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downward Lowenheim-Skolem theorem (Theorem)

Let $ L$ be a first order language, let $ \mathcal{A}$ be an $ L$-structure and let $ K\subseteq\operatorname{dom}(\mathcal{A})$. Then there is an $ L$-structure $ \mathcal{B}$ such that $ K\subseteq\mathcal{B}$ and $ \vert\mathcal{B}\vert\leq\operatorname{Max}(\vert K\vert,\vert L\vert)$ and $ \mathcal{B}$ is elementarily embedded in $ \mathcal{A}$.



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Cross-references: first order language
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This is version 2 of downward Lowenheim-Skolem theorem, born on 2002-08-29, modified 2008-06-09.
Object id is 3394, canonical name is DownwardLowenheimSkolemTheorem.
Accessed 3035 times total.

Classification:
AMS MSC03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures)

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