Back Matter

Abstract

In this book we have defined forcing notions concretely as families of sets ordered by inclusion. The standard definition treats them abstractly as preordered sets. Our version simplifies the exposition a little because the order relation is built in and does not have to be added separately. It may appear to be less general than the preorder definition, but the two are actually equivalent. The purpose of this appendix is to explain this equivalence.Recall that a preordered set is a set equipped with a relation that is reflexive and transitive, but not necessarily antisymmetric -we can have p ≤ q and q ≤ p for distinct p and q. (Beware: in the forcing literature the term "partial order" sometimes means "preorder".) In the abstract approach, it is convenient to use preorders when we get to iterated forcing. The notions of extension, compatible elements, dense set, ideal, and generic ideal all generalize straightforwardly to preordered sets, just as in Chapter 23.We should also mention that according to the usual convention an extension of p lies below p, not above it, and one is correspondingly interested in the dual notion of filters rather than ideals. We have reversed this convention in order to avoid habitually ordering sets by reverse inclusion. Having forcing notions grow downward carries no particular benefit, although it does make sense in the context of Boolean-valued forcing.Preorders do not create any genuinely greater generality over partial orders. We can always factor out the equivalence relation which sets p ∼ q if p ≤ q and q ≤ p, and it is more or less obvious that forcing with the resulting poset is equivalent in every significant sense to forcing with the original preordered set. It should also be clear that replacing one poset with another one to which it is order-isomorphic does not meaningfully affect the