TITLE: Number of Non-isomorphic models of Set Theory
QUESTION [8 upvotes]: Assume that the meta theory allows for model theoretic techniques and handling infinite sets etc (The meta theory itself is, informally, "strong as ZFC"). Also assume that I'm studying ZFC inside this meta-theory and that I'm working with the assumption that ZFC is consistent.
Let $ZFC\subset{T}$, where $T$ is a complete theory. I would like to know the number of non-isomorphic models of $T$ for some given $\kappa\geq{\aleph_{0}}$. I feel as if this should be the maximum number possible, i.e. $2^{\kappa}$.
My idea for showing this was to use Shelah's theorem that says that unstable theories have $2^{\kappa}$ many models for any $\kappa>{\aleph_{0}}$ (which establishes the result for all uncountable $\kappa$). But in order to do this I first need to show that $T$ is unstable. However I'm not sure about how to do this (or even if I'm on the right track). 
Any help would be much appreciated.

REPLY [9 votes]: As Andres requested, here is a somewhat easier argument for the case of countable models, showing that every consistent extension $T\supset\text{ZFC}$ has continuum many countable models. 
Fix the theory $T$. A simple compactness argument shows that for any $A\subset\mathbb{N}$, we can find a countable model $M$ of $T$ in which $A$ arises as the standard part of a set of natural numbers in $M$. Each such $M$ realizes only countably many such sets $A$, and so there must be continuum many distinct models of $T$ .
This argument can be described as follows: any given set of natural numbers can arise in the standard system of a countable model — the set of reals arising as the standard parts of sets in the model — and the standard system of any model uses up at most countably many reals. So there must be continuum many pairwise non-isomorphic models of the theory.