Forcing for Mathematicians

The following sections are included:

• Preface
• Contents

Forcing is a powerful technique for proving consistency and independence results in relation to axiomatic set theory. A statement is consistent with a given family of axioms if it cannot be disproven on the basis of those axioms, and independent of them if it can be neither proven nor disproven. When we have established that some assertion is consistent, there is still hope that it might actually be provable from the axioms, but once we have shown it is independent the matter is closed…

The standard formal system for reasoning about sets is ZFC, Zermelo-Fraenkel set theory with the axiom of choice. This is a pure set theory, meaning that there are no objects besides sets. Every element of a set is another set…

In this and the next two chapters we will develop the basic theory of well-ordered sets, ordinals, and cardinals. Some of this material may already be familiar to the reader. We work in ZFC, but again, we are not concerned with the details of formalization. We simply ask the reader to accept that ordinary set-theoretic reasoning can be formalized in ZFC…

An ordinal is a kind of canonical well-ordered set. Intuitively, any well-ordered set W can be converted into an ordinal by successively replacing each element x ∈ W with the set x^<. Thus the first element of W is replaced with ∅, then the second element is replaced with {∅}, then the third is replaced with {∅, {∅}}, and so on…

The finite ordinals all have distinct sizes: no two of them can be put in bijection with each other. This is no longer true of infinite ordinals; using the fact that a countable union of countable sets is countable, it is not hard to see that there are ordinals well beyond ω which can be put in bijection with ω. For instance, if we replace each element of ω with a distinct copy of ω, the result will be a countable well-ordered set whose corresponding ordinal, denoted ω², is greater than infinitely many limit ordinals. More generally, if each element of ω is replaced with any countable ordinal the result will still be countable and well-ordered. So we can concatenate ω copies of ω² to reach a countable ordinal ω³, then concatenate ω copies of ω³ to reach a countable ordinal ω⁴, and so on. We can concatenate the sequence ω,ω²,ω³,… to reach a countable ordinal denoted ω^ω. And so on…

In order to prove consistency results we propose to work with sets which play the role of miniature universes, in which the Zermelo-Fraenkel axioms hold alongside some additional statement such as the continuum hypothesis or its negation. We need to explain exactly what this means…

Recall that a sentence is a formula with no unquantified variables. In this chapter we will show that we can arrange for any true sentence to be true in some countable set M. It turns out that all we need to do is to include elements in M which verify certain existence statements. The following lemma scheme gives this construction…

From this point on, in chapters which develop the theory of forcing we will work in ZFC⁺, and in chapters focused on applications we will work in ZFC. The next several chapters take place in ZFC⁺…

We continue to work in ZFC⁺. Throughout this chapter, we fix a forcing notion P and a generic ideal G of P…

We fix a forcing notion P but do not fix a generic ideal. Now for any P-names τ₁,…, τ_n and any formula ∅ in the language of set theory with unquantified variables among x₁,…, x_n, it makes sense to ask, in ZFC⁺, for which generic ideals G the assertion ∅(τ^G₁,…, τ^G_n) holds in M[G]. Is it ever possible to infer the truth-value of ∅|M[G](τ^G₁,…, τ^G_n) directly from some structural property of G? Conceivably, there could be cases where the mere presence of a single element p ∈ P in G could be enough to ensure that ∅(τ^G₁,…, τ^G_n) holds in M[G]. In other words, it could happen that M[G] ⊨ ∅(τ^G₁,…, τ^G_n) for any generic ideal G that contains p.…

Fix a forcing notion P. We will now generalize Theorem 10.3 to arbitrary formulas. The remaining cases are easier because, working in M, we can define the forcing relation for the formula x₁ ∈ x₂ in terms of the forcing relation for x₁ = x₂, and the forcing relation for complex formulas can be constructed from the forcing relations for their constituent formulas. So we no longer need the induction on α employed in the definition of ℱ_α when the forcing relation for x₁ = x₂ was being defined in terms of itself…

Working in ZFC⁺, we can now prove that every forcing notion has a generic ideal G (Theorem 8.3) and that each axiom of ZFC holds in the generic extension M[G] associated to G (Metatheorem 11.7). According to Metatheorem 7.4, all we have left to do to establish in PA that some statement ∅ is relatively consistent with ZFC − that is, that the consistency of ZFC implies the consistency of ZFC + ∅ − is to produce a forcing notion for which we can prove in ZFC⁺ that ∅ holds in M[G], for some generic ideal G. As our first example, we will prove in this chapter that the continuum hypothesis is relatively consistent with ZFC. This result was first proven by Gödel using very different methods, but given the machinery of forcing that is now at our disposal we can make a much quicker argument…

In this chapter we will prove the relative consistency of the negation of the continuum hypothesis. This is the theorem that the technique of forcing was originally developed by Cohen to prove. Together with the result of the last chapter, it shows that CH is independent of ZFC (assuming ZFC is consistent)…

In chapters marked with an asterisk we will discuss, in ZFC, applications of set theory to problems that arise in mainstream mathematics. These chapters have some interdependence, but no unstarred chapter will depend on material from any starred chapter. Since we have just shown that the continuum hypothesis is independent of ZFC (assuming ZFC is consistent), the next few chapters will explore some consequences of CH…

The Stone-Čech compactification βℕ of ℕ is the maximal compact Hausdorff space into which ℕ densely embeds. One way to construct it is to amalgamate all functions from ℕ to [0, 1] into a single map from ℕ into a power of [0, 1], and then take the closure of the image of ℕ in this cube. The result is a compact Hausdorff space βℕ which contains a copy of ℕ and such that any function from ℕ into any compact Hausdorff space has a unique continuous extension to βℕ…

In typical applications of the continuum hypothesis we construct something uncountable (an uncountable family of entire functions, an automorphism of /fin) by means of a transfinite process which at each stage involves only countable data (a countable family of entire functions, an isomorphism between countable Boolean subalgebras of /fin). In this chapter we will apply this technique to a question about pure states on , the space of bounded linear operators from a separable infinite-dimensional complex Hilbert space H to itself…

Build a tree by starting with one base vertex at level 0, putting ℵ₀ vertices above it at level 1, putting ℵ₀ vertices above each of those at level 2, and so on, through all finite levels. This is called the complete ℵ₀−ℵ₀ tree. A vertex at level n can be reached by making a series of choices — which successor of the base vertex to select at level 1, which successor of the level 1 vertex just chosen to select at level 2, etc. — and so it is uniquely identified by a function from n = {0,…, n − 1} into ℕ. One vertex lies above another if the function corresponding to the former is an extension of the function corresponding to the latter. The infinite paths all the way up the tree correspond to functions from ℕ to itself…

The real line is dense (between any two points there is a third), unbounded (there is no least or greatest element), complete (every bounded set has a least upper bound and a greatest lower bound), and separable. Conversely, it is not too hard to show that any totally ordered set with these properties must be order-isomorphic to ℝ. Suslin's problem asks whether “separable” can be weakened to topologically c.c.c., the condition that there is no uncountable collection of disjoint open intervals. A totally ordered set that is dense, unbounded, complete, and topologically c.c.c., but not separable, is called a Suslin line…

In Chapter 16 we defined C^*-algebras to be subsets of with certain stability properties. It can also be fruitful to consider a C^*-algebra abstractly, as an object which has the structure of a Banach algebra equipped with an involution and which can be realized, up to a bijection preserving all relevant structure, as a closed subspace of some that is stable under products and adjoints. (We now allow H to be nonseparable.) From this point of view, the same C^*-algebra can be represented on different Hilbert spaces, in the following sense.

The diamond principle can be strengthened in various ways. We will prove the relative consistency of a version in which paths are blocked using vertices in a stationary set of levels. Recall that a subset of ℵ₁ is stationary if it intersects every club subset of ℵ₁.

Let be an abelian group. An extension of by ℤ is a surjective homomorphism π from an abelian group into whose kernel is isomorphic to ℤ, i.e., it is a short exact sequence

Working in ZFC⁺, we have developed a procedure for enlarging a countable transitive model M of ZFC to another countable transitive model M[G] using a forcing notion for M. But now the same procedure can be applied to M[G] using a forcing notion for M[G]. We already saw an example of this in Chapter 20, although in that case both forcing notions belonged to M. Iterating forcing constructions can be a useful technique, but as we will show, it does not actually create any greater generality…

Definitions 8.1 and 8.2 can be generalized to arbitrary partially ordered sets. Let be a poset. An extension of P is any q ≥ p; p and q are compatible if they have a common extension; a subset D of is dense if every has an extension in D; and an ideal of is a subset satisfying (1) if q ∈ G and q ≥ p then p ∈ G and (2) if p₁, p₂ ∈ G then there exists q ∈ G with q ≥ p₁, p₂. Also, we say that is c.c.c. if every set of pairwise incompatible elements is countable…

Since CH implies MA, one type of application of MA is to weaken uses of CH. That is, we might first prove a result assuming CH and then discover that it can actually be proven from MA. For instance, MA suffices to prove the existence of non-diagonalizable pure states on (Theorem 16.4). When this happens it shows that the result in question is relatively consistent with ¬CH. But a more interesting kind of use of MA occurs when CH or ⋄ settles a question in one direction and MA + ¬CH settles it in the other. We will focus on examples of this phenomenon…

We will now show that MA + “2^ℵ₀ = ℵ₂” implies there is a counterexample to Whitehead's problem, i.e., an abelian group with only trivial extensions by ℤ that is not free. This result, which complements Theorem 21.5, is also due to Shelah. Together they show that the answer to Whitehead's problem for groups of cardinality #x2135;₁ is independent of ZFC (assuming ZFC is consistent)…

Let χ be a separable metric space. An open coloring of χ is a symmetric open subset K of χ²\Δ_χ where Δ_χ = {〈x, x〉 | x ∈ χ}. A subset A of χ is 0-homogeneous if 〈x, y〉 ∈ K for all distinct x, y ∈ A and 1-homogeneous if <x, y> ∉ K for all distinct x, y ∈ A. The open coloring axiom (OCA) says that if χ is any separable metric space and K is any open coloring of χ, then either χ has an uncountable 0-homogeneous set or χ can be covered by countably many 1-homogeneous sets…

OCA + MA implies that every self-homeomorphism of βℕ \ ℕ arises from an almost permutation of ℕ. This result, which complements Theorem 15.3, is due to Veličković Together they show that the triviality of all self-homeomorphisms of βℕ \ ℕ is independent of ZFC (assuming ZFC is consistent)…

Let H be a separable infinite-dimensional complex Hilbert space. For the sake of concreteness we can take H = l², the space of square-summable complex sequences. Recall from Chapter 16 that is the space of bounded linear operators from l² to itself and is the space of compact operators, operators approximated in norm by finite rank operators. is a closed two-sided ideal of , which implies that the quotient space is also a C^*-algebra. This space is called the Calkin algebra. We write for the quotient map…

We showed in the last chapter that CH implies the Calkin algebra has outer automorphisms. Conversely, Farah proved that OCA implies all automorphisms of the Calkin algebra are inner. Thus, the existence of outer automorphisms is independent of ZFC (assuming ZFC is consistent)…

What are we to make of the independence phenomena evinced by forcing?

The two kinds of independence mentioned in Chapter 1 in geometry and number theory offer us strikingly different paradigms. In both cases there is broad agreement about the correct interpretation of the independence results. For instance, no one nowadays would consider it meaningful to ask whether the parallel postulate is “really true” in some universal sense; it simply holds in some two-dimensional geometries and fails in others…

The following sections are included:

• Appendix A: Forcing with Preorders
• Exercises
• Notes
• Bibliography
• Notation Index
• Subject Index