Zermelo–Fraenkel set theory with axiom of choice (ZFC) can be used as a foundation of mathematics. It means, that any theory can be expressed in terms of ZFC. In particular, it should be possible to formulate Euclidean geometry in terms of ZFC.
However, ZFC is based on the first-order logic and Hilbert axioms cannot be formulated in terms of the first-order logic, because of the Archimedes axiom and the axiom of completeness (for that we need to use second-order logic).
How can one combine these two facts? I have two options:
- Euclidian geometry cannot be expressed in terms of ZFC (and probably many other theories in mathematics are not expressible in ZFC because of the same reason).
- Things expressed in the second order logic can also be expressed in the first order logic. Or, less bold statement, some parts of the second order logic can be expressed in terms of the first order logic (maybe Hilbert axioms do not use all the elements of the second order logic).
My question is not about Hilbert axioms (they are used just an example). My question is about how a first order logic system can be used to emulate / model a second order logic axiomatic system. This answer does not answer my question.
