CLASSICAL SET THEORY: THEORY OF SETS AND CLASSES TARAS BANAKH Contents Introduction 4 Part 1. Naive Set Theory 5 1. Origins of Set Theory 5 2. Berry’s Paradox 6 3. The language and formulas of Set Theory 7 4. Russell’s Paradox 8 Part 2. Axiomatic Theories of Sets 10 5. Axioms of von Neumann–Bernays–G¨odel 10 6. Existence of Classes 15 7. G¨odel’s Theorem on class existence 16 8. Axiomatic Set Theory of Zermelo–Fraenkel 19 Part 3. Fundamental Constructions 21 9. Cartesian products of sets 21 10. Relations 21 11. Functions 22 12. Indexed families of classes 23 13. Cartesian products of classes 24 14. Reflexive and irreflexive relations 25 15. Equivalence relations 25 arXiv:2006.01613v1 [math.LO] 2 Jun 2020 16. Well-Founded relations 26 Part 4. Order 29 17. Order relations 29 18. Transitivity 31 19. Ordinals 32 20. Trees 35 21. Recursion Theorem 36 22. Ranks 40 23. Well-orders 41 Part 5. Foundation and Constructibility 44 24. Foundation 44 1 2 TARAS BANAKH 25. Constructibility 46 26. Inner Models 53 Part 6. Choice and Global Choice 56 27. Choice 56 28. Global Choice 63 Part 7. Ordinal Arithmetics 69 29. Transfinite Dynamics 69 30. Successors 71 31. Addition 72 32. Multiplication 74 33. Exponentiation 78 34. Cantor’s normal form 80 Part 8. Cardinals 82 35. Cardinalities and cardinals 82 36. Finite and Countable sets 84 37. Successor cardinals and alephs 89 38. Arithmetics of Cardinals 91 39. Cofinality of cardinals 94 40. (Generalized) Continuum Hypothesis 96 41. Inaccessible and measurable cardinals 98 Part 9. Linear orders 101 42. Completeness 101 43. Universality 102 44. Cuts 106 45. Gaps 108 Part 10. Numbers 112 46. Integer numbers 112 47. Rational numbers 114 48. Real numbers 115 49. Surreal numbers 117 Part 11. Mathematical Structures 122 50. Examples of Mathematical Structures 122 51. Morphisms of Mathematical Structures 127 52. A characterization of the real line 129 Part 12. Elements of Category Theory 131 53. Categories 131 54. Functors 137 55. Natural transformations 138 56. Skeleta and equivalence of categories 139 57. Limits and colimits 143 CLASSICALSETTHEORY:THEORYOFSETSANDCLASSES 3 57.1. Products and coproducts 143 57.2. Pullbacks and pushouts 145 57.3. Equalizers and coequalizers 146 57.4. Limits and colimits of diagrams 147 58. Characterizations of the category Set 149 59. Cartesian closed categories 154 60. Subobject classifiers 155 61. Elementary Topoi 158 Epilogue 161 References 162 4 TARAS BANAKH Introduction Make things as simple as possible, but not simpler. Albert Einstein This is a short introductory course to Set Theory and Category Theory, based on axioms of von Neumann–Bernays–G¨odel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-graduate) student in Mathematics should know about foundations of this science. My aim is to give strict definitions of all set-theoretic notions and concepts that are widely used in mathematics. In particular, we shall introduce the sets N, Z, Q, R of numbers (natural, integer, rational and real) and will prove their basic order and algebraic properties. Since the system of NBG axioms is finite and does not involve advanced logics, it is more friendly for beginners than other axiomatic set theories (like ZFC). The legal use of classes in NBG will allow us to discuss freely Conway’s surreal numbers that form an ordered field No, which is a proper class and hence is not “visible” in ZFC. Also the language of NBG allows to give natural definitions of some basic notions of Category Theory: category, functor, natural transformation, which is done in the last part of this book. I would like to express my thanks for the help in writing this text to: • Pace Nielsen who motivated my interests in NBG; • Ulyana Banakh (my daughter) who was the first reader of this text; • Asaf Karagila for his valuable remarks concerning the Axiom of Choice; • the participant of zoom-seminar in Classical Set Theory (Serhii Bardyla, Oleksandr Maslyuchenko, Misha Popov, Alex Ravsky and others) for many valuable comments; • To be added. Lviv (at the time of quarantine) March–May, 2020. CLASSICALSETTHEORY:THEORYOFSETSANDCLASSES 5 Part 1. Naive Set Theory A set is a Many that allows itself to be thought of as a One. Georg Cantor 1. Origins of Set Theory The origins of (naive) Set Theory were created at the end of XIX century by Georg Cantor1 (1845–1918) in his papers published in 1874–1897. Cantors ideas made the notion of a set the principal (undefined) notion of Mathematics, which can be used to give precise definitions of all other mathematical concepts such as numbers or functions. According to Cantor, a set is an arbitrary collection of objects, called elements of the set. In particular, sets can be elements of other sets. The fact that a set x is an element of a set y is denoted by the symbol x ∈ y. If x is not an element of y, then we write x∈ / y. A set consisting of finitely many elements x1,...,xn is written as {x1,...,xn}. Two sets are equal if they consist of the same elements. For example, the sets {x,y} and {y,x} both have elements x,y and hence are equal. A set containing no elements at all is called the empty set and is denoted by ∅. Since sets with the same elements are equal, the empty set is unique. The theory developed so far, allows us to give a precise meaning to natural numbers (which are abstractions created by humans to facilitate counting): 0= ∅, 1= {0}, 2= {0, 1}, 3= {0, 1, 2}, 4= {0, 1, 2, 3}, 5= {0, 1, 2, 3, 4}, . The set {0, 1, 2, 3, 4, 5,... } of all natural numbers2 is denoted by ω. The set {1, 2, 3, 4, 5,... } of non-zero natural numbers is denoted by N. Very often we need to create a set of objects possessing some property (for example, the set of odd numbers). In this case we use the constructor {x : ϕ(x)}, which yields exactly what we need: the set {x : ϕ(x)} of all objects x that have certain property ϕ(x). Using the constructor we can define some basic “algebraic” operations over sets X,Y : • the intersection X ∩ Y = {x : x ∈ X ∧ x ∈ Y } whose elements are objects that belong to X and Y ; 1Exercise: Read about Georg Cantor in Wikipedia. 2Remark: There are two meanings (Eastern and Western) of what to understand by a natural number. The western approach includes zero to natural numbers whereas the eastern tradition does not. This difference can be noticed in numbering floors in buildings in western or eastern countries. 6 TARAS BANAKH • the union X ∪ Y = {x : x ∈ X ∨ x ∈ Y } consisting of the objects that belong to X or Y or to both or them; • the difference X \ Y = {x : x ∈ X ∧ x∈ / Y } consisting of elements that belong to X but not to Y ; • the symmetric difference X△Y = (X ∪Y )\(X ∩Y ) consisting of elements that belong to the union X ∪ Y but not to the intersection X ∩ Y . In the formulas for the union and intersection we used the logical connectives ∧ and ∨ denoting the logical operations and and or. Below we present the truth table for these logical operations and also for three other logical operations: the negation ¬, the implication ⇒, the equivalence ⇔. x y x ∧ y x ∨ y x ⇒ y x ⇔ y ¬x 0 0 0 0 1 1 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 1 1 0 Therefore, we have four basic operations over sets X,Y : X ∩ Y = {x : x ∈ X ∧ x ∈ Y }, X ∪ Y = {x : x ∈ X ∨ x ∈ Y }, X \ Y = {x : x ∈ X ∧ x∈ / Y }, X△Y = (X ∪ Y ) \ (X ∩ Y ). Exercise 1.1. For the sets X = {0, 1, 2, 4, 5} and Y = {1, 2, 3, 4}, find X ∩ Y , X ∪ Y , X \ Y , X△Y . 2. Berry’s Paradox “The essence of mathematics is its freedom” Georg Cantor In 1906 Bertrand Russell, a famous British philosopher, published a paradox, which he attributed to G. Berry (1867–1928), a junior librarian at Oxford’s Bodleian library. To formulate this paradox, observe that each natural number can be described by some property. For example, zero is the smallest natural number, one is the smallest nonzero natural number, two is the smallest prime number, three is the smallest odd prime number, four is the smallest square, five is the smallest odd prime number which is larger than the smallest square and so on. Since there are only finitely many sentences of a given length, such sentences (of given length) can describe only finitely many numbers3. Consequently, infinitely many numbers cannot be described by short sentences, consisting of less than 100 symbols. Among such numbers take the smallest one and denote it by s. Now consider the characteristic property of this number: s is the smallest number that cannot be described by a sentence consisting less than 100 symbol. But the latter sentence consists of 96 symbols, which is less that 100, and uniquely defines the number s. 3The list of short descriptions of the first 10000 numbers can be found here: https://www2.stetson.edu/∼efriedma/numbers.html CLASSICALSETTHEORY:THEORYOFSETSANDCLASSES 7 Now we have a paradox4 on one hand, the numbers s belongs to the set of numbers that cannot be described by short sentences, and on the other hand, it has a short description.