Possible Thesis Topics

• There is a standard trick of Ackermann’s that makes IN look like Vω: ordain that n E m iff the nth bit of (binary) m is 1. There is a suggestion in [4] that there is an analogue for Church’s set theory CUS, that in some sense it contains no mathematics not already in ZF—that the existence of a universal set is just a trompe l’oeil trick. There is a mathematical notion of synonymy which is in play here. (The theories of Boolean rings and of Boolean algebras are synonymous, for example. See work of Visser and Friedmann.) The question is: “is CUS synonymous with ZF?” My guess is that the answer is a fairly straightforward ‘yes’—possibly with a bit of tweaking. (Mind you, i know of no proof; i did try to get David Matthai to sort this out but i was unable to twist his arm hard enough). So far so good. One can of course ask the same question about NF: “Is NF synonymous with some theory of wellfounded sets?” and that question is going to be a lot harder. It’s an interesting question beco’s there is the thought that perhaps the mathematical world painted by NF is so alien that NF cannot be synonymous with any ZF-like theory. I’m not sure what a proof of such an assertion would look like! Any result like that would provide a powerful argument for studying large sets in NF, beco’s it would mean that the large sets are not just wellfounded sets-in-drag (as they are in the Church-Oswald models) but are genuinely alien things—and what NF is telling us about them cannot be faulted on the grounds of mere inconsistency. (NF might still prove false theorems of course, but at this stage we have no means of detecting them. And the mere fact that it contradicts ZF doesn’t give any trouble because it seems that NF does not contradict what ZF has to say about wellfounded sets.)