To define a set theory in the framework of axiomatic euclidean geometry? I do not of course mean the whole ZFC...

Question

Is it possible to define a set theory, of course weaker than ZFC, in terms of axiomatic euclidean geometry eg Hilbert (corrected Euclid). My idea is that once we shall have defined such a set theory we could be able based on this to define a part of mathematics, and this part would constitute a strict definition of elementary mathematics. It is known that the concept of real numbers are defined in the framework of axiomatic set theory, but how can one define the set of real numbers?

Answer 1

Since Euclidean geometry is decidable, we can't use it to implement any undecidable theory. In particular, even basic arithmetic can't be implemented in Euclidean geometry, as a consequence of Godel's incompleteness theorem.

So the only "set theories" you'd be able to attack this way are ones too weak to even talk about finite sets in a reasonable way - I'd argue those don't deserve the name.