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Abstract:
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The thesis consists of five chapters. In the first part of Chapter 1 forcing relations for infinite logics are considered. It is shown that if in the case of infinite logic we want to extend syntactic apparatus adequately and that forcing joining stays deductive closed set which contains all logically valid formulas, then forcing joining has to formulate by "weak" formulas. In the second part of this chapter a correction of the proof of the interpolation theorem for infinite logics is presented. The result from Chapter 2 is the following: it is shown that all important properties of Robinson’s finite forcing are transmitted to n-finite forcing by corresponding "n-notions". Moreover, a construction of n-finite forcing joining by Henrard’s approximation chains is presented. The main result of Chapter 3 is that for each theory T of a language L there is an extension T' defined in the corresponding extension L' such that. Relations between a theory (the theory of dense linearly ordering with maximal and minimal elements, the theory of groups, the theory of Abelian groups, the theory of fields, full arithmetic, Peano’s arithmetic) and its corresponding n-finite forcing joins are studied in Chapter 4. Also relations between n-finite forcing joins are studied. A connection between n-finite forcing and the type theory are studied in Chapter 5, and some generalizations of the known results are given. |