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Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Forcing in model theory
 Yale University
, 1969
"... The forcing concept of Paul J. Cohen has had an immense effect on the development of Axiomatic Set Theory but it also possesses an obvious general significance. It therefore was to be expected that it would have an impact also on general Model Theory. In the present talk, I shall show that this expe ..."
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The forcing concept of Paul J. Cohen has had an immense effect on the development of Axiomatic Set Theory but it also possesses an obvious general significance. It therefore was to be expected that it would have an impact also on general Model Theory. In the present talk, I shall show that this expectation is indeed justified and
FIELDS WITH SEVERAL COMMUTING DERIVATIONS
"... Abstract. The existentially closed models of the theory of fields (of arbitrary characteristic) with a given finite number of commuting derivations can be characterized geometrically, in several ways. In each case, the existentially closed models are those models that contain points of certain diffe ..."
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Abstract. The existentially closed models of the theory of fields (of arbitrary characteristic) with a given finite number of commuting derivations can be characterized geometrically, in several ways. In each case, the existentially closed models are those models that contain points of certain differential varieties, which are determined by certain ordinary varieties. How can we tell whether a given system of partial differential equations has a solution? An answer given in this paper is that, if we differentiate the equations enough times, and no contradiction arises, then it never will, and the system is soluble. Here, the meaning of ‘enough times ’ can be expressed uniformly; this is one way of showing that the theory, mDF, of fields with a finite number m of commuting derivations has a modelcompanion. In fact, this theorem is worked out here (as Corollary 4.6, of Theorem 4.5), not in terms of polynomials, but in terms of the varieties that they define, and the functionfields of these: in a word, the treatment is geometric. The modelcompanion of mDF0 (in characteristic 0) has been axiomatized before, explicitly in terms of differential polynomials: see § 3. I attempted in [11] to characterize its models (namely, the existentially closed models of mDF0) in terms of differential
NOTIONS OF RELATIVE UBIQUITY FOR INVARIANT SETS OF RELATIONAL STRUCTURES
, 1990
"... Given a finite lexicon L of relational symbols and equality, one may view the collection of all Lstructures on the set of natural numbers co as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite twoperson games; (ii) a compact metric space; and (ii ..."
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Given a finite lexicon L of relational symbols and equality, one may view the collection of all Lstructures on the set of natural numbers co as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite twoperson games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on c. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on co is ubiquitous in the set of linear orderings on a).
Forcing in Lukasiewicz Predicate Logic
, 2008
"... In this paper we study the notion of forcing for Lukasiewicz predicate logic ( L∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for L∀, while for the latter, we st ..."
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In this paper we study the notion of forcing for Lukasiewicz predicate logic ( L∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for L∀, while for the latter, we study the generic and existentially complete standard models of L∀.
9 A NOTE ON INFINITE FORCING
"... Abstract. We consider one possible generalization of the notion of reduced product of infinite forcing systems. ..."
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Abstract. We consider one possible generalization of the notion of reduced product of infinite forcing systems.
REDUCED PRODUCTS OF INFINITE FORCING SYSTEMS
"... Abstract. We consider some basic properties of reduced products of infinite forcing systems. ..."
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Abstract. We consider some basic properties of reduced products of infinite forcing systems.
Set Theoretical Forcing in Quantum Mechanics and AdS/CFT
, 2003
"... We show unexpected connection of Set Theoretical Forcing with Quantum Mechanical lattice of projections over some separable Hilbert space. The basic ingredient of the construction is the rule of indistinguishability of Standard and some Nonstandard models of Peano Arithmetic. The ingeneric reals in ..."
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We show unexpected connection of Set Theoretical Forcing with Quantum Mechanical lattice of projections over some separable Hilbert space. The basic ingredient of the construction is the rule of indistinguishability of Standard and some Nonstandard models of Peano Arithmetic. The ingeneric reals introduced by M. Ozawa will correspond to simultaneous measurement of incompatible observables. We also discuss some results concerning model theoretical analysis of Small Exotic Smooth Structures on topological 4space R 4. Forcing appears rather naturally in this context and the rule of indistinguishability is crucial again. As an unexpected application we are able to approach Maldacena Conjecture on AdS/CFT correspondence in the case of AdS5 × S 5 and Super YM Conformal Field Theory in 4 dimensions. We conjecture that there is possibility of breaking Supersymetry via sources of gravity generated in 4 dimensions by exotic smooth structures on R 4 emerging in this context.