1. A Brief Introduction to Set Theory

2. How Many Things?

3. Mereology Theory of parts and wholes Are there bigger things than particles? Arbitrary fusions Nihilism?

4. How Many Things?

5. Lots of Little Things…

6. Some Weird Things

7. One Maximal Thing

8. Set Theory Sets are mathematical posits Any time you have some things, there is a set containing those things The set is a different thing The things it contains are its members (not parts) Since sets are things, they can be collected into sets

9. How Many Things?

16. History of Set Theory Founded by Georg Cantor in 1847. Popular ever since.

17. Names for Sets

18. Extensive Notation Bracket symbols

19. Extensive Notation Names of the set’s members John, Paul, George, Ringo

20. Extensive Notation Order doesn’t matter Paul, Ringo, John, George

21. Extensive Notation Any name will do Paul, Ringo, Dr. Winston O’Boogie, George

22. Intensive Notation Variable (your choice: x, y, z, etc.) x

23. Intensive Notation Up and down line x

24. Intensive Notation Condition that uniquely picks out the set’s members x x is a member of The Beatles

25. Intensive Notation Condition that uniquely picks out the set’s members x x had the most #1 British albums

26. Intensive Notation Condition that uniquely picks out the set’s members x x sang a song on Rubber Soul

27. Intensive Notation Condition that uniquely picks out the set’s members x x = John or x = Paul or x = George or x= Ringo

28. Set Membership The fundamental relation in set theory is the set membership relation. We write this relation with a stylized Greek epsilon: ϵ Example: John ϵ { x | x is a member of The Beatles }

29. Set Membership Only sets have members in this sense of “members”. ~(Ǝx)(x ϵ John) Sets can be members of other sets. YES: {John, Paul} ϵ { { }, {John}, {Paul}, {John, Paul} } NO: {John} ϵ {John, Paul, George, Ringo }

30. Subsets We say that A is a subset of B when all of A’s members are members of B. ( ∀ x)(x ϵ A → x ϵ B). We write: A ⊆ B. Examples: {0, 1} ⊆ {0, 1, 2} {0, 1} ⊆ {0, 1} {0, 1} ⊆ {1, 0} {0, 1} ⊆ { x | x is a number} { } ⊆ {0, 1}

31. Subsets: More Examples {1} ⊆ {1} 1 ⊄ 1 1 ⊄ {1} {1} ⊄ {{1}}

32. Axiom of Extensionality For any sets A and B: A = B if and only if A and B have the same members ( ∀ x)(x ϵ A ↔ x ϵ B) A ⊆ B & B ⊆ A

33. Extensionality Examples {0, 1} = {0, 1} {0, 1} = {1, 0} {0, 1} = { x | x is a natural number & x 2 = x} {1} = {1, 1, 1}

34. The Empty Set The empty set is the set with no members. There is only one and it is a subset of every set.

35. The Empty Set is a Member of Every Set Proof: Let a be an arbitrary object. Since { } has no members, it follows that a ∉ { } Additionally, (P → Q) is true whenever P is false. So: a ϵ { } → a ϵ B For any set B. Since a was arbitrarily selected: ( ∀ x)(x ϵ { } → x ϵ B) { } ⊆ B

36. There is Only One Empty Set Proof: Suppose A and B are sets with no members. Then A ⊆ B (from previous proof). And B ⊆ A (same proof). So A = B by the Axiom of Extensionality.

38. Further Notation Sometimes { } is written: ∅

39. Set Theoretic Operations Let A and B be sets. Union: A ∪ B = { x | x ϵ A or x ϵ B} Intersection: A ∩ B = { x | x ϵ A and x ϵ B}

40. Comparison of Laws (A ∪ B) ∪ C = A ∪ (B ∪ C) (A ∩ B) ∩ C = A ∩ (B ∩ C) A ∪ B = B ∪ A A ∩ B = B ∩ A A ∪ A = A A ∩ A = A A ∩ (A ∪ B) = A A ∪ (A ∩ B) = A (A v B) v C ↔ A v (B v C) (A & B) & C ↔ A & (B & C) A v B ↔ B v A A & B ↔ B & A A v A ↔ A A & A ↔ A A & (A v B) ↔ A A v (A & B) ↔ A

41. Comparison of Laws A ∪ { } = A A ∩ { } = { } A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A v (P & ~P) ↔ A A & (P & ~P) ↔ (P & ~P) A v (B & C) ↔ (A v B) & (A v C) A & (B v C) ↔ (A & B) v (A & C)

42. Power Sets For any set A, A’s power set is defined as follows: POW(A) = { x | x ⊆ A } If A has N members, then POW(A) has 2 N members. That’s why it’s called a power set. Sometimes people write 2 A to denote POW(A).

43. POW({0, 1, 2}) 012 {0, 1, 2}YYY {0, 1}YYN {0, 2}YNY {0}YNN {1, 2}NYY {1}NYN {2}NNY { }NNN

44. Russell’s Paradox

45. The Naïve Comprehension Schema Basic idea of set theory: When you have some things, there is another thing, the collection of those things. For every predicate F: (Ǝy)( ∀ x)(x ϵ y ↔ Fx)

46. Bertrand Russell One of the founders of analytic philosophy (contemporary Anglophone philosophy). One of the greatest logicians of the 20 th Century Showed that the basic idea of set theory can’t be right.

47. Russell’s Paradox Consider the predicate: ~x ϵ x

48. Russell’s Paradox According to Comprehension: (Ǝy)( ∀ x)(x ϵ y ↔ ~x ϵ x) Let’s call “y” here “R” for Russell’s Paradox Set.

49. Russell’s Paradox R = { x | ~x ϵ x } Question: R ϵ R?

50. Russell’s Paradox Let’s suppose: R ϵ R. Then: R ϵ { x | ~x ϵ x } ~R ϵ R

51. Russell’s Paradox Let’s suppose: ~R ϵ R. Then: R ϵ { x | ~x ϵ x } R ϵ R

52. Russell’s Paradox The Naïve Comprehension Schema leads to a contradiction. Therefore it is false. There are some properties with no corresponding set of things that have those properties.

53. Fixing Set Theory

54. Regularity Part of our problem seems to arise from the weirdness of sets that can have themselves as members. So we can legislate that away: Axiom of Regularity:

55. Restricted Comprehension That doesn’t solve the paradox though! Q: Why?

56. Restricted Comprehension New strategy: start with the basic elements and then specify explicitly which sets exist. Instead of: For every predicate F: (Ǝy)( ∀ x)(x ϵ y ↔ Fx) We have: For every predicate F: ( ∀ z)(Ǝy)( ∀ x)(x ϵ y ↔ Fx & x ϵ z)

57. Axioms of Set Theory Pair Axiom: For any x and y, the set { x, y } exists. Union Axiom: For any sets A and B, A ∪ B exists. Power Set Axiom: For any set A, { x | x ⊆ A } exists. There are more axioms, but you get the point.

58. The Universe of Sets

59. Are we happy with this solution?

60. Cantor’s Diagonal Proof

61. Numbers vs. Numerals

62. Decimal Representations A decimal representation of a real number consists of two parts: A finite string S 1 of Arabic numerals. An infinite string S 2 of Arabic numerals. It looks like this: S 1. S 2

63. We can’t actually write out any decimal representations, since we can’t write infinite strings of numerals. But we can write out abbreviations of some decimal representations. 1/4 = 0.25 1/7 = 0.142857 π = ? _______

64. We will prove that there cannot be a list of all the decimal representations between ‘0.0’ and ‘1.0’. A list is something with a first member, then a second member, then a third member and so on, perhaps continuing forever.

65. Choose an Arbitrary List 1.‘8’‘4’‘3’‘0’ … 2.‘2’‘5’‘6’‘2’‘5’‘6’‘2’‘5’… 3.‘7’‘9’‘2’‘5’‘1’‘0’‘7’‘2’… 4.‘9’‘8’‘0’‘6’‘4’‘2’‘8’‘1’… 5.‘3’ … 6.‘4’‘3’‘7’ ‘1’‘0’‘2’‘0’… 7.‘8’ ‘1’‘3’‘2’‘9’ ‘6’… 8.‘1’‘6’‘1’‘6’‘1’‘6’‘1’‘6’… …

66. Find the Diagonal 1.‘8’‘4’‘3’‘0’ … 2.‘2’‘5’‘6’‘2’‘5’‘6’‘2’‘5’… 3.‘7’‘9’‘2’‘5’‘1’‘0’‘7’‘2’… 4.‘9’‘8’‘0’‘6’‘4’‘2’‘8’‘1’… 5.‘3’ … 6.‘4’‘3’‘7’ ‘1’‘0’‘2’‘0’… 7.‘8’ ‘1’‘3’‘2’‘9’ ‘6’… 8.‘1’‘6’‘1’‘6’‘1’‘6’‘1’‘6’… …

67. Diagonal = 0.85263096… Add move each numeral ‘1 up’– so ‘8’ becomes ‘9’, ‘5’ becomes ‘6’, etc. New Representation = 0.96374107…

68. New Number Not on the List ‘9’‘6’‘3’‘7’‘4’‘1’‘0’‘7’… 1.‘8’‘4’‘3’‘0’ … 2.‘2’‘5’‘6’‘2’‘5’‘6’‘2’‘5’… 3.‘7’‘9’‘2’‘5’‘1’‘0’‘7’‘2’… 4.‘9’‘8’‘0’‘6’‘4’‘2’‘8’‘1’… 5.‘3’ … 6.‘4’‘3’‘7’ ‘1’‘0’‘2’‘0’… 7.‘8’ ‘1’‘3’‘2’‘9’ ‘6’… 8.‘1’‘6’‘1’‘6’‘1’‘6’‘1’‘6’… …

69. Doesn’t Help to Add It In! ‘9’‘6’‘3’‘7’‘4’‘1’‘0’‘7’… 1.‘8’‘4’‘3’‘0’ … 2.‘2’‘5’‘6’‘2’‘5’‘6’‘2’‘5’… 3.‘7’‘9’‘2’‘5’‘1’‘0’‘7’‘2’… 4.‘9’‘8’‘0’‘6’‘4’‘2’‘8’‘1’… 5.‘3’ … 6.‘4’‘3’‘7’ ‘1’‘0’‘2’‘0’… 7.‘8’ ‘1’‘3’‘2’‘9’ ‘6’… 8.‘1’‘6’‘1’‘6’‘1’‘6’‘1’‘6’… …

70. Discussion Questions Does this prove you can’t list all the real numbers? How do we fix the proof? Can you use a similar proof to show that the rational numbers aren’t countable? Can you list the powerset of the natural numbers?