Model theory and Philosophy

The problem with model theory is: in what system are there results of model theory carried out ? Is model theory nothing but an internal reflection-into-self of ZF(C) itself ? If so, it is interesting, but hardly philosophically conclusive or as weighty as sometimes suggested. Also then all results of model theory can then all be represented in a countable model.  There is a certain analogy between the Downwards Löwenheim-Skolem theorem and the topological property of $\sigma$-compactness.

Theory and metatheory

ZF(C) is also arguably a mirror,  a microcosm which can reflect and represent second and higher-order logic.  In what system is proven the categoricity of certain second-order theories (naturals, reals) ? 

How arbitrary and full of presuppositions it is to consider first-order models are representing the world ! For this simply involves doing a Henkin model move involving an arbitrary restriction on power sets and relations. No, we should be prepared to consider full second-order models and not necessarily as reflected into set theory. Then we can know what we are talking about when we define and talk about $\mathbb{N}$ and $\mathbb{R}$.  There is a huge difference from the conclusion that we cannot deduce everything about these objects (true) from the claim that we cannot know anything for certain about mathematical objects at all (false).

If model theory be considered to set-theoretic, its category theoretic version needs to be considered. Just as in model theory we are lead naturally to consider classes of models, so too in category theory it is very natural to consider categories of categories (on specific kinds). Thus a 'category' is both the generalization of a model and a generalization of a class of models.