Is it possible to base set theory on Euclidean geometry, by carefully defining various notions from set theory in terms of geometric objects, so that the ZFC axioms can be shown to hold for them? Analytic geometry allows Euclidean geometry to be based on set theory, and I am curious about whether the same can be done in reverse.



Can set-like objects obeying ZFC be constructed in Euclidean geometry?
**No, this can't be done.**




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The key concept here is "simulation" - when is one theory strong enough to understand, in some sense, another? There are various versions of this (in particular, the term "interpretability" is very relevant). Below I'll give one which is fairly simple and applicable to this situation; it has drawbacks in many contexts, but those aren't relevant here.



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> Say that a theory $T$ *can simulate* a theory $S$ if there is some computable function $f$ from sentences in the language of $S$ to sentences in the language of $T$ such that for each $\varphi$ we have $S\vdash\varphi$ iff $T\vdash f(\varphi)$.
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This is an *extremely* broad notion: for example, (first-order) PA can simulate ZFC and vice versa. However, by the same token the dividing lines it produces are quite strong.


In particular, we have:



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> If $T$ is decidable and $S$ is not - that is, the set $\{\varphi: T\vdash\varphi\}$ is computable but the set $\{\psi: S\vdash\psi\}$ is not computable - then $T$ cannot simulate $S$.
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*Proof:* If $T$ could simulate $S$ via $f$ we would have $$\{\psi: S\vdash\psi\}=\{\psi: T\vdash f(\psi)\},$$ and the latter of these is computable since $f$ is computable and $T$ is decidable. $\quad\Box$ 


Now ZFC is not a decidable theory - indeed, Godel's incompleteness theorem ([as strengthened by Rosser](https://en.wikipedia.org/wiki/Rosser%27s_trick)) shows that basic mathematics is in fact undecidable - but Euclidean geometry *is* (as a consequence of [the decidability of real closed fields](https://en.wikipedia.org/wiki/Real_closed_field#Model_theory:_decidability_and_quantifier_elimination)). So there's no real way to go from geometry to set theory. 


And there are of course other examples of natural theories which at first glance may look foundationally promising but turn out to be decidable, like [set-theoretic mereology](http://jdh.hamkins.org/set-theoretic-mereology-the-theory-of-the-subset-relation-is-decidable/) or [arithmetic of naturals with addition alone](http://www.lsv.fr/~haase/documents/h18.pdf). By contrast, even [a very basic theory of addition and multiplication of naturals](https://en.wikipedia.org/wiki/Robinson_arithmetic) is already undecidable, so these "promising" theories can't simulate much at all.




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EDIT: Of course, this depends on what exactly we understand as "geometry." Certainly the geometry of Euclid is reducible to the theory of real closed theories, and so the above applies to it; on the other hand, arguably things like [tiling problems](https://en.wikipedia.org/wiki/Wang_tile) and [cellular automata](https://en.wikipedia.org/wiki/Cellular_automaton) are "geometric" in nature and these are of course complex enough to encode Turing machines. 


