A note on the existence of the integers and rationals

摘要

We examine the role of the replacement axiom and the power set axiom in proving the existence of theintegers (Z) and the rationals (Q). We show that without the power set axiom, the replacement axiom is sufficient for theexistence of Z and Q but if both axioms are removed from Zermelo-Fraenkel set theory, then no infinite Cartesian productscan be proved to exist and thus the existence of Z and Q cannot be proved.