Hereditarily finite sets can be constructed by the rules "construct 0" and "from a and b construct a{b}". For short we write {a,b,c} for 0{a}{b}{c}, e.g. Those sets together with the element relation satisfy the axioms of ZFC without the axiom of infinity. Certain of those sets can be considered to be natural numbers. We investigate an obviously consistent rule system which, however, is not a formal one. It containes a rule with infinitely many premises. By the rules of that system there is deducible the theory of hereditarily finite sets. From this result and a theorem of Jaques Herbrand it follows that a weakened version, zfc*, of ZFC is consistent. By an axiom of it there exists a set containing all natural numbers at least. From zfc* we infer some elementary facts of nonstandard analysis and also consider delta functions.