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.


1 Answer 1


What we rather have for Hilbert's axioms of geometry (and the same is the case for Peano's axioms for natural numbers) is that we can model them withing the framework of ZFC.

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.

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    $\begingroup$ This makes totally sense. The primitive notions will probably occur then as definition, won't it? Similarly for the primitive relations. $\endgroup$
    – user342207
    Oct 6, 2016 at 18:25

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