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

Let $\mathrm{IC}$ be the sentence asserting that a strongly inaccessible cardinal exists, and begin withDefine the iterated consistency progressionThe construction is effective, so every $T_n$ and $T_{\infty }$ is a recursively axiomatized first-order theory extending ZFC.

Because $T_{n+1}$ extends $T_n$, every theorem of $T_n$, including every formal consistency statement it proves, is a theorem of $T_{n+1}$; hence $T_n{\le }_{\mathrm{Cons}}{T}_{n+1}$. The theory $T_{n+1}$ proves $\mathrm{Con}(T_n)$ by construction, whereas a consistent $T_n$ cannot prove its own consistency by Gödel second incompleteness theorem. ThereforeLikewise $T_{\infty }$ extends every $T_n$ and contains $\mathrm{Con}(T_n)$ as an axiom already at stage $n+1$, while $T_n$ does not prove it. Consequentlyassuming the stated consistency hypotheses.