In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4's completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong

more »

... ness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space. 546 PHILIP KREMER S4 for the class of subspaces of the irrational line: any consistent theory is satisfiable in this class, where a theory is a set of formulas containing all the theorems of S4 and closed under both modus ponens and necessitation. These results leave open the strong completeness of S4 for R, for C, and for any arbitrary dense-in-itself metric space. §1. Outline. 1. 1. An interior map strategy. The completeness of S4 for a given dense-in-itself metric space X is typically proved by showing that any finite rooted reflexive transitive Kripke frame is the image of an interior map from X . Section §3, below, strengthens the completeness of S4 for Q to strong completeness by showing that any countable rooted reflexive transitive Kripke frame is the image of an interior map from Q. We pursue this strategy in two steps. Firstly, we observe that any countable rooted reflexive transitive Kripke frame can be unravelled into the infinite binary tree, 2 <ω (Lemma 3.3): thus S4 is strongly complete for 2 <ω (Lemma 3.4). Secondly, we observe that there is an interior map from Q onto 2 <ω : thus S4 is strongly complete for Q. This is an instance of a general strategy. First, show that S4 is strongly complete for some master space Y , in this case 2 <ω . Then transfer the strong completeness of S4 for the master space to the strong completeness of S4 for X , backwards via a surjective interior map, as in Figure 1 . The interior map strategy just outlined shows the strong completeness of S4 for a particular space by showing that every countable rooted reflexive transitive Kripke frame is the image of that space under some interior map. But this strategy is not applicable to every dense-in-itself metric space: for example, the Kripke frame N, ≤ is not the image of any interior map from R-see Lemma 4.6. 2 Neither is 2 <ω -see Corollary 4.8. This observation has motivated the strong suspicion that there is a set of formulas satisfiable in N, ≤ , and therefore consistent, but not satisfiable in R. This suspicion is refuted by our main result. Section §4 also considers another interior map strategy and shows that it too fails. Algebraic semantics. Sections §5ff are devoted to the main project: proving that S4 is strongly complete for any dense-in-itself metric space. Section §5 generalizes the topological semantics for S4 to an algebraic semantics. Any topological space X generates an interior algebra I(X ) = P(X ), ⊆, I nt X , where I nt X is the interior operator on subsets of X . In general, an interior algebra is a triple I = A, ≤, I , where A, ≤ is a Boolean algebra, and I is a unary interior function on A satisfying certain conditions. 3 Section §5 generalizes topological models to algebraic models based on interior algebras and defines what it is for S4 to be strongly complete for an interior algebra I: the strong completeness of S4 for the topological space X is then equivalent to the strong completeness of S4 for the interior algebra I(X ). So, to prove the strong completeness S4 for any dense-in-itself metric space X , it suffices to transfer the strong completeness of S4 from I(2 <ω ) to I(X ). With topological spaces, the strong completeness of S4 is transferred backwards from the range of a surjective interior map onto the domain as in Figure 1 . With interior algebras, the strong completeness of S4 is transferred forwards from the domain of an embedding to 2 I owe this observation to Guram Bezhanishvili, David Gabelaia, and Valentin Shehtman. 3 These conditions are the duals of the Kuratowski axioms for a closure function, introduced in Kuratowski (1922). See Section §5, below. Strong completeness for dense-in-themselves metric spaces. Section §7 defines, for any dense-in-itself metric space X , a function f X : X → 2 ≤ω and shows that this function is continuous. Section §8 shows that, if X is a complete dense-in-itself metric space, then f X is a surjective interior map. 7 Thus, when X is a complete dense-in-itself metric space, we can transfer the strong completeness of S4 from 2 ≤ω to X backwards via f X . An aside, the existence of an interior map from any complete dense-in-itself metric space onto 2 ≤ω might be of broader interest, for example, in studying logics of the real line and of other complete dense-in-themselves metric spaces in other contexts, such as dynamic topological logic (see Kremer & Mints, 2005) or bimodal logic in twodimensional topological semantics (see van Benthem et al., 2006; Kremer, xxxx). Unfortunately, there are incomplete dense-in-themselves metric spaces X such that the function f X : X → 2 ≤ω is neither an interior map nor surjective. In such cases, we cannot simply transfer the strong completeness of S4 from 2 ≤ω to X backwards via f X . Fortunately, even in such cases, the function f X induces an embedding, say h * , from the subalgebra J U of I(2 ≤ω ) into I(X ). So the strong completeness of S4 is transferred first from I(2 ≤ω ) to J U via the embedding h U , and then from J U to I(X ) via the embedding h * : see Figure 3 . Note that h * •h U is an embedding from I(2 <ω ) into I(X ). This suffices for the strong completeness of S4 for X . §2. Notation, terminology, and main result. We begin by fixing notation and terminology. We assume a propositional language with a countable set PV of propositional variables; standard Boolean connectives &, ∨, and ¬; and one modal operator, . We abbreviate ¬ ¬A as ♦A and (¬A ∨ B) as (A ⊃ B). A finite set of formulas is consistent iff either it is empty or the negation of the conjunction of the formulas in it is not a theorem of S4; and an infinite set of formulas is consistent iff every finite subset is consistent. Given a nonempty set X , a topology on X is a family τ of subsets of X , such that (1) ∅, X ∈ τ ; (2) if S, S ∈ τ , then S ∩ S ∈ τ ; and (3) if σ ⊆ τ , then σ ∈ τ . Dugundji (1966) and Engelking (1989) are standard references on topology. The members of τ are the open subsets of X (in the topology τ ). A basis for τ is any set σ ⊆ τ such that every member of τ is the union of members of σ . A topological space is an ordered pair X, τ , 7 After I showed him the construction of an interior map from R onto 2 ≤ω , David Gabelaia conjectured that the construction could be generalized to any complete dense-in-itself metric space. He was right. An anonymous referee has alerted me to Lando (2012), which constructs an independently discovered interior map from R onto 2 ≤ω . Proof of Lemma 7.1. Suppose that X is a dense-in-itself metric space, that O ⊆ X is nonempty and open and that ε > 0. By induction on n ∈ N, we will define, for each n ∈ N, three disjoint subsets of O: L n (L for left), R n (R for right), M n (M for middle), so that for each n ∈ N,