That is said more exactly, as I did in my previous post: The axiom of regularity provides that each set is in the cumulative hierarchy.

Again, no, that's wrong. Axioms don't prevent paradoxes. It is by REFRAINING from certain axioms that we get by without contradiction.

There is no finite limit ordinal anyway. And the existence of a limit ordinal falls right out of the axiom of infinity. And the poster didn't show anything. He's confused.: There DOES exist an enumeration of the sets he mentions.

There's no such thing in ZFC.

We don't need forcing merely to conceive the notion of strongly inaccessible cardinals. The notion is definable in rather simple set theoretical terms. As to existence: To prove that there exist inaccessible cardinals requires an axiom beyond ZFC. No forcing or anything else will provide the existence of inaccessible cardinals in ZFC; only by extending the ZFC axioms do we prove that there exist inaccessible cardinals.