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The Continuum Hypothesis and the Axioms of Set Theory

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The Continuum Hypothesis and the Axioms of Set Theory The Continuum Hypothesis and the Axioms of Set Theory Rachel Minster November 2015 Advisor: Dr. Alan Dow Mathematics Department Honors Program University of North Carolina at Charlotte 1 Introduction This research project will address the independence of the Continuum Hypothe- sis from the axioms of set theory. In order to do this, we will explore fundamental set theory, G¨odel'sconstructible universe, and Cohen's method of forcing. 1.1 Independence and Consistency A statement is independent if neither it nor its negation are provable from the same system. In practice, this is equivalent to both the statement and its negation being consistent. This involves two separate arguments, and if both are satisfied, the result is independence. The axiom system of set theory, the ZFC axioms, is a basis for all proofs in set theory. We take the axioms to be true when proving statements in set theory, so we assume that the axioms together are consistent. With this, we are interested in what the ZFC system can or cannot prove. For instance, the axioms cannot prove that they are consistent. This is due to G¨odel'sSecond Incompleteness Theorem, which states that a sufficiently expressive system cannot be used to prove its own consistency. 1.2 Basic Definitions We take a model of ZFC using 2 and =. This is our set-theoretic universe, V. Definition 1.1. A set is an element of V. Definition 1.2. A class is a collection of all x that satisfy a formula φ(x; y1; y2; :::; yn). Classes can be sets if they occur in V, but some do not. 1 Definition 1.3. A set x is transitive if every element of x is a subset of x. If y 2 z and z 2 x, then y 2 x. Definition 1.4. A well-ordering is a linear order where every nonempty subset has a least element. 1.3 The Axioms The Zermelo-Frankel Axioms with the Axiom of Choice, abbreviated to ZFC Axioms, are the basis for set theory. ZFC fulfills G¨odel'srequirements for a sufficiently rich axiom system. His First Incompleteness Theorem suggests that, in a good axiomatic system, there will be at least one statement true but not provable. Also, axioms should be "obviously" true. [7] An interesting fact about the axioms is that they are an infinite system. The axioms stated below make up the standard list, but it has been proven that ZFC is not finitely axiomatizable, so there is no actual finite collection of axioms equivalent to the ZFC system. All the axioms provide a basis for the existence of various sets. Using these, we can build different sets and know they are sets instead of classes. The standard axioms are listed below, and a more intuitive definition is included after each axiom. Axiom (Axiom of Existence). 9z z = ; The empty set is a set, so the universe is not empty. Axiom (Axiom of Extensionality). 8x 8y [x = y $ 8u(u 2 x $ u 2 y)] Two sets are equal if they contain the same elements. Axiom (Axiom of Pairing). 8x 8y 9z (x 2 z ^ y 2 z) For any two sets, another set exists which contains both. Axiom (Union Axiom). 8x [x 6= ; ! 9z z = fw :(9y 2 x)(w 2 y)g] A set of sets exists and is a set itself. The class fw :(9y 2 t)(w 2 y)g is referred to as the "big union"for any term t, denoted as S t. 2 Axiom (Axiom of Foundation). 8x [x 6= ;! (9y 2 x)(x \ y = ;)] Each set is well-founded. This means that each non-empty set has a minimal element under the 2 relation. Axiom (Replacement Axiom Scheme). For each formula φ(x; u; v; w1; :::; wk), 8w1; :::; 8wk 8x [8u 2 x 9!v φ ! 9z z = fv : 9u 2 x φg] The image of a set under any formula defining a function will also be in a set. Axiom (Comprehension Axiom Scheme). For each formula φ(x; y; w1; :::; wk), 8w1; :::; wk 8x 9z z = fy : y 2 x ^ φ(x; y; w1; :::wk)g All elements in a set satisfying a certain property or formula comprise a set. Axiom (Power Set Axiom). 8x 9z z = fy : y ⊆ xg We denote fy : y ⊆ xg as P(x). We call P(x) the power set of x, and the axiom states that P(x) is a set itself. Axiom (Axiom of Infinity). 9x[x 6= ; ^ 8y y 2 x ! 9z 2 x y 2 z] This allows for the existence of infinite sets. Axiom (Axiom of Choice). 8X [(8x 2 X 8y 2 X(x = y $ x \ y = ;)) ! 9z (8x 2 X 9!y y 2 x \ z)] A set can be created from one or more elements of pairwise disjoint sets. Impli- cations of this axiom include that every set can be well-ordered. This becomes important in the construction of new sets and models. Not every theorem in set theory needs the assumption of the Axiom of Choice, but we will frequently assume it for the existence of different types of sets. 3 1.4 Ordinals and Cardinals Definition 1.5. A set is an ordinal if it is transitive and well-ordered by the 2 relation. We denote the class of all ordinals by ON. There are two types of ordinals. A successor ordinal is an ordinal α of the form β [ fβg = β + 1, which we call the successor of some ordinal β. If α = succ(α), then α is called a limit ordinal. The class of ordinals, ON, is well-ordered by the Axiom of Foundation. Thus there is no infinite descending sequence of ordinals, and there is a least element of any non-empty set of ordinals. Theorem 1.6. Let A be a well-ordereable set. Then A, under its well-ordering, is order-isomorphic to a unique ordinal α. Due to the transitive nature of ordinals, we can define α < β $ α 2 β for any two ordinals α; β. We can also define addition, multiplication, and exponentiation on the ordinals through transfinite recursion. [7] Let α; β 2 ON and δ be a limit ordinal. Addition: α + 0 = α α + 1 = succ(α) α + succ(β) = succ(α + β) α + δ = supfα + β : β < δg Multiplication: α · 0 = 0 α · succ(β) = (α · β) + α α · δ = supfα · β : β < δg Exponentiation: α0 = 1 αsucc(β) = (αβ) · α αδ = supfαβ : β < δg Definition 1.7. The α such that a well-orderable set A is order-isomorphic to α 2 ON is called the cardinality of A, denoted by jAj. jAj is defined for every A under the Axiom of Choice. Mentioning jXj assumes that X can be well-ordered. Informally and traditionally, cardinality refers to the size of a set. Two sets have the same size if there is a bijection between them. Definition 1.8. An ordinal α is a cardinal if jαj = α. 4 The ordinal referring to the set of natural numbers is denoted by !, which we use frequently as it is one of the smallest infinite sets. ! is also a cardinal as j!j = !. Definition 1.9. A set is countable if its cardinality is at most !, and uncountable if otherwise. There are many countable finite sets, but we are more interested in the infinite ones. Countable infinite sets all have cardinality !. Cantor's discovery of uncountable sets has profound implications, as there are thus infinite sets of different sizes. Notation. We denote the least cardinal greater than an ordinal α by α+. We tend to denote the least cardinal greater than ! as !1, which is the first uncount- able cardinal. The second is !2, and so on. Now we can state the Continuum Hypothesis, abbreviated CH, which Georg Cantor proposed. It seems very plausible, but we cannot prove it to be true or false. [6] Both the statement and its negation are consistent with ZFC, which we will show. Theorem 1.10 (Continuum Hypothesis). jP(!)j = !1 Cantor's theorem, which we discuss later, states that jP(!)j ≥ !1. The hypothesis goes a step further, but this extra step cannot be verified in set theory. 1.5 History Every branch of mathematics has a foundation of axiomatic statements assumed to be true. These express properties and rules for that branch. All mathematical concepts can be expressed in terms of sets and the membership relation. The axiom system of set theory, ZFC, can then be used to derive all mathematics, which makes set theory the foundation of mathematics. The only problems come from infinite sets, as they behave differently than finite sets. This leads to questions about the behavior of infinite sets. Once the behavior of certain finite sets is determined, the succeeding question asks if the same is true for a larger set. This continues into the infinite. If something is true for countable sets, is it then true for uncountable sets? Set theorists strive to answer these questions. Major questions in set theory spark philosophical debates among mathemati- cians. There are multiple theorems and statements that, if true or false, would have huge implications. Unfortunately, they are still undecidable, so we cannot know whether they are true or false. To see the implications, set-theorists take a model of ZFC with whatever statement or combination of statements they wish to use. For example, ZFC +:CH can be used to see what would happen if the Continuum Hypothesis was not true. Another important statement, Martin's Axiom, is used the same way.
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