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Thread: the class of all models

  1. #1
    Oct 2007

    the class of all models

    Explain me please, why if the set of first-order sentences $\displaystyle \Gamma$ has an infinite model, then the set (class) of all models of $\displaystyle \Gamma$, $\displaystyle Mod(\Gamma)$, is strictly bigger than any set $\displaystyle X$?

    Here is a passage from a textbook:
    ... $\displaystyle Mod(\Gamma)$ is sometimes unbounded. It is unbounded precisely when $\displaystyle \Gamma$ has an infinite model. By unbounded we mean that for any set $\displaystyle X$, $\displaystyle Mod(\Gamma)$ is strictly bigger than $\displaystyle X.$ ...
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  2. #2
    MHF Contributor
    Oct 2009

    Re: the class of all models

    This probably has to do with the (upward) Löwenheim–Skolem Theorem, according to which Γ has models of every infinite cardinality. Then the question reduces to showing that the class containing all cardinals is bigger than any set.
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