I know that I’ve said this before, but given that the issue has come up again, I think it bears repeating.
The standard example people use of something that’s intuitive but not actually true is Frege’s Axiom V. And that axiom, remember, says that any predicate has a set for its extension. Apparently this is intuitive. I don’t find it even approximately intuitive. I’ve been introspecting a bit, and I’ve noticed the following things.
- Two days a week, my intuitions are nominalist and so don’t accept any sets, let alone sets for any predicates.
- Two days a week, my intuitions have an iterative conception of sets, so there couldn’t possibly be a set of all sets. (How could it get built, since it would have to have itself as a constituent? That’s very counterintuitive.)
- Two days a week, my intuitions are modest and say that set theory is too hard for them, and that they aren’t going to issue verdicts about anything to do with set theory.
- And on Sundays I watch football.
I don’t doubt that there are some things that are true but counterintuitive. Some of the simple results about comparative advantage in trade theory are counterintuitive but true, for instance. It’s counterintuitive, but I think true, that Mary learns nothing when leaving the black and white room. But I’m very sceptical that Axiom V is one of the counterintuitive truths.