Relative consistency ($\mathrm{Con}(T)\Rightarrow \mathrm{Con}(S)$)

A relative-consistency result constructs a model of a first-order theory $S$, or interprets its proofs, from the assumed consistency of a background theory $T$. It establishes $\mathrm{Con}(T)\Rightarrow \mathrm{Con}(S)$ in the stated metatheory, not an unconditional proof of consistency. For independence of an axiom $\phi$ from $S$, obtain consistent models of both $S+\phi$ and $S+\neg \phi$, relative to the background assumption.