TITLE: Why is there a need for ordinal analysis?
QUESTION [9 upvotes]: Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence theorem, shouldn't this suffice to prove the consistency of first order arithmetic? Why is Gentzen's proof necessary?

REPLY [12 votes]: In addition to the reasons Andreas gives, Gentzen's theorem gives additional information that's interesting even if you don't have any qualms about consistency.
In particular, ordinal analysis gives a fairly precise characterization of the provably total computable functions of a theory (and, along with it, a lot of information about the structure of proofs in PA).