Given a formal logic system W with it's set of axioms, if it is capable of 'handling' basic arithmetic then W can not be used to 'prove' its own consistency. In other words there will always be a statement H in W that is 'true' yet not provable. You could call system W a STATIC logic system . Could there also be a definable and rigorously constructed 'Semi-Variant' Logic system with a set of 'base' axioms that are invariant and a set of 'variable' axioms that can 'change' if presently needed to change for the sake of consistency or completeness. Could the logical foundations of Physics be called a 'Semi-variant' Logic system?
Are all true mathematical statements provable?
An interesting look for Gödel Theorem.... Check :
What do you mean by "W1 does not have H"? I thought that W1 was derived from W by adding some axioms. (Incidentally, I don't know why you're adding "certain relevant and useful axioms" as opposed to just doing the obvious thing and adding H itself as an axiom.) In particular, then, the basic grammar of W1 is the same as that of W, so any statement in W is still a statement in H.
Perhaps what you meant is that in W, H is no longer unprovable. Continuing along, you have a sequence (W,W1,W2,...) where, as you go down the list, you keep adding axioms.
This is not a way around Godel's theorem. Let X be the union of all the sets of axioms from all the Wi. Then (by Godel's theorem!), if X is recursive then your logical system still contains true statements about arithmetic that are not derivable from X.
(Of course if X is not recursive --- for example, if you chose at some point along the way to add every true statement about arithmetic as an axiom --- then Godel does not apply.)