No, the question here is not "are there statements in _this axiom system_ that cannot be proved by this axiom system" (what incompleteness theorem says), but "a

No, the question here is not "are there statements in _this axiom system_ that cannot be proved by this axiom system" (what incompleteness theorem says), but "are there statements about _the positive integers_ that cannot be proved by this axiom system". In other words, is this system of axioms equivalent to the axioms that define positive integers.