# the class of all models

• December 23rd 2012, 02:26 AM
andrei
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:
Quote:

... $Mod(\Gamma)$ 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 $X.$ ...
• December 24th 2012, 10:42 AM
emakarov
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.