On the consistency of Euclidean geometry

In class today I pointed out that non-Euclidean geometries are logically consistent if the Euclidean one is, and it sure seems to be.  Jon asked whether Hilbert hadn't in fact proven the consistency of Euclidean geometry.  Not having the facts loaded up in my head, I gave a hand-waving argument that he did not:  Consistency proofs are always relative to some background theory that is assumed consistent, so at most Hilbert could have given a relative consistency proof like those of non-Euclidean geometry relative to Euclidean, perhaps taking set theory as the background.  Here is a more concrete answer from the Stanford Encylopedia:

"For the axioms of geometry, [Hilbert proved consistency] by providing an interpretation of the system in the real plane, and thus, the consistency of geometry is reduced to the consistency of analysis. The foundation of analysis, of course, itself requires an axiomatisation and a consistency proof. Hilbert provided such an axiomatisation in (1900b), but it became clear very quickly that the consistency of analysis faced significant difficulties, in particular because the favoured way of providing a foundation for analysis in Dedekind's work relied on dubious assumptions akin to to those that lead to the paradoxes of set theory and Russell's Paradox in Frege's foundation of arithmetic."  (http://plato.stanford.edu/entries/hilbert-program)

So rather than giving a set-theoretic model, as I suggested he might have, Hilbert gave a model in analysis, but the point is the same.

One might still take issue with my blanket statement, 'Consistency proofs are always relative.'  In fact, it is sometimes possible to give a purely syntactic consistency proof, as Hilbert later tried to do -- a proof, that is, that some particular system of symbolic expressions and transformation rules can never lead to an expression of the form 'P and not P'.  This is easily done, for example, if neither the axioms nor the rules of inference contain the expression 'not'!  But such a proof is not very meaningful unless the formal system of logic is complete, i.e., unless all logical consequences of a theory can be derived from it syntactically.  (This is usually cashed out in model-theoretic terms:  P is a logical consequence of T if P is true in all models where T is true.)  

But even given such a syntactic consistency proof in a complete logic, we might still say that the proof is relative, for we have then reduced the question of the consistency of our theory to that of an informal theory of formalisms, i.e., of symbols and transformations of symbol strings.  As Hilbert thought, such a theory might be so simple and clear as to escape any serious worry of inconsistency.  But this brings us full circle, for as I intended to suggest, Euclidean geometry is already beyond any serious worry of inconsistency.  It would seem that the most one can do is reduce it to other theories that seem even less questionable.