To prove a theorem in mathematics we usually use our axioms, definitions and other already proved theorems. Suppose we wante to prove a specific theorem and we haven't prove any other theorem and also we don't use any other definition except the definitions that are contained in the statement we want to prove. That is we want to prove from scratch (using just the axioms) the statement "If P then Q" where we have only defined what P, Q and the conditional mean. Are the axioms (using also rules of inference) enough to prove that statement in the system?
Does adding definitions change the system?
Suppose the statement "The numbers 2,4 are even" where we have defined "even=any number greater than 2". This clearly shows that just by changing one definition we have at least change the truth value of a statement. In other words theorems in an axiomatic system are also sensitive to definitions. That is the system changes not only by axioms but also from the definitions. So does this mean that we are not allowed to add definitions in a system or we can just only such definitions that don't contradict the previous irrespective of the fact that these new definitions may alter the number of theorems that can be proved?