Notion of internality in model theory     Good evening,
Can someone explain to me the notion of internality in model theory (what it is,
where it comes from...) ?
Thank you
 A: The standard use in model theory is something like this.  A partial type $p$ is internal to a type $q$ if
there is a definable function $f$ such that any realization of $p$ is equal to $f(c_1,\dots,c_m)$ where $c_1,\dots,c_m$ are realizations of $q$.
A typical example from differential fields:  Let $X$ be the set of solutions of a linear differential equation of order $n$. 
Then $X$ is internal to the constants.  Let $a_1,\dots,a_n$ be a fundamental system of solutions.  Let $f(c_1,\dots,c_n)=\sum c_ia_i$.  Then every element of $X$ is the image of an
$n$-tuple of constants.
A: The use of 'internality' in model theory that is most familiar to me is its use in nonstandard analysis.  Look at 'internal' in the wikipedia article on non-standard analysis and see if that is what you remember.
A: Could you be thinking of Skolem's Paradox?  


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*http://en.wikipedia.org/wiki/Skolem%27s_paradox
It is sometimes explained in terms of "internal sets", the sets that a model can "see".  Example:


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*http://en.wikipedia.org/wiki/Second-order_logic#Non-reducibility_to_first-order_logic
That's a description of why a theory containing the power set of the integers still has a countable model.  The model doesn't actually contain the full powerset, but it also doesn't contain a bijection between its integers and its sets of integers, so "internally" the power set is uncountable even though it's countable "externally".
If that's what you're looking for, then http://math.stackexchange.com is probably a better place than here for follow-up discussion.
