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On the Intermediate Logic of Open Subsets of Metric Spaces Timofei Shatrov

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On the Intermediate Logic of Open Subsets of Metric Spaces Timofei Shatrov On the intermediate logic of open subsets of metric spaces Timofei Shatrov abstract. In this paper we study the intermediate logic MLO( ) of open subsets of a metric space . This logic is closely related to Medvedev’sX logic X of finite problems ML. We prove several facts about this logic: its inclusion in ML, impossibility of its finite axiomatization and indistinguishability from ML within some large class of propositional formulas. Keywords: intermediate logic, Medvedev’s logic, axiomatization, indistin- guishability 1 Introduction In this paper we introduce and study a new intermediate logic MLO( ), which we first define using Kripke semantics and then we will try to ratio-X nalize this definition using the concepts from prior research in this field. An (intuitionistic) Kripke frame is a partially ordered set (F, ). A Kripke model is a Kripke frame with a valuation θ (a function which≤ maps propositional variables to upward-closed subsets of the Kripke frame). The notion of a propositional formula being true at some point w F of a model M = (F,θ) is defined recursively as follows: ∈ For propositional letter p , M, w p iff w θ(p ) i i ∈ i M, w ψ χ iff M, w ψ and M, w χ ∧ M, w ψ χ iff M, w ψ or M, w χ ∨ M, w ψ χ iff for any w′ w, M, w ψ implies M, w′ χ → ≥ M, w 6 ⊥ A formula is true in a model M if it is true in each point of M. A formula is valid in a frame F if it is true in all models based on that frame. The set of formulas which are valid on every Kripke frame is intuitionistic logic Int. An intermediate logic is an extension of Int closed under modus ponens and substitution. Every consistent intermediate logic is contained in classical logic CL. The set L(F ) of all formulas valid in a given frame F forms an interme- diate logic, which allows us to define a logic just by constructing its Kripke Advances in Modal Logic, Volume 7. c 2008, the author(s). 306 Timofei Shatrov frame (although not every intermediate logic can be constructed in this way). For example, the logics ML and ML1, which will be often mentioned in this paper are defined as follows: X DEFINITION 1. Let X be a set, then P1(X) is the Kripke frame (2 , ) (non-empty subsets of X ordered by converse inclusion). Medvedev’s logic\ ∅ ⊇ of finite problems ML is L(P (X)) X is finite [1] and the logic of infinite { 1 | } problems ML1 is L(P1(ω)) (or L(P1(X)) for any infinite X) [6]. T It is well-known that ML ML, but it is an open problem whether 1 ⊆ ML1 = ML or not. Let be a topological space, and O( ) a set of non-empty open sets in . ThenX (O( ), ) is a Kripke frameX with the least element . Logic X X ⊇ X MLO( ) is defined as the logic of this frame: X DEFINITION 2. MLO( ) = L(O( ), ). We will often use O( ) to denote the frame (O( ), ) asX a simple notation.X ⊇ X X ⊇ Finally, a few words on why this logic is interesting. We can understand information as a subset of some basic set Ω. An information type is an arbi- trary set of informations (these notions were introduced by Yuri Medvedev in [2]). A type σ is called regular iff E E E σ & E E E σ ∀ 1∀ 2 1 ∈ 2 ⊆ 1 ⇒ 2 ∈ As described in [5] regular information types form a structure that is natu- rally connected to Medvedev’s logic of finite problems ML (in case of finite Ω), or logic of infinite problems ML1 in case of infinite Ω. Namely, the set of all regular types I(Ω) ordered by inclusion is a Heyting algebra, and it is isomorphic to the Heyting algebra of upward-closed subsets of P1(Ω). Thus, the logic of Heyting algebra I(Ω) is L(P1(Ω)). Now, for a topological space it may be more natural to understand information as an open subset of X , and not an arbitrary one. For example, every measurement in a physicalX experiment has a margin of error, so the only kind of information we can get about the measured value is that it is within some interval (open subset of R). In this case regular information types correspond to upward-closed subsets of O( ) and thus form a Heyting X algebra of MLO( ). X 2 Useful facts and definitions In this section we briefly recall facts related to Kripke semantics of inter- mediate logics. DEFINITION 3. Let u F be a point of a Kripke frame (F, ). Then the Kripke frame F u = v ∈F u v with the ordering is called≤ a cone. { ∈ | ≤ } ≤ DEFINITION 4. The surjective mapping h from frame F to frame G is called a p-morphism iff u F h(F u) = Gh(u), that is ∀ ∈ 1. u,v F (u v) h(u) h(v) ∀ ∈ ≤ ⇒ ≤ On the intermediate logic of open subsets of metric spaces 307 2. u F, w G (h(u) w) v u : h(v) = w ∀ ∈ ∈ ≤ ⇒ ∃ ≥ If there is a p-morphism from F to G, this is denoted by F ։ G. LEMMA 5. 1. L(F ) L(F u). ⊆ 2. If F G then L(F ) L(G). ։ ⊆ Let F be a finite Kripke frame with the least element (called the root and denoted by 0F ). Then it is possible to construct a propositional formula X(F ), called Jankov (characteristic) formula with the following properties: LEMMA 6. 1. For any propositional formula A, X(F ) (Int + A) F = A ∈ ⇐⇒ 6| 2. For any frame G, G = X(F ) u G : Gu F 6| ⇐⇒ ∃ ∈ ։ 3 MLO( ) as a subset of Medvedev’s logic X In this section we construct a p-morphism from the Kripke frame of open subsets of to the Kripke frame of ML1, thus proving that MLO( ) is a X X subset of ML1 and ML. THEOREM 7. Let be an infinite metric space. Then MLO( ) ML1. X X ⊆ Proof. To prove this we construct a p-morphism f : O( ) P (ω). X ։ 1 X can be split into 3 disjoint subsets: = 1 2 3. = x ε y ,y = x ρ(x,yX ) >X ε∪Xcontains∪X all isolated points. X1 { ∈ X |∃ ∀ ∈X 6 ⇒ } 2 = x xn n∞=1 1,xn x, n 1 contains the limits of isolatedX { points.∈ X |∃{ } ⊂ X → → ∞} \ X = ( ) contains the limit points which are not the limits of X3 X \X1 \X2 isolated points. A point of 3 is always a limit of other points in 3, thus is either empty or infinite.X X X3 If is empty then cannot be finite (since otherwise is empty, and X3 X1 X2 is finite). We can choose a sequence xn 1, nX = 0, 1, 2,... where all x are different.∈X Let D = x , i = 1, 2, 3,... , n i { i} D0 = x0 x 1 n x = xn . Each of Di is open, since each xi is an isolated{ } point.∪ { ∈ Let X |∀G 6 be a} nonempty open subset. If G D = , ⊂ X ∩ i ∅ i = 0, 1, 2,... then G 1 = , so G 2, since 3 is empty. But since each point in is a limit∩X of isolated∅ points⊆X from X, G has to contain some, X2 X1 which is a contradiction. Thus, there always exists Dn which intersects G. We define f(G) = n ω G Dn = , which is easily checked to be p-morphism. { ∈ | ∩ 6 ∅} Let be nonempty. It is open, since its complement = X3 X \X3 X1 ∪X2 is clearly closed (a limit of points in 2 is also in 2). Choose an arbitrary x and let D = x 1 X< ρ(x,x ) <X1 , n = 2, 3,... If we 0 ∈X3 n { ∈ X3| n+1 0 n } exclude empty Dn, there would be still an infinite number of them left (since 308 Timofei Shatrov x is a limit point of ). Also define D = x ρ(x,x ) > 1 , D = . 0 X3 1 { ∈X3| 0 2 } 0 X1 Each open subset G intersects one of Dn. After renumbering Dn such that all of them are non-empty⊂X we can define f(G) = n ω G D = , { ∈ | ∩ n 6 ∅} which is a p-morphism. It should be noted that by definition ML1 = MLO(ω). For a metric space , MLO( ) would be equal to ML1 if it is possible to construct a p-morphismX fromX P (Y ) to O( ) for some infinite set Y . 1 X For example, consider = 0 1/n n N with the usual metric and X { } ∪ { | ∈ } Y = N 0, 1 . Open sets of are either arbitrary subsets of 0, or cofinite× subsets { } of . We defineXh : P (Y ) O( ) as follows: X \ X 1 → X h(E) = 1/n j (n, j) E if E = ω • { |∃ ∈ } |X \ | h(E) = 1/n j (n, j) E 0 n (n, 1) E if E < ω • { |∃ ∈ } ∪ { |∃ ∈ } |X \ | It is easy to check that h is a p-morphism. The following lemma is a generalization of this construction. LEMMA 8. Suppose is an infinite metric space consisting only of isolated points and limits of isolatedX points ( = using the notation from X X1 ∪X2 the proof of Theorem 7). Then ML1 = MLO( ). X Proof. Let Y = 1 ( 2 x0 ) (x0 1), and for y = (y1,y2) Y φ(y) = y , ψ(y) = yX .× DefineX ∪h {: P}(Y ) ∈O X( ) as follows: ∈ 1 2 1 → X h(E) = int(φ(E) ( ψ(E))) ∪ X2 ∩ φ(E) is always in h(E), and we only add appropriate limit points from ψ(E) so that the result is an open set.
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