# downward Lowenheim-Skolem theorem

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

Title | downward Lowenheim-Skolem theorem |
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Canonical name | DownwardLowenheimSkolemTheorem |

Date of creation | 2013-03-22 13:00:42 |

Last modified on | 2013-03-22 13:00:42 |

Owner | Evandar (27) |

Last modified by | Evandar (27) |

Numerical id | 5 |

Author | Evandar (27) |

Entry type | Theorem |

Classification | msc 03C07 |