Set Theory

Notes on set theory, mainly forcing. The first four sections closely follow the lecture notes Williams [8] and the book Kunen [4]. The last section covers topics from various sources, as indicated there. Hopefully, all errors are mine. 1. The Suslin problem 1.1. The Suslin hypothesis. Recall that R is the unique dense, complete and separable order without endpoints (up to isomorphism). It follows from the separability that any collection of pairwise disjoint open intervals is countable. Definition 1.1.1. A linear order satisfies the countable chain condition (ccc) if any collection of pairwise disjoint open intervals is countable. Hypothesis 1.1.2 (The Suslin hypothesis). (R, <) is the unique complete dense linear order without endpoints that satisfies the countable chain condition. We will not prove the Suslin hypothesis (this is a theorem). Instead, we will reformulate it in various ways. First, we have the following apparent generalisation of the Suslin hypothesis. Theorem 1.1.3. The following are equivalent: (1) The Suslin hypothesis (2) Any ccc linear order is separable Proof. Let (X, <) be a ccc linear order that is not separable. Assume first that it is dense. Then we may assume it has no end points (by dropping them). The completion again has the ccc, so by the Suslin hypothesis is isomorphic to R. Hence we may assume that X ⊆ R, but then X is separable. It remains to produce a dense counterexample from a general one. Define x ∼ y if both (x, y) and (y, x) are separable. This is clearly a convex equivalence relation, so the quotient Y is again a linear order satisfying the ccc. We claim that each class is separable. Indeed, let Iα be a maximal pairwise disjoint family of intervals in the class. Then it is countable, and each Iα is separable. If x < y in Y, then for any pre-images x̃ and ỹ in X, (x̃, ỹ) is not separable. Let Ik be a maximal collection of pairwise disjoint open sub-intervals (it is countable by the ccc). If the image of each Ik is either x or y, then each is separable, with some dense countable subset Dk. Let D = ∪kDk. Then D is a countable set, and if it avoids some open interval, then this interval can be added to the Ik, contradicting maximality. □ Definition 1.1.4. A Suslin order is a non-separable linear order that has the ccc.