Dusko Pavlovic points out that various data types representing the natural numbers by posets are useful. That is true but hardly related to tree representations in set theory. (For example, these posets are generally not "extensional" in the sense of membership trees.) When I remarked that the operation x|--> {x} has no role in mathematical practice and does not exist in categorical set theory, Dusko replied:
but didn't joyal and moerdijk actually write a book about it?
Joyal and Moerdijk wrote a book on algebraic characterization of models of ZF. They use an operation with formal properties like Peano's singleton. So I have to admit the singleton operation does figure in practice, when the "practice" is to describe ZF and related set theories. Not otherwise. When I said membership trees "obviously play no role in ordinary math practice" he replied
the words "obviously" and "practice" don't go together well. 20 years ago, it seemed obvious that complexity theory was mostly an academic whim. nowadays, the security infrastructure built upon it is a critical part of the engineering practices, and the very life of the net. large cardinals may still find unexpected applications, say in establishing the new tax policies =;0
Membership trees are hardly the same as the study of large cardinals. The large cardinals I know of are all described by isomorphism invariant properties (measurable: an uncountable set k which admits a non-principle k-complete ultrafilter). So the definitions that ZF set theorists give do not rely on membership, they are already definitions in categorical set theory. As to "obvious", we might wish that everything about practice was obscure. It would free up 'debate' wonderfully. But it is obvious right now that membership trees in set theory are used only for a handful of technical theorems in the foundations of set theory. Categorical set theorists also use them, for equiconsistency results with ZF. I don't claim to *prove* they will never have any other use. Perhaps one day they will be central to work in PDEs. Perhaps one day (as Philip Johnson predicts) Bible based biology will produce far greater advances than materialist science as practiced in recent centuries. I only say such claims are arbitrary. best regards, Colin