Can an axiomatic system (which is capable of expressing arithmetic) be complete and consistent?
Let me explain my motivation a little bit (though it can be a kind of a mess...) I'm aware of Goedel's Theorems, and I started thinking about this question when I happened to read about the so-called "true arithmetic". Well, assuming Peano arithmetic is consistent (if we accept that the natural numbers structure is its model), we can take all statements which hold in this model as axioms (which will contain PA), can't we? And this system will be consistent and complete (and not decidable)? How about proving its own consistency? Can it even express it? I read Goedel's original work and there the statement of consistency is formalized as $\exists x [Form(x) \land \lnot Bew(x)]$ (1) which means that there exists some formula $x$ (this is expressed by $Form(x)$) which is not provable ($\lnot Bew(x)$) (though, roughly speaking, the same can be of course said about any set of sentences - it is consistent if and only it doesn't prove everything). So what I am trying to make sense of, is it that this "true arithmetic" can't even express its consistensy in the form (1)? Is it simply a statement we know about this system on the basis of, say, $ZF$?