Replacement and Other Axioms

Abstract

We have just about reached one of the goals we set in Chapter 4, which was to prove all the “naive” results of Chapter 2 from the axioms of Zermelo. Only a couple of minor points remain, but they are significant: they will reveal that Zermelo’s axioms are not sufficient and must be supplemented by stronger principles of set construction. Here we will formulate and add to the axiomatic theory ZDC the Axiom of Replacement discovered in the early 1920’s, a principle of set construction no less plausible than any of the constructive axioms (I) – (VI) but powerful in its consequences. We will also introduce and discuss some additional principles which are often included in axiomatizations of set theory.