Minimalist set theory David Pierce June , Minimalist Set Theory This work is licensed under the Creative Commons Attribution–Noncommercial–Share-Alike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ CC BY: David Pierce $\ C Mathematics Department Middle East Technical University Ankara Turkey http://metu.edu.tr/~dpierce/ [email protected] Preface The previous edition of this book was for use in Math (Set Theory) at METU in the spring semester of /. I edited and added to the book throughout the semester; the last version for the course was dated May , . The book was based on notes I had used in teaching the course in the past; but I had rewritten many sections. This is true now. In particular, I have split what was Chapter into the two chapters and of this edition, and I have made many adjustments to these chapters. The material now in Chapter was either missing or relegated to appendices before. The catalogue description of Math is: Language and axioms of set theory. Ordered pairs, relations and func- tions. Order relation and well ordered sets. Ordinal numbers, transfi- nite induction, arithmetic of ordinal numbers. Cardinality and arith- metic of cardinal numbers. Axiom of choice, generalized continuum hypothesis. The set theory presented in this book is a version of what is called ZFC: Zermelo–Fraenkel set theory with the Axiom of Choice. I call the presentation minimalist, as in the title, for several reasons: . The only basic relation between sets is membership; equality of sets is a defined notion. Classes as such have no formal existence: they are not individuals in the theory, though we can treat them in some respects as if they were. item:only . Axioms are introduced only when further progress is otherwise hin- dered. The form of many axioms, namely that such-and-such a class is a set, is used even for the Axiom of Infinity: the class ω of natural numbers is obtained without first assuming that it exists as a set. See Appendix D for further discussion. Contents Preface . Introduction . Logic .. Collections .......................... .. Theclassofallsets. .. Logic.............................. .. Formulas ........................... .. Recursionandinduction . .. Sentences ........................... .. Formalproofs......................... . Classes .. Classesandrelations . .. Relations between classes and collections . .. Setsasclasses......................... .. Thelogicofsets ....................... .. Operationsonclasses. . Numbers .. The collection of natural numbers . .. Relationsandfunctions . .. The class of formal natural numbers . .. Arithmetic........................... .. Orderings ........................... .. Finitesets........................... . Ordinality .. Well-orderedclasses . .. Ordinals............................ .. Limits ............................. .. Transfiniterecursion . .. Suprema............................ .. Ordinaladdition .. ..... ..... ..... ..... .. Ordinal multiplication . .. Ordinal exponentiation . .. Baseomega .......................... . Cardinality .. Cardinality .......................... .. Cardinals ........................... .. Cardinal addition and multiplication . .. Cardinalities of ordinal powers . .. TheAxiomofChoice. .. Exponentiation . .. TheContinuum. . Incompleteness .. Formulasassets . .. Incompleteness . .. Models............................. .. Completeness . .. Settheories .......................... .. Compactness ......................... .. Secondincompleteness . . Models .. Thewell-foundeduniverse . .. Absoluteness ......................... .. Collections of equivalence classes . .. Constructiblesets. .. The Generalized Continuum Hypothesis . . Independence .. Models............................. A. The Greek alphabet B. The German script Contents C. The Axioms D. Other set theories and approaches Bibliography Contents List of Tables .. The two basic truth tables . .. The filling-out of a truth table . .. Atruthtable ......................... .. Aformalproof ........................ .. The lexicographic ordering of 4 × 6 ............. .. The right lexicographic ordering of fs(βα).......... D.. Lemmon’s Introduction to axiomatic set theory . List of Figures .. The figure of the Pythagorean Theorem . .. Thedodecahedron,aboutacube . .. Thedigits........................... .. Aparsingtree......................... .. The parsing tree of a quantifier-free sentence . .. Thelogicofsets ....................... .. The logic of R ......................... .. A homomorphism of iterative structures . .. A bijection from a natural number to another . .. ON × ON,well-ordered . .. Towards the Cantor set . .. Thewell-foundeduniverse . B.. The German alphabet by hand . . Introduction In this book, we—the writer and the reader—shall develop an axiomatic theory of sets: . We shall study sets, which are certain kinds of collections. We shall do so by the axiomatic method. There are various reasons why one might want to do this. I see them as follows. item:sets . All concepts of mathematics can be defined by means of sets. Among such concepts are the numbers one, two, three, and so on—numbers that we learn to count with at an early age. item:elegance . Theorems about sets can be as beautiful, as elegant, as any other theorems in mathematics. The axiomatic method is of general use in mathematics, and set theory is an example of its application. item:sets-for-logic . Set theory provides a fundamental or foundational example of the axiomatic method, in the following sense. The field of mathematics called model theory can be considered as a formal investigation of the axiomatic method, as it is used in ordinary mathematics. In model theory, one de- fines structures, which are to be considered as models of certain theories. All of these notions—structure, model, theory—are defined in terms of sets. By the axiomatic method, I mean: ) the identification of certain fundamental properties of some math- ematical structure (or kind of structure); ) from these alone, the derivation of other properties of the structure. In a strict sense, a theory is a kind of collection of formal sentences. Sentences are not normally considered as sets, though they can be. A structure is a set considered with certain basic relations, that is, subsets of the Cartesian powers of the set. The theory of a structure is the collection of true sentences about the structure. The determination of these sentences is made by considering the interactions of the basic relations. All of this requires some notion of sets. It is true that not everybody likes the set-theoretic conception of model theory: see Angus Macintyre’s speculative paper, ‘Model theory: geometrical and set-theoretic aspects and prospects’ []. The fundamental properties are called axioms (or postulates). For example, groups are studied by the axiomatic method. The struc- ture of interest in group theory is based on the set of permutations or symmetries of a given set. If the given set is A, then the symmetries of A are just the invertible functions from A to itself. These symmetries compose a set, which we may call Sym(A). This set always has at least one element, the identity on A, which can be denoted by idA or simply id. Every element σ of Sym(A) has an inverse, σ−1. Any two elements σ and τ of Sym(A) have the composites σ ◦ τ and τ ◦ σ. In short, we have a structure on Sym(A), which we may denote by (Sym(A), id, −1, ◦). (.) eqn:sym Some fundamental properties of this structure are that, for all x, y, and z in Sym(A), x ◦ id = x, x ◦ x−1 = id, x ◦ (y ◦ z)=(x ◦ y) ◦ z. (.) eqn:group-ax We have been referring to sets, such as A and Sym(A); this illustrates reasons and above to study sets. Everything in this example can be reduced to sets as follows. Every element of Sym(A) is a certain kind of subset of the Cartesian product A × A. This product is the set {(x,y): x ∈ A & y ∈ A} of ordered pairs of elements of A. An ordered pair (a,b) is the set {{a}, {a,b}}. This set has just two elements, namely {{a} and {a,b}}. These elements are themselves sets, {a} having the unique element a, and {a,b} having just two elements, a and b (which may however be the same). id or idA is the set {(x,x): x ∈ A}. −1 is the set {(x,x−1): x ∈ Sym(A)}. σ−1 is the set {(y,x): (x,y) ∈ σ}. ◦ is the set {((x,y),x ◦ y): (x,y) ∈ Sym(A) × Sym(A)}. σ ◦ τ is the set {(x,z): ∃y ((x,y) ∈ τ &(y,z) ∈ σ)}. . Introduction The properties in (.) are the group axioms. Suppose (G, e, ∗, ·) is a structure satisfying these axioms: that is, for all x, y, and z in G, x · e= x, x · x∗ =e, x · (y · z)=(x · y) · z. Then (G, e, ∗, ·) is called a group. There is a set A such that (G, e, ∗, ◦) embeds in (Sym(A), id, −1, ◦); that is, the former structure can be con- sidered as a substructure of the latter. Indeed, A can be chosen as G itself, and the embedding of G in Sym(G) is the function that takes an element g of G to the symmetry {(x,g · x): x ∈ G} of G. This result is known as Cayley’s Theorem. It shows that all groups, as defined by the axioms, have the structural properties of groups of permutations. Thus the theorem provides an example of complete success with the axiomatic method. Axiomatic set theory does not have the same success. Here there is no result corresponding to Cayley’s Theorem. We can prove that there is no such result, by proving the Incompleteness Theorem (which we shall do in §.). This theorem can be taken to illustrate reason to study