Did Cohen need regularity?

Question

In his article "THE INDEPENDENCE OF THE CONTINUUM HYPOTHESIS", Paul Cohen writes in the beginning:

We shall work with the usual axioms for Zermelo-Fraenkel set theory, and by Z-F we shall denote these axioms without the Axiom of Choice, (but with the Axiom of Regularity).

Thus the meta-theory he is working in is the Zermelo-Fraenkel set theory without AC, but with the axiom of regularity. In this meta-theory, Cohen proves his famous result that CH is indepedent of ZFC. Now I wonder:

Did he use the regularity axiom of his meta-theory in his proof?

Answer 1

If he did use Regularity in some of the argument, then he still didn't have to, because he could just have carried out everything relative to WF, the class of hereditarily well-founded set. WF satisfies ZF with Regularity even if the ambient universe does not satisfy Regularity.

Becuase the consistency of this theory or that is just a matter of arithmetic (thanks to Gödel) and $\omega$ is the same within WF as outside it, this would lift any relative consistency result from WF to a weaker meatheory.

Answer 2

When $M$ is a ground model and $P\in M$ is a poset and $A$ is a $P$-name over $M, $ then $A$ is a set of ordered pairs $(x,y')$ where $x\in P$ and $y'$ is a $P$-name over $M.$ If $G$ is a $P$-generic filter over $M$ then $A_G=\{y'_G: \exists x\in G\;((x,y')\in A\}.$ Now we can't assume $A_G$ exists without Replacement: By Comprehension we have the existence of $B=\{(x,y')\in A: x\in G\}.$ And $\forall (x,y')\in B\;\exists ! z \;(z=y'_G\}.$ But we must apply Replacement to this formula to conclude that $A_G$ exists.

Remark: Cohen's paper considered only forcing over $P$ \ min$(P)$ when $P$ is a complete Boolean algebra. But all forcing can be done this way; it is not always convenient to do so.