7 problems

Consider an inferentialist analysis of second-order arithmetic, induction principles, and the definability of concepts within a framework grounded in the structure of atomic system…

Let $P_{\alpha}$ be the ramified analytical hierarchy, with fixed point $P_{\beta_{0}}=P_{\beta_{0}+1}$, and let $\underline{P}_{\beta_{0}}$ denote the corresponding full structure…

Let $W(r,l)$ denote the least number $n$ such that every $r$-coloring of $n$ has a homogeneous arithmetic progression of length $l$. Provability conjecture. Van der Waerden's theor…

Rathjen's conjecture. Adding full Separation to CZF should elevate the theory's proof-theoretic strength from $\mathrm{ID}$ to full Second Order Arithmetic.

Ordinal-analysis conjectures. The following three equalities are conjectured:

Let $\mathrm{DNR}$ denote the diagonally nonrecursive axiom and let $\mathrm{WWKL}_0$ denote the corresponding subsystem of second-order arithmetic. For theories $T_1$ and $T_2$, w…

Bi-Interpretability Conjecture. The Turing definable relations are exactly those whose information content is describable in second-order arithmetic. Moreover, every description of…