FOM: really basic questions

[F. Xavier Noria]

 Dear FOMers,
 
 I've just taken a degree in Mathematics from the University of Barcelona
 and, in spite of the high level of the usual posts, I'd like to explain
 some basic doubts concerning f.o.m. that I think I must have pretty clear:
 
 
 1) I don't understand why I'm supposed to be surprised because of the
 incompleteness of PA.
 
 I mean, G, where G stands for the axioms of Group Theory, is trivially an
 incomplete theory because neither AxAy(xy = yx) nor its negation are in
 Ded(G).
 
 G\"odel's first theorem says that PA is incomplete, and what? This is the
 fact, is a theorem, what is the surprise?
 
 Perhaps, G is incomplete because G has *too many* models and this seemed
 not to be the case of PA, so my question is: "Historically, was PA created
 to *catch* N?" If this were the case, why aren't reactions restricted to
 something like: "Well, actually it doesn't."? We know there are a
 continuum of non-isomorphic models of PA with universe \omega, indeed.
 
 Perhaps, what I don't understand have to do with the next question:
 
 
 2) Sometimes I've heard comments like: "The famous conjecture X of Number
 Theory might be undecidable" and I'd want to fix what is exactly their
 scope.
 
 [I'll argue within first order calculus.]
 
 On the one hand, f.o.m. asserts that all the work done by number theorists
 can be formalized in ZFC, doesn't it?, on the other hand, it seems to me
 that number theorists look for theorems in Th(N), neither in Con(PA) nor
 in Con(ZFC). [Let me suppose that we all agree on what on earth is N.]
 
 But, Th(N) is complete as far as is the theory of a model, and therefore
 I cannot see in what sense Goldbach's Conjecture, for instance, could be
 an undecidable theorem of Number Theory. Of course, GC might be an
 undecidable sentence of PA... where is the link?

 ---

 I'd be very grateful if you emailed any clarifications.
 
 With best wishes,
 
 -- Xavier