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The Löwenheim-Skolem theorem for models with standard part
... A sequence (a i : i ∈ N) of elements of U is called A-indiscernible if any order-preserving map f : u → u ′ between two finite subsets of N extends to an automorphism of U fixing A. Using Ramsey's theorem and compactness, one shows that if (b i : i ∈ N) is any sequence, there exists an indiscernible sequence (a i : i ∈ N) such that for any formula φ(x, y), if φ(b i , b j ) holds for all i < j, then φ(a i , a j ) holds for all i < j. A theorem of Morley's [36] asserts the same thing with the formulas φ(x, y) replaced by types, provided N is replaced with a sufficiently large cardinal. For certain points (outside the main line), we will use Morley's theorem as follows. ...
... Define a sequence of elements a i (i < ω1 ), and sets A i = M ({a j : j < i}), with tp L (a i /A i ) wide. By Morley's theorem [36], there exists an indiscernible sequence (c i : i < ω + 2) such that for any n, for some j 1 < . . . < j n , tp(c 1 , . . . ...
We note a parallel between some ideas of stable model theory and certain
topics in finite combinatorics related to the sum-product phenomenon. For a
simple linear group G, we show that a finite subset X with |X X \^{-1} X |/ |X|
bounded is close to a finite subgroup, or else to a subset of a proper
algebraic subgroup of G. We also find a connection with Lie groups, and use it
to obtain some consequences suggestive of topological nilpotence. Combining
these methods with Gromov's proof, we show that a finitely generated group with
an approximate subgroup containing any given finite set must be
nilpotent-by-finite. Model-theoretically we prove the independence theorem and
the stabilizer theorem in a general first-order setting.
... As Morley made clear these methods could interpreted in terms of 'Hanf numbers' (If there is a model with property P of cardinality H (P), there are arbitrarily large such models.). These results [19,23] were rendered into a beautiful lecture series which was written up by Vivienne Morley [21,25]. These Hanf numbers have two interpretations: 'a type can be omitted in a model of cardinality κ' or 'a sentence in an infinitary language has a model of cardinality κ'. ...
En 1956, Andrzej Ehrenfeucht et Andrzej Mostowski ont décrit un procédé permettant d'engendrer, à partir de chaînes, des modèles d'une théorie donnée : Ces modèles sont obtenus comme enveloppe, par des fonctions de Skolem, d'une chaîne d'éléments (ou de uples) indiscernables, qui leur sert d'épine dorsale. J'expose icí les principales applications de cette construction à la logique et à l'algèbre universelle contemporaines : ceci comprend l'existence de modèles d'Ehrenfeucht-Mostowski pour les langages finis et infinis; la construction de beaucoup de modèles de théories instables; la construction de beaucoup de groupes localement finis universels; les indiscernables de Silver et le réel 0# en théorie des ensembles; le théorème de Löwenheim-Skolem (celui "Vers le haut") pour les langages infinis; le théorème des deux cardinaux et quelques autres.
In analogy to ω -logic, one defines M-logic for an arbitrary structure M (see [5],[6]). In M -logic only those structures are considered in which a special part, determined by a fixed unary predicate, is isomorphic to M . Let L be the similarity type of M and T its complete theory. We say that M -logic is κ - compact if it satisfies the compactness theorem for sets of < κ sentences. In this paper we introduce the related notion of compactness for expandability : a model M is κ - compactly expandable if for every extension T′ ⊇ T of cardinality < κ , if every finite subset of T′ can be satisfied in an expansion of M , then T′ can also be satisfied in an expansion of M . Moreover, M is compactly expandable if it is ∥ M ∥ ⁺ -compactly expandable. It turns out that M -logic is κ -compact iff M is κ -compactly expandable.
Whereas for first-order logic consistency and finite satisfiability are the same, consistency with T and finite satisfiability in M are, in general, no longer the same thing. We call the model M κ - expandable if every consistent extension T ′ ⊇ T of cardinality < κ can be satisfied in an expansion of M . We say that M is expandable if it is ∥ M ∥ ⁺ -expandable. Here we study the relationship between saturation, expandability and compactness for expandability. There is a close parallelism between our results about compactly expandable models and some theorems of S. Shelah about expandable models, which are in fact expressed in terms of categoricity of PC-classes (see [7, Th. VI.5.3, VI.5.4 and VI.5.5]). Our results could be obtained directly from these theorems of Shelah if expandability and compactness for expandability were the same notion.
We prove here theorems of the form: if T has a model M in which P 1 ( M ) is κ 1 -like ordered, P 2 ( M ) is κ 2 -like ordered …, and Q 1 ( M ) is of power λ 1 , …, then T has a model N in which P 1 ( M ) is κ 1 ′ -like ordered …, Q 1 ( N ) is of power λ 1 ′ , …. (In this article κ is a strong-limit singular cardinal, and κ ′ is a singular cardinal.)
We also sometimes add the condition that M, N omits some types. The results are seemingly the best possible, i.e. according to our knowledge about n -cardinal problems (or, more precisely, a certain variant of them).
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