Cofinality of System T Dialogue-Tree Heights

Determine whether the heights of dialogue trees D(t) associated with closed System T terms t:(ι→ι)→ι are cofinal in ε₀, the supremum of the finite ω-exponential towers.

Background

The paper proves that every dialogue tree D(t) generated by a closed System T term t:(ι→ι)→ι has height strictly below a finite ω-exponential tower, with the tower height computed syntactically from the type levels occurring in the source term. It also identifies the supremum of these finite towers with ε₀. The unresolved question is whether the collection of actual dialogue-tree heights reaches arbitrarily high ordinals below ε₀, equivalently whether these heights are cofinal in ε₀. The paper does not establish such cofinality and explicitly leaves it as a question for further determination.