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Register a new userOmitting types for infinitary [0, 1]-valued logic
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Authors
Christopher J. Eagle
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We describe an infinitary logic for metric structures which is analogous to $L_{omega_1, omega}$. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.
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We consider model-theoretic properties related to the expressive power of three analogues of $L_{omega_1, omega}$ for metric structures. We give an example showing that one of these infinitary logics is strictly more expressive than the other two, but also show that all three have the same elementary equivalence relation for complete separable metric structures. We then prove that a continuous function on a complete separable metric structure is automorphism invariant if and only if it is definable in the more expressive logic. Several of our results are related to the existence of Scott sentences for complete separable metric structures.
Let 2<nleq l<m< omega. Let L_n denote first order logic restricted to the first n variables. We show that the omitting types theorem fails dramatically for the n--variable fragments of first order logic with respect to clique guarded semantics, and for its packed n--variable fragments. Both are modal fragments of L_n. As a sample, we show that if there exists a finite relation algebra with a so--called strong l--blur, and no m--dimensional relational basis, then there exists a countable, atomic and complete L_n theory T and type Gamma, such that Gamma is realizable in every so--called m--square model of T, but any witness isolating Gamma cannot use less than $l$ variables. An $m$--square model M of T gives a form of clique guarded semantics, where the parameter m, measures how locally well behaved M is. Every ordinary model is k--square for any n<k<omega, but the converse is not true. Any model M is omega--square, and the two notions are equivalent if M is countable. Such relation algebras are shown to exist for certain values of l and m like for nleq l<omega and m=omega, and for l=n and mgeq n+3. The case l=n and m=omega gives that the omitting types theorem fails for L_n with respect to (usual) Tarskian semantics: There is an atomic countable L_n theory T for which the single non--principal type consisting of co--atoms cannot be omitted in any model M of T. For n<omega, positive results on omitting types are obained for L_n by imposing extra conditions on the theories and/or the types omitted. Positive and negative results on omitting types are obtained for infinitary variants and extensions of L_{omega, omega}.
The Omitting Types Theorem in model theory and the Baire Category Theorem in topology are known to be closely linked. We examine the precise relation between these two theorems. Working with a general notion of logic we show that the classical Omitting Types Theorem holds for a logic if a certain associated topological space has all closed subspaces Baire. We also consider stronger Baire category conditions, and hence stronger Omitting Types Theorems, including a game version. We use examples of spaces previously studied in set-theoretic topology to produce abstract logics showing that the game Omitting Types statement is consistently not equivalent to the classical one.
Fix 2<n<omega. Let L_n denote first order logic restricted to the first n variables. CA_n denotes the class of cylindric algebras of dimension n and for m>n, Nr_nCA_m(subseteq CA_n) denotes the class of n-neat reducts of CA_ms. The existence of certain finite relation algebras and finite CA_ns lacking relativized complete representations is shown to imply that the omitting types theorem (OTT) fails for L_n with respect to clique guarded semantics (which is an equivalent formalism of its packed fragments), and for the multi-dimensional modal logic S5^n. Several such relation and cylindric algebras are explicitly exhibited using rainbow constructions and Monk-like algebras. Certain CA_n constructed to show non-atom canonicity of the variety SNr_nCA_{n+3} are used to show that Vaughts theorem (VT) for L_{omega, omega}, looked upon as a special case of OTT for L_{omega, omega}, fails almost everywhere (a notion to be defined below) when restricted to L_n. That VT fails everywhere for L_n, which is stronger than failing almost everywhere as the name suggests, is reduced to the existence, for each n<m<omega, of a finite relation algebra R_m having a so-called m-1 strong blur, but R_m has no m-dimensional relational basis. VT for other modal fragments and expansions of L_n, like its guarded fragments, n-products of uni-modal logics like K^n, and first order definable expansions, is approached. It is shown that any multi-modal canonical logic L, such that $K^nsubseteq Lsubseteq S5^n$, L cannot be axiomatized by canonical equations. In particular, L is not Sahlqvist. Elementary generation and di-completeness for L_n and its clique guarded fragments are proved. Positive omitting types theorems are proved for L_n with respect to standard semantics.
We intend to investigate the metalogical property of omitting types for a wide variety of quantifier logics (that can also be seen as multimodal logics upon identifying existential quantifiers with modalities syntactically and semantically) exhibiting the essence of its abstract algebraic facet, namely, atom-canonicity, the last reflecting a well known persistence propery in modal logic. In the spirit of universal logic , with this algebraic abstraction at hand, the omitting types theorem OTT will be studied for various reducts extensions and variants (possibly allowing formulas of infinite length) of first order logic. Our investigatons are algebraic, addressing (non) atom canoicity of varieties of algebra of relations. In the course of our investigations, both negative and positive results will be presented. For example, we show that for any countable theory $L_n$ theory $T$ that has quantifier elimination $< 2^{omega}$ many non-principal complete types can be omitted. Furthermore, the maximality (completeness) condition, if eliminated, leads to an independent statement from $sf ZFC$ implied by Martins axiom. $sf OTT$s are approached for other algebraizable (in the classical Blok-Pigozzi sense) reformulations or/ a
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