Университеттің 85 жылдығына арналған Қазіргі заманғы математика
Университеттің 85 жылдығына арналған «Қазіргі заманғы математика
жүктеу/скачать 12,2 Mb. Pdf көрінісі
|
| Университеттің 85 жылдығына арналған «Қазіргі заманғы математика:
проблемалары және қолданыстары» III халықаралық Тайманов оқуларының материалдар жинағы, 25 қараша, 2022 жыл 9 and its classification made in [4], [5], [6], [8] (Pillay-Steinhorn, Marker, Mayer, Marker- Steinhorn, 1986–1994), to take the constants for K ∂ xψ ( x,y ¯) from an infinite indiscernible sequence I = ⟨ α n ⟩ n<ω over M and α n from p (ɲ). Taking into consideration that if K ψ ( x,y ¯) ( ɲ, , a ¯ ,α ¯ n ) ∩ M = ∅ , then there is a finite number irrational cuts (1-types over M) such that for any such 1-type r ∈ S 1 ( M ), K ψ ( x,y ¯)( ɲ ,a ¯ ,α ¯ n ) is a subset of QV r ( α ¯ n ):={ β ∈ r (ɲ)| there exists an Mα ¯ n -1-formula Θ( x,α ¯ n ), such that β ∈ Θ(ɲ ,α ¯ n ) ⊂ r (ɲ)}. B.S. Baizhanov in 1996 obtained a classification of 1-types over a subset of a model of weakly o-minimal theory and solved the problem of expanding a model of weakly-o-minimal theory by a unary convex predicate in the preprint "Classifications of 1-types in weakly o- minimal theories and its applications" and submitted in the JSL, that revised version published in [9](2001). We say that ɱ + the expansion by all externally definable subsets admits quantifier elimination , if for any formula ϕ ( y ¯) of Σ + there exists Σ- formula K ϕ ( y ¯ ,z ¯), there exists α ¯ ∈ N \ M such that for any a ¯ ∈ M the following holds: ɱ + |= ϕ ( a ¯) ⇔ = K ϕ ( a ¯ ,α ¯) . Approach of Shelah. In his paper, S. Shelah [10] (2004) considered a model of NIP theory and proved that the expansion by all externally definable subsets ad- mits quantifier elimination and thereby is NIP. The key problem here is eliminating quantifier "there exists in the submodel". In his proof in the way of contradiction Shelah used an indiscernible sequence ⟨ b n : n<ω ⟩ in order to show that if eliminating quantifier "there exists x in the submodel" θ ( x,a ¯) fails, then θ ( α, ¯ b n ) holds i ff n is even, for some α , which implies the independence property, for a contradiction. V.V. Verbovskiy [11] (preprint 2005) found a somewhat simplified account of Shelah’s proof, namely by using noting of a finitely realizable type. A. Pillay [12](preprint 2006) gave two re-proofs of Shelah’s theorem, the first going through quantifier-free heirs of quantifier-free types and the second through quantifier-free coheirs of quantifier-free types. The analysis of approaches shows that the using the theory of orthogonalitywe can control the set of realizations of one-types. The generalization of notionsofquasi- neighborhoodandneighborhooditispossibletoformulatethenext жүктеу/скачать 12,2 Mb. Достарыңызбен бөлісу:
|