...Some mathematicians look for a way to be sure that a contradiction will never arise whatever the developments...
Mathematicians like a minimum set of primitives from which the rest can be derived. Ideally the primitives are so obvious that no reasonable person could contest them. OTOH, there are cases where a primitive is assumed, but is not entirely logical. Playfair's axiom, for example, is a case in point. Mathematicians tried for centuries to derive it from the other axioms, to no avail. Then some mathematicians questioned "What happens if we make another assumption" and completely consistent non-euclidean geometries were born.
This is virtually impossible due the Gödel's theorem.
Gödel's theorem does not preclude consistent mathematics. It states that in a sufficiently powerful consistent mathematical system, there are true statements that can't be proven. Mathematics is built axiomatically from a set of primitives and rules which, given certain statements that are acknowledged as true, produce other statements that must be true. (In computer grammars they refer to these as production rules.) There is an alternate statement of it which, IIRC, is something like "Any *complete* mathematical system cannot be consistent."
