TITLE: Why is "points exist" not an axiom in geometry?
QUESTION [26 upvotes]: I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.

REPLY [3 votes]: One point that is missing from the other answers is that of vacuous truth. If formulated properly, we do not need points for the theorems in geometry to hold. If there are no right angled triangles then the Pythagorean theorem trivially holds for all of them, if there are no circles, Thales' theorem hold for all of them and so on. There might be a few theorems that cannot be restated in the form "For all objects x of class Y we have that ...", but even those could be reformulated as "If there exists a point, then...".
So, say you take a model-theoretic point of view of geometry and look at it just as a "set of theorems that hold for any model which satisfies those axioms". Then adding "points exists" as an axiom will only exclude the empty set as a model, which was kind of a boring model anyway. There might be a reason to do so for convenience and readability, in the same way as for example most authors exclude $\{0\}$ from the fields, but the differences in the resulting theory will be rather minor.