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Completeness theorem affirms that a formal first order system is consistent iff it has a model. The FOL number theory(PA) or First Order Arithmetic has a model, which is the natural numbers structure. So PA is consistent. But then,

why one needs proofs of the consistency of PA such (such those by Gentzen, Ackermann etc... obviously not in PA)? Is it only for the sake of proving the result "syntactically" and not by using semantic tools such as models, or what else?

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The mathematical principles needed to prove the completeness theorem are considerably stronger than PA itself, and according to some, stronger than what is really needed to do mathematics. So in a philosophical sense, there's no real meaning to such a proof of the consistency of PA. –  Zhen Lin Mar 14 '12 at 12:42
    
could you please cite a reference for these consistency proofs (by Gentzen, Ackermann, ect)? I am not sure I know what you are referring at –  magma Mar 14 '12 at 15:43
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up vote 7 down vote accepted

One reason is that the construction of the natural number structure (and the proof that it is a model of PA) actually takes place in a much stronger system: (a fragment of) ZFC. Since you are using ZFC to construct a model of PA, this amounts to "ZFC $\vdash$ PA is consistent."

Now, what happens if (Xenu forbid!) (that fragment of) ZFC is itself inconsistent? In that case since ZFC proves everything, the fact that it proves the consistency of PA is rather worthless. What we can glean from this is that the consistency of ZFC implies the consistency of PA; an example of a relative consistency result, but does not give the absolute consistency of PA.

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There is a little thing I don't understand completely. Usually, when treating with models, we don't want the structure to be exactly a structure described in a formal way. So it is immaterial whether the natural number structure take place in ZFC or not. It is just a non-formal structure we can talk about without referring to ZFC, isn't it? –  Oo3 Apr 6 '12 at 9:34
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