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Axiomatic Set Theory Appendix A Axiomatic Set Theory Nothing will come of nothing. Shakespeare, King Lear Standard sources include [165] (with its prequel, [164]), [133], and [216]. In the first-order logic quantification is allowed only over the elements of the domain of discourse, and in the second-order logic quantification is allowed over subsets, relations, and functions on the domain of discourse. The Zermelo–Fraenkel Set theory with the Axiom of Choice (ZFC) is a theory of first-order logic in language whose only non-logical symbol is binary relation symbol, ∈. Its models are structures (M, E), where E is the interpretation of ∈. It is a distinguished partial ordering on M. A model of ZFC satisfies every theorem of ZFC (and therefore all of mathematics, as we presently know it). Since ZFC is a first-order theory, it is consistent if and only if each of its finite subsets is consistent.1 A.1 The Axioms of ZFC The first axiomatization of set theory was introduced by Zermelo in order to formalize his proof that the Axiom of Choice is equivalent to the assertion that every set can be well-ordered. The Axiom of Replacement has been added by Frænkel (see [148]). 1After ZFC has been developed in a rudimentary language, it is used to properly define syntax, semantics, and all of model theory (Section D.1). Only then one can consider ZFC as a formal first-order theory. King Lear may not have been referring to this issue, but his statement stands nonetheless. © Springer Nature Switzerland AG 2019 455 I. Farah, Combinatorial Set Theory of C*-algebras, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-27093-3 456 A Axiomatic Set Theory We will use the logical connectives ¬ (negation), ∧ (and), ∨ (or), → (implies), ↔ (if and only if), and quantifiers ∀ (for all) and ∃ (there exists). The axioms are more conveniently stated using the following common abbreviations: (∃!x)ϕ(x) stands for “there exists a unique x that satisfies ϕ(x)” (in symbols, (∃x)(ϕ(x) ∧ (∀y)(ϕ(y) → x = y)), (∀x ∈ y)ϕ stands for “every x in y satisfies ϕ”, i.e., (∀x)(x ∈ y → ϕ), and (∃x ∈ y)ϕ stands for “some x ∈ y satisfies ϕ”, i.e., (∃x)(x ∈ y ∧ ϕ). Another common abbreviation is x ⊆ y,for(∀z)(z ∈ x → z ∈ y). The universal closure of a formula ϕ is the sentence (∀x0)(∀x1)...(∀xn−1)ϕ, where x0,...,xn−1 is the list of all variables freely occurring in ϕ. ZFC consists of universal closures of the following axioms and axiom schemes.2 1. (Extensionality) (∀z)(z ∈ x ↔ z ∈ y) → x = y. 2. (Foundation, or Regularity) (∃y)(y ∈ x) → (∃y)(y ∈ x ∧ (∀z)(z∈ / x ∨ z/∈ y). 3. (Comprehension Scheme) For every formula ϕ of the language of ZFC in which y does not occur freely (∃y)(∀x)(x ∈ y ↔ (x ∈ z ∧ ϕ(x))). 4. (Pairing) (∃z)(x ∈ z ∧ y ∈ z). 5. (Union) (∃y)(∀z)(∀t)((t ∈ z ∧ z) ∈ x → t ∈ y). 6. (Replacement Scheme) For every formula ϕ of the language of ZFC in which t does not occur freely (∀y ∈ x)(∃!z)ϕ(y, z) → (∃t)(∀y ∈ x)(∃z ∈ t)ϕ(y,z). 7. (Infinity) (∃x)(∅∈x ∧ (∀y ∈ x)(y ∪{y}∈x)). 8. (Power Set) (∃y)(∀z)(z ⊆ x → z ∈ y). 9. (The Axiom of Choice, AC) For every set x all of whose elements are nonempty sets there exists a function f with domain x such that f(y)∈ y for all y ∈ x. Translations of selected axioms of ZFC into the English language follow. “Formula” stands for “first-order formula in the language of ZFC”. 10. Comprehension Scheme: Every subset of a set that is definable by a formula is also a set. 11. Union: If x is a set, then there exists a set that includes x, the union of all elements of x.3 12. Replacement Scheme: If x is a set and a formula ϕ defines a function f with domain x via f(y) = z if and only if ϕ(y,z), then the image ϕ[x]:={f(y): y ∈ x} exists. 13. Infinity: There exists a set whose elements are {0, 1, 2,...,n,n+ 1,...}. 14. Power Set: It literally states that for every set x there exists a set y that includes the power set P(x) of x.4 15. AC: For every set x all of whose elements are nonempty sets there exists a function f with domain x such that f(y)∈ y for all y ∈ x. 2Each of (3)and(6) is not a single axiom, but a scheme that associates an axiom to every formula of the language of set theory. It is known that ZFC is not finitely axiomatizable. 3By applying the Comprehension Scheme one proves that x exists. 4By applying the Comprehension Scheme one proves that P(x) itself exists. A.2 Well-Foundedness, Transfinite Induction, and Transfinite Recursion 457 By ZF we denote the axioms of ZFC without the Axiom of Choice. Since by Gödel’s theorem one cannot prove the consistency of ZFC within ZFC, metamathematical considerations usually start from a “sufficiently large finite fragment of ZFC”. This fragment depends on the context and it is denoted ZFC*. A proof of the following can be found in every introductory book on set theory and most sufficiently old introductory functional analysis texts. Theorem A.1.1 The following statements are provably equivalent in ZF.5 1. The Axiom of Choice. 2. Zorn’s Lemma: If a partially ordered set P contains an upper bound for every linearly ordered subset, then it has at least one maximal element. 3. Hausdorff’s maximality principle: If P is a partially ordered set, then every linearly ordered subset of P is contained in a maximal (under the inclusion) linearly ordered subset. 4. Tychonoff’s theorem: the product of compact topological spaces is compact. 5. The Cartesian product of any family of nonempty sets is nonempty. With the axioms of ZFC in place, one proceeds to define ordered pairs, Cartesian products, functions, and all other mathematical objects, as sets. Using the axioms of ZFC, one can then prove the existence of these objects. The details can be found in any basic text on axiomatic set theory. A.2 Well-Foundedness, Transfinite Induction, and Transfinite Recursion Let i be an element of j and let j be an element of i.6 Anonymous (as shared by Alan Dow) Suppose E is an ordering (i.e., a transitive, antisymmetric, and irreflexive binary relation) and Y is a subset of its domain. Some a ∈ Y is a minimal element of Y if there is no b ∈ Y such that b E a.AnorderingonX is well-founded if every nonempty subset of X has a minimal element. An ordering which is both linear and well-founded is a well-ordering. The Axiom of Choice implies that E is well-founded if and only if there is no infinite decreasing sequence of elements of its domain. The Axiom of Foundation implies that ∈ is well-founded. The utility of well-foundedness is most obvious from the principles of transfinite induction and transfinite recursion. We state only a weak variant of each one of them (see Remark A.3.1). 5Notably, the justification of this theorem was Zermelo’s motivation for introducing axiomatic set theory; see [185]. 6This is a set theorist’s way of saying “Let ε<0”, only worse. 458 A Axiomatic Set Theory Proposition A.2.1 (Transfinite Induction) If E is a well-founded relation on X, ϕ(x) is a formula, and (∀x ∈ X)(∀y E x)ϕ(y) → ϕ(x), then (∀x ∈ X)ϕ(x). Every object in a model of ZFC is a set and any formula of the language of ZFC can be evaluated at any set. An example of the effect of this flexibility on the readability of formulas is (inadvertently) provided in the Definition by Transfinite Recursion stated below. The formula ϕ occurring in it has two parameters. One of them is “the function defined up to x” and the other is “the value of the function at x”. By f X we denote the restriction of a function f to a subset X of its domain. Proposition A.2.2 (Definition by Transfinite Recursion) Suppose E is a well- founded relation on a set X, ϕ(x,y) is a formula, and for every x there exists a unique y such that ϕ(x,y) holds. Then there exists a unique function f : X → Z for some set Z, such that ϕ(f {y ∈ X : y E x}, f (x)) holds for all x ∈ X. Some applications of this theorem are the Mostowski Collapsing Theorem (Theorem A.6.2) and the definition of the rank of a well-founded tree (Propo- sition B.2.8). Both Propositions A.2.1 and A.2.2 are theorems of ZF. All of the recursive constructions presented in this book rely on the following variant of the latter. Its proof requires the Axiom of Choice. Proposition A.2.3 Suppose E is a well-founded relation on a set X, ϕ(x,y) is a formula, and for every x there exists y such that ϕ(x,y) holds. Then there exists a function f : X → Z, for some set Z, such that ϕ(f {y ∈ X : y E x}, f (x)) holds for all x ∈ X. If the set {y : (∃x)ϕ(x,y)} as in Proposition A.2.3 is equipped with a well- ordering ≤, then ϕ(x, y) := ϕ(x,y) ∧ ((∀z)ϕ(x, z) → y ≤ z) satisfies the assumptions of Proposition A.2.2, and the particular instance of this proposition does not require the Axiom of Choice.
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