Definition of Locus in Synthetic Geometry?

Question

How is "locus" defined in Euclidean geometry? I know that locus is defined as a set of all points satisfying a certain condition. But how do we define "locus" within an axiomatic system. The "points" and "lines" are primitive concepts and set theory is not buit into the system.

Answer 1

Fix your language, and let $\varphi(P)$ be a formula in the language, where $P$ is a variable point. Then, for each model $\mathfrak{A}$, the set $\{P \text{ is a point in } \mathfrak{A}: \varphi(P)\}$ is the locus of all points that satisfy $\varphi(P)$.

The formula $\varphi(P)$ is a syntactic object that represents the locus of all points $P$ with the property that $\varphi(P)$ is true.