Set Theory Jason Filippou CMSC250 @ UMCP 06-20-2016 Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 1 / 56 Outline 1 Branches of Set Theory 2 Basic Definitions Single sets Two or more sets 3 Proofs with sets 4 An application: Formal languages 5 Paradoxes in Set Theory Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 2 / 56 Branches of Set Theory Branches of Set Theory Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 3 / 56 Formal Proofs! Branches of Set Theory Naive Naive set theory is typically taught even at elementary school nowadays. Only kind of set theory till the 1870s! Consists of applications of Venn Diagrams. Very intuitive, suitable for graphical applications Not an ounce of formality. Cannot be used for... Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 4 / 56 Branches of Set Theory Naive Naive set theory is typically taught even at elementary school nowadays. Only kind of set theory till the 1870s! Consists of applications of Venn Diagrams. Very intuitive, suitable for graphical applications Not an ounce of formality. Cannot be used for... Formal Proofs! Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 4 / 56 Branches of Set Theory Naive Based entirely on Venn Diagrams. Ω Α Β C Figure 1: An example Venn Diagram. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 5 / 56 Branches of Set Theory Axiomatic (Cantor & Dedekind) First axiomatization of Set Theory. Understanding of infinite sets and their cardinality. Figure 2: Georg Cantor, 1870s Figure 3: Richard Dedekind, 1900s Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 6 / 56 Also: Power set operation. Cantor's paradise. Branches of Set Theory Famous Result 1874 Cantor paper: \On a Property of the Collection of All Real Algebraic Numbers" The set of real numbers is uncountable. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 7 / 56 Branches of Set Theory Famous Result 1874 Cantor paper: \On a Property of the Collection of All Real Algebraic Numbers" The set of real numbers is uncountable. Also: Power set operation. Cantor's paradise. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 7 / 56 Branches of Set Theory Russel's Paradox Consider the following set: S = fxjx2 = xg Then, does S 2 S ? Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 8 / 56 Branches of Set Theory So why do we care? If Axiomatic (Cantorian) Set Theory is \broken", why do we study it? Rough answer: Just because a theory is \broken" (i.e leads to contradictions) doesn't mean we shouldn't study it. Theories are specialized (more stuff is added to them) in order to avoid contradictions all the time. Non-Euclidean geometries. Zermello - Fraenkel Set Theory. Qualitative answer: it gives us background necessary to discuss: 1 Limitations of computers as a whole! 2 Some fundamental results on countability and uncountability of infinite sets. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 9 / 56 Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck ::: Branches of Set Theory Branches of Axiomatic Set Theory Zermelo-Fraenkel (answers Russel's paradox through the axiom of choice). Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck ::: Branches of Set Theory Branches of Axiomatic Set Theory Zermelo-Fraenkel (answers Russel's paradox through the axiom of choice). Kripke-Platek Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Morse-Kelley Tarski-Grothendieck ::: Branches of Set Theory Branches of Axiomatic Set Theory Zermelo-Fraenkel (answers Russel's paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Tarski-Grothendieck ::: Branches of Set Theory Branches of Axiomatic Set Theory Zermelo-Fraenkel (answers Russel's paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 ::: Branches of Set Theory Branches of Axiomatic Set Theory Zermelo-Fraenkel (answers Russel's paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Branches of Set Theory Branches of Axiomatic Set Theory Zermelo-Fraenkel (answers Russel's paradox through the axiom of choice). Kripke-Platek Von Neumann - Bernays - G¨odel Morse-Kelley Tarski-Grothendieck ::: Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 10 / 56 Basic Definitions Basic Definitions Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 11 / 56 Basic Definitions Single sets Single sets Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 12 / 56 Definition (Ordered Set) An ordered set is a pair (S; ≤), where S is a set and ≤ is a total order. Total orders: binary relations that are antisymmetric, transitive and total. Examples: ≤, ⊆, lexicographic ordering. Definition (Multiset) A multiset is a collection of objects with repetitions. We won't really care about those, or about ordered multisets. Basic Definitions Single sets Definitions Definition (Set) A set is a collection of objects without repetitions. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56 Definition (Multiset) A multiset is a collection of objects with repetitions. We won't really care about those, or about ordered multisets. Basic Definitions Single sets Definitions Definition (Set) A set is a collection of objects without repetitions. Definition (Ordered Set) An ordered set is a pair (S; ≤), where S is a set and ≤ is a total order. Total orders: binary relations that are antisymmetric, transitive and total. Examples: ≤, ⊆, lexicographic ordering. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56 Basic Definitions Single sets Definitions Definition (Set) A set is a collection of objects without repetitions. Definition (Ordered Set) An ordered set is a pair (S; ≤), where S is a set and ≤ is a total order. Total orders: binary relations that are antisymmetric, transitive and total. Examples: ≤, ⊆, lexicographic ordering. Definition (Multiset) A multiset is a collection of objects with repetitions. We won't really care about those, or about ordered multisets. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 13 / 56 Since the chief operation is membership, and sets have unique elements, how would you implement them in computer memory? Basic Definitions Single sets Membership Chief operation on sets: membership (2). If Ω is a domain of choice and S is a set, any element e of Ω can either belong to A (e 2 A) or not (e2 = A). Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 14 / 56 Basic Definitions Single sets Membership Chief operation on sets: membership (2). If Ω is a domain of choice and S is a set, any element e of Ω can either belong to A (e 2 A) or not (e2 = A). Since the chief operation is membership, and sets have unique elements, how would you implement them in computer memory? Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 14 / 56 1 Curly braces: S = f0; 2; 4; 6g, Z = fAshley; John; Markg, F = f1; 2; 3; 5; 8; 13; 21;::: g 2 Definition: A = fz 2 Z j z ≥ −2g 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A [ B, P(f0; 1g) Basic Definitions Single sets Defining a set Three ways. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 2 Definition: A = fz 2 Z j z ≥ −2g 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A [ B, P(f0; 1g) Basic Definitions Single sets Defining a set Three ways. 1 Curly braces: S = f0; 2; 4; 6g, Z = fAshley; John; Markg, F = f1; 2; 3; 5; 8; 13; 21;::: g Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A [ B, P(f0; 1g) Basic Definitions Single sets Defining a set Three ways. 1 Curly braces: S = f0; 2; 4; 6g, Z = fAshley; John; Markg, F = f1; 2; 3; 5; 8; 13; 21;::: g 2 Definition: A = fz 2 Z j z ≥ −2g Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 4 An operation (union, superset, etc): C = A [ B, P(f0; 1g) Basic Definitions Single sets Defining a set Three ways. 1 Curly braces: S = f0; 2; 4; 6g, Z = fAshley; John; Markg, F = f1; 2; 3; 5; 8; 13; 21;::: g 2 Definition: A = fz 2 Z j z ≥ −2g 3 Agreed upon symbol: N, P, etc Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 Basic Definitions Single sets Defining a set Three ways. 1 Curly braces: S = f0; 2; 4; 6g, Z = fAshley; John; Markg, F = f1; 2; 3; 5; 8; 13; 21;::: g 2 Definition: A = fz 2 Z j z ≥ −2g 3 Agreed upon symbol: N, P, etc 4 An operation (union, superset, etc): C = A [ B, P(f0; 1g) Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 15 / 56 Corollary 8A ⊆ Ω; jAj ≥ 0 |{−10; 0; 10gj = ? 3 jfn 2 Njn < 100gj = ? 100 ∗ jfn 2 N jn < 100gj = ? 99 jNj = ? @0 (???) Basic Definitions Single sets Cardinality Definition (Cardinality of a set) Let A be a finite set.a Then, the number of elements of the A, denoted jAj 2 N is called the cardinality of A. aHold your horses, please. Jason Filippou (CMSC250 @ UMCP) Set Theory 06-20-2016 16 / 56 |{−10; 0; 10gj = ? 3 jfn 2 Njn < 100gj = ? 100 ∗ jfn 2 N jn < 100gj = ? 99 jNj = ? @0 (???) Basic Definitions Single sets Cardinality Definition (Cardinality of a set) Let A be a finite set.a Then, the number of elements of the A, denoted jAj 2 N is called the cardinality of A.