We consider the problem whether the ∀Σ₁b theorems of the fragments T₂ⁿ form a strictly increasing hierarchy. We shall show a link to some results about the lengths of proofs in predicate logic that supports the conjecture that the hierarchy is strictly increasing.
"Fragments of bounded arithmetic and the lengths of proofs." J. Symbolic Logic 73 (4) 1389 - 1406, December 2008. https://doi.org/10.2178/jsl/1230396927