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Abstract:
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Part1. Basic notions of model theory are given.
Part2. Dual notions in categories of Boolean algebras and Stone spaces are studied in respect to natural contra-variant functor. The cellularity number of a Boolean algebra B, celB is studied, certain cardinal properties are proved, e.g. it is consistent with ZFC that celB is attained for every Boolean algebra B.
Part3. Lindenbaum algebras of first-order theories are studied in details. It is proved that every Boolean algebra is isomorphic to the Lindenbaum algebra B1 of Σ1 formulas of certain first-order complete theory. Stability number ST(k) of a first-order theory T is studied, and it is shown that ST(k) = Ku(k), where Ku(k) is the Kurepa number (Kurepa introduced it in 1935) and T is the theory of dense linear ordering without end-points, while the cardinality of the Stone space of B1(A), A is a model of T, is equal to ded(A), the Dedekind number of the ordering A. Ku(k)= sup{ded(A): A is a model of T, |A|=k}.
Part4. Σn Πn ramifications of various notions in model theory are defined and studied, e.g. elementary embeddings, completeness, chains, direct limits, diagram properties, etc. Preservation theorems for these types of formulas are proved. Examples for including ordered structures and algebraic fields are given.
Part5. Model completions and elimination of quantifiers are studied. As an application, it is proved that by means of model theory that the classes of Boolean algebras and distributive lattices with the least and the greatest elements are Jonsson’s classes. Algebraic description of saturated models of submodel-complete theories are given, unifying results of Haussdorff (dense linear ordering), Erdös, Gillman (ordered fields) and Boolean algebras (Negrepontis) for homogeneous-universal models.
Part6. Here is studied what model-theoretic properties are absolute in ZF in the sense introduced by Levy, i.e. in which cases strong hypothesis (AC, GCH, V=L) can be eliminated from the proof of these properties. It is shown that the following properties of first-order theories are absolute: the consistency, completeness, model-completeness and elimination of quantifiers. These gives new light on model-theoretic proofs of these properties. |