Iterated elementary embeddings and the model theory of infinitary logic

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• 作者：John T. Baldwin^a ; ¹ ; ^{jbaldwin@uic.edu" class="auth_mail" title="E-mail the corresponding author} ; Paul B. Larson^b ; ² ; ^{larsonpb@miamioh.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：03C48 ; 03E15 ; 03E57 ; 03C45
• 刊名：Annals of Pure and Applied Logic
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：167
• 期：3
• 页码：309-334
• 全文大小：688 K

文摘

We use iterations of elementary embeddings derived from countably complete ideals on ω₁${\omega }_1$ to provide a uniform proof of some classical results connecting the number of models of cardinality 18e4ccac3142607dcfeb0c568f2bc" title="Click to view the MathML source">ℵ₁${\mathrm{\aleph }}_1$ in various infinitary logics to the number of syntactic types over the empty set. We introduce the notion of an analytically presented abstract elementary class (AEC), which allows the formulation and proof of generalizations of these results to refer to Galois types rather than syntactic types. We prove (Theorem 0.4) the equivalence of e052ffdd63fb2b08649078d4a" title="Click to view the MathML source">ℵ₀${\mathrm{\aleph }}_0$-presented classes and analytically presented classes and, using this, generalize (Theorem 0.5) Keisler's theorem on few models in 18e4ccac3142607dcfeb0c568f2bc" title="Click to view the MathML source">ℵ₁${\mathrm{\aleph }}_1$ to bound the number of Galois types rather than the number of syntactic types. Theorem 0.6 gives a new proof (cf. [5]) for analytically presented AEC's of the absoluteness of 18e4ccac3142607dcfeb0c568f2bc" title="Click to view the MathML source">ℵ₁${\mathrm{\aleph }}_1$-categoricity from amalgamation in e052ffdd63fb2b08649078d4a" title="Click to view the MathML source">ℵ₀${\mathrm{\aleph }}_0$ and almost Galois ω-stability.