I am thinking about completeness and incompleteness of theory's, and to illustrate both properties i am thinking of how to build an complete system, and then turn it into an incomplete one.
Example. Let the theory $\mathfrak{T}$ be the set of formulas that represent what a child (named raul) can speak in some point of the time. Let $D = \{ paul, raul \}$, $Functions = \{ fatherof \}$. Then the interpretation of the system affirms that raul knows who is his father. At some point, raul get's old and learns lots of new things. Then, if we update the interpretation of the system to reflect this change in the domain and don't update it's axioms, we should have a simple example of completeness/incompleteness in theories, right?