Iterated consistency progression
= Iterated consistency progression
Starting from a recursively axiomatized theory $T_0$, define
$$
T_{n+1}=T_n+\operatorname{Con}(T_n).
$$
Assuming consistency, <Gödel second incompleteness theorem> makes every step strict in the <consistency-strength preorder>, while the effective union $T_\omega=\bigcup_{n<\omega}T_n$ lies strictly above every finite stage.