The words "necessarily" and "possibly" do not denote extensional sets. — Leontiskos

Can you say what you mean by this?

Do you mean a sentence with these terms cannot have a truth value, or do you mean they fail at substitutivity always?

...a Tarskian interpretation fixes the domain of quantification and the extensions of all the predicates. Pretty clearly, however, to capture necessity and possibility, one must be able to consider alternative “possible” domains of quantification and alternative “possible” extensions for predicates as well. — From Tarskian to Possible World Semantics.

The trouble with Tarski's system is that there is but one domain, and one interpretation. Kripke's move was to notice that if we consider multiple domains and interpretations, we can use Tarski's approach to analyse modal statements.

It might have been the case that Algol did not become one of John's pets. That would be a change in the interpretation, but not in the domain. The extension of "Is John's pet" would no longer be { Algol, BASIC }, but just { BASIC }.

And in the previous example the domain was { John, Algol, BASIC }. Now it might have been the case that instead John has a pet canary — COBOL (I'm not choosing these names!). The domain would then be { John, Algol, BASIC, COBOL }. Some of the sentences we used would here keep their truth value - that Algol is one of John's pets would remain true. Others would change - that all of John's pets are dogs would no longer be true.

Notice that this latter instance is also a change in the interpretation. The interpretation is a list of which individuals are assigned to which predicates. Adding an individual to the domain changes the interpretation.

That's all a possible world amounts to. A different interpretation of the symbols in a Tarskian system.

In one possible world, the interpretation has the pets as Algol and BASIC. That's the possible world in which it is true that John's pets are Algol and BASIC. In another, the pets are Algol, BASIC and COBOL. In another, Algol is not one of John's pets.

Notice that extensionality survives within, but not between, these worlds.

Here's were we can explain and overcome the accusations from Quine and others that modal logic cannot be treated extensionally.