2 · · · 8vn9y8x (x 2 y , (x 2 z ^ ')) 4 Pairing 8x8y9z (x 2 z ^ y 2 z) 5 Union 8x9z8y8v ((v 2 y ^ y 2 x) ) v 2 z) 6 Replacement 8x8v18v2 · · · 8vn (8y (y 2 x ) 9!z') ) 9w8y (y 2 x ) 9!z(y 2 w ^ ')) 7 Infinity 9x (; 2 x ^ 8y (y 2 x ) (y [ fyg) 2 x)) 8 Power Set 8x9y8z (z ⊆ x ) z 2 y) 9 Choice 8x (; 62 x ) 9f : x ! S x; 8a 2 x (f (a) 2 a)) Christof Spanring, PhDs Tea Talk Choice and Argumentation 7 / 16 Example (Basis Theorem for Vector Spaces) Every vector space has a basis. Example (Well-ordering Theorem) Every set can be well-ordered. Example (Zorn’s Lemma) If any chain of a non-empty partially ordered set has an upper bound then there is at least one maximal element. Choice and Companions Example (The Axiom of Choice) Every set of non-empty sets has a choice function, selecting exactly one element from each set. Christof Spanring, PhDs Tea Talk Choice and Argumentation 8 / 16 Example (Well-ordering Theorem) Every set can be well-ordered. Example (Zorn’s Lemma) If any chain of a non-empty partially ordered set has an upper bound then there is at least one maximal element. Choice and Companions Example (The Axiom of Choice) Every set of non-empty sets has a choice function, selecting exactly one element from each set. Example (Basis Theorem for Vector Spaces) Every vector space has a basis. Christof Spanring, PhDs Tea Talk Choice and Argumentation 8 / 16 Example (Zorn’s Lemma) If any chain of a non-empty partially ordered set has an upper bound then there is at least one maximal element. Choice and Companions Example (The Axiom of Choice) Every set of non-empty sets has a choice function, selecting exactly one element from each set.