Good evening,
Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ?
Thank you
Good evening, Can someone explain to me the notion of internality in model theory (what it is, where it comes from...) ? Thank you 


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. 


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 nonstandard analysis and see if that is what you remember. 


Could you be thinking of Skolem's Paradox? It is sometimes explained in terms of "internal sets", the sets that a model can "see". Example: 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 followup discussion. 

