Wilhelm Ackermann has proved that Zermelo-Fraenkel set theory in which the axiom of infinity is replaced by its negation is equiconsistent to Peano's first order arithmetic. The latter theory has been proved to be consistent by Gerhard Gentzen. We show that from those results and a theorem of Jacques Herbrand - it follows that a modification of ZFC is consistent. Then we consider some consequences of that axiom system.

Especially, we introduce an extension of the usual set of natural numbers and show that it containes an infinitely large element which satisfies a principle of Leibniz, i.e. has all pertinent properties that belong to all sufficiently large natural numbers.

We also introduce an extension of the usual set of rational numbers and show that it is discrete on the one hand and has some similar properties as the set of reals in ZFC on the other.