We have many different systems of axioms in geometry and from observations we most often use Euclidean ones. Euclid's postulates are insufficient, but the Hilbert system seems overloaded and redundant. (For example, the axiom of two triangles is very similar to the theorem of the equality of triangles, and it is the same with the axioms of the congruence of angles and segments.) There is a school system of postulates (where some can be derived from others, but for the sake of simplicity they are shown as axioms) and in general Birkhoff's system, in which there are only four axioms, but clarity is lost). When studying these, there is a feeling of loss of theorems, when there are already 20 axioms.
Which system of axioms is used most often in modern geometry?
Is there a system of axioms in algebra and in other parts of mathematics?