As SCOTT has shown, the Replacement. scheme of Z F derives a large part, of its strength from the Extensionality axiom. For in the absence of the latter, the supply of demonstrably functional formula matrices is relatively ineagei..l) I n this situation, u-e can restore some of Replacemmt's vigor by increasing the intensional resources of our language, i.e. by augmenting our ability to individuate properties by means of formulas which charactmize tht-m. (I say "properties " rather than "sets " because we are assuming th.:. absence of Extensionalit,y.)z) I would like to point' out three strategies for strengthening a property theory, two of v.hic!i involve a strengthening of Replacement:
- We could adopt an axiom (or axiom scheme) which, just by itself, supplies us wit.11 extensions whose existence could not previously be demonstrated. For example, we could add an axipm of infinity to a theory which previously only guaranteed the existence of finite prop-rties.
2 .
We could adopt an axiom (or axioni scheme) which, just by itself, do,-s not supply us with any new extensions, but which allows us to individuate proporties which could not previously be distinguished from their extensional equivalcnt,s. Such an axioin could eith.?r expand our stock of demonstrably functional predicates or allow us to prove that the range of some such predicate contains desirable extensions. I n ,either case, it might supply us indirectly, via Replacement, with new extensions.
We could adopt an axiom (or axiom scheme) which. just by itself, neither grants u s new extensions nor allows us to fix upon individuals from any category of previouslg indistinguishablo propwties, but which allows us to individuate old propsrties in nrw ways: by means of a ncw category of intensions. Sucb an axiom might supply us indirectly, via Replacement', with new extensions as in strztegy 2.