Completeness under a stronger finite verification condition

Establish completeness of the natural deduction system NSF with respect to the class of strict finitistic models in which, for every predicate symbol P and every node k, the extension P^{v(k)} at k is finite, replacing the paper’s current finite verification condition that only finitely many predicates are forced at a node.

Background

The paper adopts a finite verification condition requiring that only finitely many predicates are forced at any node, to reflect manageability of actually verified atomic facts. The author notes that, in the propositional setting of Yamada (2023), a more faithful analogue is to bound, per node, the number of variables forced true.

For the first-order setting here, a natural strengthening is to require that for each predicate P and node k the extension P^{v(k)} is finite. The author reports being unable to prove completeness of NSF under this stronger per-predicate finiteness, leaving open whether the completeness result can be recovered for this refined semantic class.