Given Peano Arithmetic PA, is it possible to form the theory $T_1=PA$+{all true but unprovable statements of PA}?
Then define $T_{n+1}=T_n$+{all true but unprovable statements of $T_n$}
Then let $Q=T_{\infty}$. Is it possible to say anything about Q? Is it a well-defined theory? Is it consistent? Does Godel's second theorem apply to it? Is it possible to construct a true unprovable statement of Q?
Also, what is the system in which we deduce a statement to be true but unprovable in another system, what are the additional axioms?