lo.logic - What are some results in mathematics that have snappy proofs using model theory?

Plane geometry is decidable. That is, we have a computable algorithm that will tell us the truth or falsity of any geometrical statement in the cartesian plane.

This is a consequence of Tarski's theorem showing that the theory of real closed fields admits elimination of quantifiers. The elimination algorithm is effective and so the theory is decidable. Thus, we have a computable procedure to determine the truth of any first order statement in the structure (R,+,.,0,1,<). The point is that all the classical concepts of plane geometry, in any finite dimension, are expressible in this language.

Personally, I find the fact that plane geometry has been proven decidable to be a profound human achievement. After all, for millennia mathematicians have struggled with geometry, and we now have developed a computable algorithm that will in principle answer any question.

I admit that I have been guilty, however, of grandiose over-statement of the situation---when I taught my first logic course at UC Berkeley, after I explained the theorem some of my students proceeded to their next class, a geometry class with Charles Pugh, and a little while later he came knocking on my door, asking what I meant by telling the students "geometry is finished!". So I was embarrassed.

Of course, the algorithm is not feasible--its double exponential time. Nevertheless, the fact that there is an algorithm at all seems amazing to me. To be sure, I am even more surprised that geometers so often seem unaware of the fact that they are studying a decidable theory.