  However, the expression “Ordinary mathematics” is disputed turf. On another front there are logicians that are fond of making the point that some of the further reaches of large cardinal theory, routinely disregarded – and even disparaged – by many mathematicians nevertheless have implications for a kind of low-level mathematics whose core significance is not in dispute. Inevitably the question of the adequacy of NF for “Ordinary mathematics” will overlap with this debate. However there is a further complication to be considered. All these big collections that implement the collection of all cardinals (or of all ordinals etc.) are nevertheless sets, so there are questions arising from their sethood that we can ask about them, and some of these questions have no parallel in ZF. What is the cofinality of the wellordering of the ordinals? Can we map onto \(V\) the set of those sets that do not map onto \(V\)? No doubt there will be people who say that these questions are not proper mathematical questions at all, but are mere pathological artefacts of a misconceived system of axioms. However, there is the intriguing possibility that answers to these questions could have implications for the study of the kind of small sets that appear in “ordinary mathematics”. No such implications have been found so far, but then no-one has looked very hard. We know of subclasses of the naturals definable by formulæ that have big sets as parameters, so we know where to start looking. It is tempting to speculate what the consequences might be for NF studies were any to be discovered.