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A list of recommended books for studying set theory, along with some problems and definitions from the math 771 course at the university of wisconsin-madison. Topics covered include extensionality, pairing, union, and comprehension scheme, as well as classes, finite and infinite sets, and wellorderings.

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Download Set Theory: A List of Recommended Books and Problems for Math 771 - Prof. Joseph S. Miller and more Study notes Mathematics in PDF only on Docsity! A.Miller M771 Jan 1998 http://www.math.wisc.edu/∼miller/m771 A selected list of books about set theory Beginning Roitman, Introduction to modern set theory Vaught, Set theory, an introduction Monk, Introduction to set theory Moschovakis, Notes on set theory Just and Weese, Discovering modern set theory I, II Hausdorff, Set theory Intermediate Kunen, Set theory Jech, Set theory Handbook of mathematical logic, chapter on set theory Ciesielski, Set theory for the working mathematician Advanced Bartoszynski and Judah, Set theory, on the structure of the real line Kanamori, The higher infinite Shelah, Proper forcing Shelah, Cardinal arithmetic Devlin, Constructibility Jech, The axiom of choice Fremlin, Consequences of Martin’s axiom Prerequisites The prerequisite for this course is Math 770, Introduction to Mathematical Logic. Usually the following topics are among those covered in 770: Naive set Theory: cardinal and ordinal arithmetic, axioms of Zermelo-Fraenkel set theory with the axiom of choice. First order logic: formulas, sentences, theories, structures, models, satisfaction, the compactness, completeness and incompleteness theorems. What follows is excerpted from a manuscript which can be found on my home page. 1 The Axioms of Set Theory Here are some. The whole system is known as ZF for Zermelo-Fraenkel set theory. When the axiom of choice is included it is denoted ZFC. It was originally developed by Zermelo to make precise what he meant when he said that the well-ordering principal follows from the axiom of choice. Latter Fraenkel added the axiom of replacement. Another interesting system is GBN which is Gödel-Bernays-von Neumann set theory. Empty Set: ∃x ∀y(y /∈ x) The empty set is usually written ∅. Extensionality: ∀x∀y(x = y ⇐⇒ ∀z(z ∈ x ⇐⇒ z ∈ y)) Hence there is only one empty set. Pairing: ∀x∀y∃z∀u(u ∈ z ⇐⇒ u = x∨u = y) We usually write z = {x, y}. Union: ∀x ∃y (∀z(z ∈ y ⇐⇒ (∃uu ∈ x∧ z ∈ u)) We usually write y = ∪x. A ∪ B abbreviates ∪{A,B}. z ⊆ x is an abbreviation for ∀u (u ∈ z → u ∈ x). Power Set: ∀x ∃y ∀z(z ∈ y ⇐⇒ z ⊆ x) We usually write y = P (x). For any set x , x+ 1 = x ∪ {x}. Infinity: ∃y (∅ ∈ y ∧∀x(x ∈ y → x+ 1 ∈ y)) The smallest such y is denoted ω, so ω = {0, 1, 2, . . .}. Comprehension Scheme: ∀z∃y∀x[x ∈ y ⇐⇒ (x ∈ z ∧ θ(x))] The comprehension axiom is being invoked when we say given z let y = {x ∈ z : θ(x)}. The formula θ may refer to z and to other sets, but not to y. In general given a formula θ(x) the family {x : θ(x)} is referred to as a class, it may not be a set. For example, the class of all sets is V = {x : x = x}. 2 1.30 For any set X let [X]ω be the countably infinite subsets of X. Show that |Rω| = |[R]ω| = c. 3 1.31 Show that the cardinality of the set of open subsets of R is c. 1.32 Show that the set of all continuous functions from R to R has size c. 1.33 Show that ωω has cardinality c. Wellorderings A linear order (L,≤) is a wellorder iff for every nonempty X ⊆ L there exists x ∈ X such that for every y ∈ X x ≤ y (x is the minimal element of X). For an ordering ≤ we use < to refer to the strict ordering, i.e x < y iff x ≤ y and not x 6= y. We use > to refer to the converse order, i.e. x > y iff y < x. 2.1 Prove: Let (L,≤) be any well-ordering and f : L → L an increasing function (∀x, y ∈ L (x < y → f(x) < f(y))). Then for every x in L x ≤ f(x). 2.2 For two binary relations R on A and S on B we write (A,R) ' (B, S) iff there exists a one-to-one onto map f : A→ B such that for every x, y in A (xRy iff f(x)Sf(y)). Such a map is called an isomorphism. If (L1,≤1) and (L2,≤2) are well-orders and (L1,≤1) ' (L2,≤2) then show the isomorphism is unique. 2.3 Let (L,≤) be a wellorder and for any a ∈ L let La = {c ∈ L : c < a}. Show that (L,≤) is not isomorphic to (La,≤) for any a ∈ L. 2.4 (Cantor) Show that for any two wellorders exactly one of the following occurs: they are isomorphic, the first is isomorphic to an initial segment of the second, or the second is isomorphic to an initial segment of the first. Hint: Let (L,≤) and (K,≤′) be two wellorders. Let G = {〈a, b〉 : (La,≤) ' (Kb,≤′)} Show that G is the graph of an order preserving bijection whose domain is all of L or whose range is all of K. Axiom of Choice (AC) Axiom of Choice: For every family F of nonempty sets there exists a choice function, i.e. a function f with domain F such that for every x in F , f(x) ∈ x. (WO) Well-ordering Principle : Every nonempty set can be well ordered. 3See previous footnote. 5 (TL) Tuckey’s Lemma: Every family of sets with finite character has a maximal ele- ment. A family of sets F has finite character iff for every set X, X ∈ F iff for every finite Y ⊆ X, Y ∈ F . (MP) Maximality Principle: Every family of sets closed under the unions of chains has a maximal member. (ZL) Zorn’s Lemma: Every family of sets contains a maximal chain. 3.1 Show that ZL implies MP. 3.2 Show that MP implies TL. 3.3 Show that TL implies AC. 3.4 (Zermelo) Show that AC implies WO. Hint: Given X use AC to get a function f : P (X) 7→ X such that f(A) ∈ X \ A for all A a proper subset of X. Call a well ordering (L,≤) respectful iff L ⊆ X and for every a ∈ L f(La) = a. Show that the union of all respectful well orderings is a well ordering of X. 3.5 Given a nonempty family F let < be a strict well-ordering of F . Say that a chain C ⊆ F is greedy iff for every a ∈ F if {b ∈ C : b < a} ∪ {a} is a chain, then either a ∈ C or b < a for every b ∈ C. Show that the union of all greedy chains is a maximal chain. Conclude that WO implies ZL. Ordinals A set X is transitive iff ∀x ∈ X (x ⊆ X). A set α is an ordinal iff it is transitive and strictly well ordered by the membership relation (define x ≤ y iff x ∈ y or x = y, then (α,≤) is a wellordering). We also include the empty set as an ordinal. For ordinals α and β we write α < β for α ∈ β. The first infinite ordinal is written ω. We usually write 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, . . . , n = {0, 1, . . . , n− 1}, · · · , ω = {0, 1, 2, . . .} 4.1 Show: If α is an ordinal then so is α+ 1. (Remember α = α ∪ {α}.) Such ordinals are called successor ordinals. Ordinals that are not successors are called limit ordinals. 4.2 Show: If α is an ordinal and β < α, then β is an ordinal and β ⊆ α and β = {γ ∈ α : γ ∈ β}. Axiom of Regularity: ∀x x 6= ∅ → ∃y(y ∈ x∧¬∃z (z ∈ y ∧ z ∈ x)) 6 Another way to say this is that the binary relation R = {(u, v) ∈ x × x : u ∈ v} has a minimal element, i.e., there exist z such that for every y ∈ x it is not the case that zRy. Note: a minimal element is not the same as a least element. 4.3 Show α is an ordinal iff α is transitive and linearly ordered by the membership relation. 4.4 For any ordinals α and β show that α ∩ β = α or α ∩ β = β. Show any two ordinals are comparable, i.e., for any two distinct ordinals α and β either α ∈ β or β ∈ α. 4.5 The union of set A of ordinals is an ordinal, and is sup(A). 4.6 Show that the intersection of a nonempty set A of ordinals is the least element of A, written inf(A). Hence any nonempty set of ordinals has a least element. 4.7 Prove transfinite induction: Suppose φ(0) and ∀α ∈ ORD if ∀β < α φ(β), then φ(α). Then ∀α ∈ ORD φ(α). Replacement Scheme Axioms: ∀a ([∀x ∈ a∃!y ψ(x, y)] → ∃b∀x ∈ a∃y ∈ b ψ(x, y)) The formula ψ may refer to a and to other sets but not to b. Replacement says that for any function that is a class the image of a set is a set. If F is a function, then for any set a there exists a set b such that for every x ∈ a there exists a y ∈ b such that F(x) = y. 4.8 (von Neumann) Let (L,≤) be any well-ordering. Show that the following is a set: {(x, α) : x ∈ L, α ∈ ORD, and (Lx,≤) ' α}. Show that every well ordered set is isomorphic to a unique ordinal. Let ORD denote the class of all ordinals. Transfinite Recursion: If F is any function defined on all sets then there exists a unique function G with domain ORD such that for every α in ORD G(α) = F(G α). This is also referred to as a transfinite construction of G. 4.9 Suppose F : V → V, i.e., a class function. Define g (an ordinary set function) to be a good guess iff dom(g) = α ∈ ORD, g(0) = F(∅), and g(β) = F(g β) for every β < α. Show that if g is a good guess, then g β is good guess for any β < α. 4.10 Show that if g and g′ are good guesses with the same domain, then g = g′. 4.11 Show that for every α ∈ ORD there exists a (necessarily unique) good guess g with domain α. 4.12 (Fraenkel) Prove transfinite recursion. 4.13 Explain the proof of WO implies ZL in terms of a transfinite construction. 7