6 problems

Let ${\sf PA}_n$ denote the set of theorems of Peano Arithmetic of complexity at most $\Pi_n$. A completion of ${\sf PA}_{n+2}$ is a complete consistent extension of this theory, a…

Axiomatization conjecture. The axioms and schemata $(1)$–$(17)$ axiomatize $_2$.

Sound reflection hierarchy classification conjecture. For every $n\in\mathbb{N}$, both of the following hold: (1) there exists a $0^{(n)}$-recursive sequence $(T_n)_{n<\omega}$ of…

Monotone-operator classification conjecture. For some $\beta\preceq\alpha$ and some true sentence $\varphi$, for every $\psi$ such that $T+\psi\vdash\varphi$,

Weakest-extensional-operator conjecture. The consistency operator is the weakest extensional operator that is stronger than the identity operator.

Consistency-operator canonicity conjecture. The consistency operator and its iterates are canonical among recursive extensional functions.