FOM: really basic questions
F. Xavier Noria
fxn at cambrabcn.es
Fri Sep 18 11:42:07 EDT 1998
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
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