# Zermelo-Fraenkel set theory and Hilbert's axioms for geometry

The Zermelo-Fraenkel axioms with the axiom of choice (ZFC) are nowadays by many considered as the foundation for contemporary mathematics, and according to several mathematicians all mathematics is a logical consequence of this type of set theory.

For example, the Peano axioms for the natural numbers can be constructed from Zermelo-Fraenkel set theory and deriving classical subjects in mathematics from Peano's axioms is well-known.

I was wondering whether Euclidean geometry can also be constructed from Zermelo-Fraenkel set theory. To be more precise: can Hilbert's axioms for geometry be derived from Zermelo-Fraenkel set theory in a similar fashion as Peano's axioms? I would think this is possible since Zermelo-Fraenkel set theory is, as already mentioned above, considered as the basis of all extant mathematics, but I never came across such a derivation.

Any comment is appreciated.

So to repeat the case of Peano: In ZFC we can show the existence of $\omega$, the smallest infinite ordinal. This $\omega$ models Peano's axioms, where $\emptyset$ represents the natural number $0$ and $x\mapsto x\cup\{x\}$ represents the successor function $S$.
For geometry, the probably simplest way is to note that ZFC gives us a model of $\Bbb R$, and of $\Bbb R^3$, the elements of which can be thought of as points by means of a Cartesian coordinate system. The Hilbert axioms hold for these points in a straightforward fashion.