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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
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.
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.
A FEW REMARKS ON nINFINITE FORCING COMPANIONS
"... Abstract. We show that the basic properties of Robinson’s infinite forcing companions are naturally transmitted to the so called ninfinite forcing companions and start with the examination of mutual relations of ninfinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the art ..."
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Abstract. We show that the basic properties of Robinson’s infinite forcing companions are naturally transmitted to the so called ninfinite forcing companions and start with the examination of mutual relations of ninfinite forcing companions of Peano arithmetic. 1. Preliminaries Throughout the article L is a first order language. In general discussions mostly it is irrelevant whether it is with equality or not; however, in some cases, for instance when it comes to finite models, the supposition of the existence of the equality relation could be of significance – see 2.6. For a theory T of the language L, µ(T) will be the slass of all its models (as usual, by a theory we assume a consistent deductively closed set of sentences – thus, T ϕ means ϕ ∈ T). By Σnformula we mean any formula equivalent to a formula in prenex normal form whose prenex consists of n blocks of quantifiers, the first one is the block of existential quantifiers (Πnformulas are defined analoguosly). The models (of the language L) will be denote by A,B..., while their domains will be A,B,.... For a model A, Diagn(A) is the set of all Σn, Πnsenteneces of the language L(A) (the simple expansion of the language L obtained by adding a new set of constants which is in one to one correspendence with domain A) which hold in A. In particular, for n = 0, Diag0(A) is not the diagram of A in the sense in which it is used in model theory, but this difference is of no importance for the text (the same situation we had when we were dealing with the generalization of finite forcing). As usual, we will not distinguish an element a from A and to it the corresponding constant. If A is a submodel of B and (B, a)a∈A Diagn(A), we say that A is an nelementary submodel of B (i.e., that B is an nelementary extension of A), in notation A ≺n B. In general, A is nembedded in B if for some embedding f of A into B, f(A) is an nelementary submodel of B. A Σn+1chain of models is a chain of models A0 < A1 < · · · < Aα < · · · , α < γ, where for each