Axiom V and Intuitions

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.

6 Replies to “Axiom V and Intuitions”

  1. I think it’s worth distinguishing here between Axiom V and naive comprehension. Of course, the former entails the latter really straightforwardly, but it’s striking that the former does even now seem quite a bit more plausible than the latter. (Boolos even claims that the former is ‘obvious’, though I take it that’s to be taken in roughly the same sense as when people claim that we used to ‘know’ that the Earth is flat.) This doesn’t make a difference for some of your points, but it might for the second. It’s not clear that one couldn’t both have iterative conceptiony intuitions and find Axiom V intuitive, given that the clash between these isn’t so immediate as is the clash between the former and comprehension.

    By the way, I worry about the example too, at least as I’ve usually encountered it being presented. But I think that Axiom V is at least a stronger example that naive comprehension, and it’s worth discussing the point in its strongest form.

  2. Just out of curiosity, is what bears repeating that you don’t have the intuition that any predicate has a set for its extension, or that this is not intuitive to you? Or might you be suggesting that no one has this intuition, or that it’s not intuitive for anyone (or not intuitive, period)?

  3. I agree with you to some extent, but try to remember what Law V actually says. It does not say “any predicate has a set for its extension.” It is not about language at all (—is that what you mean by predicates?—), and even when relegated to concepts or properties, it doesn’t actually explicitly make an existence claim.

    It reads:

    The extension of F = the extension of G if and only if, for all x, Fx iff Gx.

    (Rough gloss: All and only coextensive concepts have the same extension,)

    [Actually, Frege’s version is slightly more general than this, but that’s the version most people have in mind when they mention this.]

    The thing is, this statement doesn’t really say that there are extensions, explicitly. It’s more that it presupposes it. In a classical logic such as Frege’s, if you have a term, the term must refer. And to pronounce it as “the extension of F” does seem to give it a kind of meaning which would make Law V analytic.

    So it’s harder to make the case that it isn’t intuitively true — because the existence of extensions is a presupposition, not a direct assertion, of the Law, when you hear it, it does sound eminently plausible. It takes some reflection to see what the problem is. So I guess I tend to think it is very intuitive: it rather shows how and why our intuitions can deceive us.

  4. I think Kevin and Aidan are right. What I should have said is that I don’t find naive comprehension very intuitive. I go back and forth between thinking naive comprehension is counterintuitive, and thinking it is something that intuition is silent on, but I can’t get into a frame of mind of thinking it is intuitive. And without having done any psychological studies, I’m actually a little surprised if other people find it intuitive.

    But that doesn’t extend really to axiom V. As Kevin says, if you state axiom V the right way, so its metaphysical implications are hidden in the right way, it does sound better than (standard statements of) naive comprehension.

  5. My general thought about Basic Law V is that it seems to be the intuitive way that extensions should behave, if they exist. I definitely have moments where my intuitions go the first or the third way that Brian mentions (either nominalist or saying “set theory is too hard for intuitions”) but these are normally intuitions that are shaped by the results of the past century of paradoxes and logic. I don’t think I ever have a strong intuition in favor of the iterative conception (where things are “built up” by applying the power set operation), but there are definitely times where something like an axiom of foundation is part of my intuition. However, I think when I was first learning about sets, talk about “the set of all sets” seemed perfectly reasonable to me, and perhaps slightly exciting precisely because it follows from one intuition (the idea of sets as “extensions”) while going directly against this foundation-like intuition.

    Basically, what I want to say is that there are a couple sets of intuitions going on here, with a foundation-type intuition and an extension-type intuition. But only the latter seems to be sufficient to base set theory just on an enumeration of the intuitions (as Frege did). To build a set theory on the foundation-type intuitions seems to require some more technical work giving the full idea of the iterative conception (or technical work leading to a type-theoretic conception, or something else that bans self-membership).

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