Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 116 / 2 / a / Solution

For a first-order theory $R$ extending ZFC, let $C_R$ be its set of formal consequences and let $\mathrm{Cons}$ denote the class of formal consistency statements $\mathrm{Con}(Q)$ for recursively axiomatized extensions $Q$ of ZFC. Using Gödel numbering to code proofs and theories, these objects and the following comparison are definable in the base theory ZFC.

The consistency-strength preorder isThus every consistency assertion provable in $T$ is also provable in $S$. Its strict part is