OK, so what you’re saying is that any predicate that uses a variable in multiple places is recursive. That seems a very unusual notion of “recursive”. Such situations are very common and not at all problematic. What if you’re working with digraphs, with the possibility of self-loops? You can’t handle talking about which nodes do or don’t have edges pointing to themselves? That’s fundamentally all that’s going on here. Not being able to handle that, not being able to look at diagonal conditions, seems pretty broken. Not to mention that any predicate that is a conjuction or disjunction may well include the same variable in multiple places!
It may be possible to make what he is doing more precise or more usable. If he’s objecting to impredicativity then maybe’s he be ok with ZFC or NBG but not KM? (I know that KM is proper with respect to NBG but don’t know much else about it. I’m not sure how relevant this is.)
OK, so what you’re saying is that any predicate that uses a variable in multiple places is recursive. That seems a very unusual notion of “recursive”. Such situations are very common and not at all problematic. What if you’re working with digraphs, with the possibility of self-loops? You can’t handle talking about which nodes do or don’t have edges pointing to themselves? That’s fundamentally all that’s going on here. Not being able to handle that, not being able to look at diagonal conditions, seems pretty broken. Not to mention that any predicate that is a conjuction or disjunction may well include the same variable in multiple places!
It may be possible to make what he is doing more precise or more usable. If he’s objecting to impredicativity then maybe’s he be ok with ZFC or NBG but not KM? (I know that KM is proper with respect to NBG but don’t know much else about it. I’m not sure how relevant this is.)