This comment was posted to reddit on Jul 05, 2021 at 11:49 pm and was deleted within 16 hour(s) and 59 minutes.

The particular notions that you'd be interested are those of relative consistency and interpretability.

We say that X is consistency *relative to Y* if the consistency of Y guarantees the consistency of X. So when you use the ZFC axioms to show that a complete ordered field exists, you've shown that the complete ordered field axioms are consistent *relative to ZFC*.

An "interpretation" is basically a way of translating one mathematical theory into the language of another mathematical theory. This is exactly what you're doing when you "encode" real numbers in terms of sets. Interpretations a common way of establishing relative consistency results.