FORCING ROWAN JACOBS Abstract. ZFC, the axiom system of set theory, is not a complete theory. That is, there are statements such that ZFC + ' and ZFC + :' are both consistent. The Continuum Hypothesis| the statement that the size of the real numbers is the first uncountable cardinal @1|is a famous example. This paper will investigate why ZFC is incomplete and what structures satisfy its axioms. The proof technique of forcing is a powerful way to construct structures satisfying ZFC + ' for a wide range of '. This technique will be developed and its applications to the Continuum Hypothesis and the Axiom of Choice will be demonstrated. 1. Ideas Behind Forcing 1.1. Models of ZFC. During the late 19th and early 20th century, set theory played the role of a solid foundation underneath other fields, such as algebra, analysis, and topology. However, without an axiomatization, set theory opened itself up to paradoxes, such as Russell's Paradox. If every predicate defines a set, then we can define a set S of all sets which do not contain themselves. Does S contain itself? If so, then it must not contain itself, since it is an element of S, and all elements of S are sets that do not contain themselves. If not, then it must contain itself, since it is a set that does not contain itself, and all sets that do not contain themselves are in S. With this and similar paradoxes, it became clear that the naive form of set theory, in which any predicate over the entire universe of sets can define a set, could not stand as a foundation for other fields of mathematics. Thus several axiomatizations of set theory were introduced in order to formalize the field of set theory and eliminate its contradictions and ambiguities. The only one to have any traction outside the world of logicians was ZFC, or Zermelo-Fraenkel with the Axiom of Choice. The universe of ZFC is often implicitly taken to be the background of many areas of mathematics, for it allows us to define and rigorously work with the structures of algebra, analysis, and topology, among others. Like any other set of axioms, e.g. the axioms satisfied by a group, a field, or a topological space, the axioms of ZFC describe a particular type of structure|in this case, a universe of set theory. There is of course more than one isomorphism class of groups, fields, or rings. If we wish to work inside `the universe of ZFC,' then the question arises, is the same true of universes of set theory, or do the axioms of ZFC describe a unique (or essentially unique) object? In order to answer this question, we will restrict ourselves to first-order logic, that is, logic in which predicates and functions cannot be quantified over or taken as arguments for other predicates or functions. It is worth noting that ZFC is a first-order axiomatization of set theory, and does not contain any higher-order axioms. We study first-order axioms because models of first-order logic are much more amenable to study than those of higher-order logics. First-order logic gives us several tools such as the theorems of Compactness, Godel's Completeness Theorem, and the Lowenheim-Skolem Theorem, which are not available in higher-order logic, making the study of higher-order models much more difficult. Given any set of first-order axioms over some language (i.e. some set of function, relation, and constant symbols), called a theory, we call a collection of objects with interpretations for each symbol that satisfy all of the axioms a model of our theory. (If M is a model of T , we write M j= T .) Thus, our question is equivalent to the question of whether there is more than one model of ZFC, up to isomorphism. 1 2 ROWAN JACOBS The answer turns out to be that there is. It is a theorem of model theory, known as the Lowenheim-Skolem Theorem, that any theory with infinite models has infinite models of any car- dinality above the cardinality of the language. The language of set theory consists only of the relations of = (equality) and 2 (containment), so every infinite cardinality is above the cardinality of our language. So there must exist models of ZFC that are very, very small. In particular, there exists some countable model of ZFC. How can this happen, when it is a theorem of ZFC that there exists an uncountable set, and all its elements must also belong to the model? If we view our countable model M as a subset of the canonical model of ZFC, and we let X be a set such that M j= X is uncountable, then there is a bijection f : ! ! X in V , but no such bijection is an element of M. Thus, it is consistent both that the universe of M is a countable set, and that it models the statement \there exists an uncountable set." Not only does ZFC have models of different cardinalities, but it also has models which satisfy dif- ferent statements. This follows from Godel's First Incompleteness Theorem, which states that any axiom system sufficient to express elementary arithmetic cannot be both consistent and complete. ZFC is one such axiom system, and if it is consistent, then it cannot be a complete theory, meaning that there are statements ' such that ZFC must be consistent with both ' and :'. Thus there is a model of ZFC M j= ', and another model N j= :', so these models are not isomorphic. This paper will show how to prove that ZFC is incomplete, using the famous example of the Continuum Hypothesis (CH): the statement that jRj = @1. The proof technique displayed here, known as forcing, is the most common way to show that a given statement is consistent with ZFC, and it can be used for many statements other than CH. This paper will primarily follow Thomas Jech's presentation of forcing in chapters 14 and 15 of Set Theory: The Third Millenium Edition Revised and Expanded. 2. The Machinery of Forcing 2.1. Canonical Models. The Von Neumann universe, denoted V . V is constructed hierarchically, as follows. V0 is the empty set. If β is an ordinal for which Vβ is defined, then Vβ+1 is the power set of Vβ, i.e. Vβ+1 = P(Vβ). If λ is a limit ordinal, then Vλ is the union of all Vα for α < λ. V is then defined as the union of Vα for all ordinals α. (It is worth noting that in ZFC, defining sets by quantifying over all ordinals is not allowed. So V is not a set itself, but is called a proper class. This prevents Russell's Paradox, among other problems.) In V , all sets can be classified by rank. The rank of a set S is the minimum α such that S 2 Vα. The construction of V is not the only way to hierarchically build a model. There is a very important submodel of V called the constructible universe and denoted L. We define L0 = ; and S for λ a limit ordinal, Lλ = α<λ Lα, as before. However, instead of letting Lβ+1 be the power set of Lβ, we define Lβ+1 as only the subsets of Lβ which are definable by a formula in the language f=; 2g of set theory with parameters and quantifiers restricted to Lβ (that is, we are not allowed to refer to sets of rank higher than β in constructing Lfβ + 1g). Like V , L is a model of ZFC. In fact, the two may be equal, a hypothesis referred to as V = L. If V 6= L, then V and L may model many different statements. For instance, L always models the continuum hypothesis, whereas (as will be shown later) it is consistent that V 2 CH. It is worth noting that L is the minimal proper class model of ZFC which contains all the ordinals of V . Both L and V are proper classes, but models can also be sets. Let M be a set that models ZFC. Then M is called a transitive model if its underlying set M is a transitive set, i.e. if for any element x, x 2 M ) x ⊂ M. The property of transitivity of a model is necessary for substructuress and superstructures of that model to behave properly.1 1More technically, bounded formulas are only guaranteed to be absolute relative to a class of models if those models are transitive. FORCING 3 There are some sets that every transitive model of ZFC will contain. For instance, the Axiom 2 of Infinity ensures that the natural numbers N (alternately called ! when viewed as an ordinal number) will be in every transitive model. In particular, not only does every model M have some interpretation of N, but all of these interpretations are necessarily the same set. By contrast, every transitive model of ZFC must contain some interpretation of R, by taking the power set of N. However, what subsets of N exist will vary from model to model, so the interpretations of R will not necessarily be identical. In general this will be true of all power set constructions—different models may disagree on which subsets of some given set exist, so the power sets of that set may be different. In order to build a model containing new sets, we will first build a poset that `approximates' the set we are trying to construct. Generally this will be a finite or countable (relative to M) approximation of a final set that will be uncountable.