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Math Help - the class of all models

  1. #1
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    Moldova
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    the class of all models

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

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