It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of hereditarily finite sets, developed by Previale in "Induction and foundation in the theory of hereditarily finite sets." This solves some problems stated there. The analysis is undertaken using for each of these results a limited fragment of the relevant theory.
"Finiteness Axioms on Fragments of Intuitionistic Set Theory." Notre Dame J. Formal Logic 48 (4) 473 - 488, 2007. https://doi.org/10.1305/ndjfl/1193667705