Constructing the real numbers and consistency?

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.

/r/math Thread