We are continuing to discuss Luca’s Chapter 5. The naive comprehension principle — for every property F, there is a set which is its extension — seems intuitively appealing but leads to paradox. So how about modifying the principle along the following lines: for every good property F, there is a set which is its extension (a set of Fs)? Such a principle might inherit something of the intuitive appeal of the unmodified naive principle, but (with a suitable choice of what counts for goodness) avoid contradiction. So what could make for goodness, here? One suggestion that goes back to Cantor, Russell, and von Neumann, is that a property F is good if not too many things fall under it — in other words, we should modify naive comprehension by imposing what Russell called a ‘limitation of size’. How should the story then go?
In §5.2 and §5.3 Luca carefully explores the roots of the Cantorian idea that F is a good property if there are fewer Fs than ordinals. In §5.4. we then meet a proposal inspired by remarks of von Neumann’s: F is good if there are fewer Fs than sets. Luca then goes on to discuss one familiar way of implementing the von Neumann approach, famously explored by Boolos. We add to second order logic a Frege-like abstraction principle that says (roughly) that, when F and G are good, if everything which is an F is a G and vice versa, then the set of Fs is the set of Gs.
But how exactly are we to formulate the required abstraction principle (a ‘New V’ to replace Frege’s disastrous unrestricted Old Axiom V)? And then just how strong a set theory does the resulting Frege-von Neumann theory yield? §5.5 reviews Boolos’s own discussion, explaining Boolos’s New V, and noting that we get as a result e.g. Separation, Choice and Replacement, plus versions of Foundation and Union, but we don’t get Powerset or Infinity. [There is a sense-destroying typo in the displayed formula on p. 144, which also distractingly uses the same variable both free and quantified.] In §5.6, Luca then considers an objection to Boolos’s version of New V (that it allows pathological cases which we don’t want), and introduces a revised version due to Alex Paseau, New V—, which gets round that problem, but still leaves us without Powerset or Infinity. Indeed, Russell had noted this very issue about a limitation of size principle: it tells us that the universe isn’t too big — but this leaves it open that the universe is very small.
Now, one reasonable response to this observation could be “Fair point: but then the pure iterative conception doesn’t by itself tell us how high the universe goes either. The idea that sets are built up in levels leaves it open how many levels there are; and in this respect a limitation of size approach is on a par in leaving it open how big the universe is (other than not being too big!). Both conceptions need supplementation by further thoughts, in particular to give us axioms of infinity.” And this is indeed Luca’s response in §5.7. Which some may find a bit of a surprise, given that the author of §2.1 does seem to build into the iterative conception the idea that there are infinite stages in the iterative hierarchy which are indexed by limit ordinals; and indeed I grumbled a bit in my comments about that section that this seemed to be running together the basic iterative conception with a further thought about the height of the universe. But now Luca seems happy to separate those thoughts more emphatically than he did at the outset. So I’m with him on that! However, he doesn’t tell us what kinds of axiom of infinity would sit most naturally inside the second-order framework of Boolos-style Frege-von Neumann set theory.
Back to fundamentals. For Cantor-style and von-Neumann-style limitation of size principles, the choice of ‘the yardstick of excessive bigness’ seems to be baldly motivated by the requirement that we avoid contradiction. So, it might be argued, limitation of size theories don’t reflect an explanation of what gives rise to the paradoxes in the the way that the iterative conception does: it is just a brute fact that Bigness is Bad. In §5.8, Luca — correctly, I think — defends this line of criticism against an argument from Linnebo that purports to show that developing the iterative conception must also implicitly rely on a limitation of size principle. So, he concludes, “the limitation of size conception’s explanation of the paradoxes is less attractive than the iterative one, because it ultimately rests on the fact that supposing certain properties to determine a set leads to paradox. Following Boolos, this means that the limitation of size conception is not natural.” Though I suppose we might well wonder: does being not-so-explanatory imply being not natural? (Being less explanatory is already a shortcoming, and a charge I find it easier to get my head around than ‘lack of naturalness’).
In §5.9, Luca considers a different way of elaborating the thought that only good properties have extensions, turning to the idea that F is good if it is definite, i.e. not indefinitely extensible — another idea that goes back to Russell. So what if we modify a Boolos-style FN set theory by adopting a revised New V, call it Def(V) — if F and G are definite, the set of Fs is the set of Gs if and only if every F is a G and vice versa? I found Luca’s discussion of this proposal very interesting; and it engages with a thought-provoking recent paper by Linnebo which was new to me exploring ‘Dummett on Indefinite Extensibility’ (Philosophical Issues, 2018). I won’t try to summarize the ins and outs of the discussion here. But Luca concludes that it remains obscure how much set theory we’d get from adopting Def(V), and in any case it is the iterative conception which “explains the key insight of the definite conception that, when faced with the paradoxes, we ought to retain indefinite extensibility at the expense of universality”. [Note, by the way, that there are two potentially troubling misprints in the discussion on his p. 155. The ‘Y’ in the second displayed formula should be another ‘X’; and before the third displayed formula we should read ‘if and only if there is a function \(\delta\) such that the following holds’.]
This chapter leaves me with two related questions. (1) Having raised the possibility of adding an axiom of infinity to a Boolos-style FN framework to give us a more competent set theory, Luca doesn’t tell us more about the options here, and whether the resulting theory would then be stronger than standard (second-order?) ZFC. I’d have really liked to hear more about the how the further story would go here. What would second-order FN set theory with enough of Powerset/Infinity look like? (2) How does a Boolos-style FN relate to the more usual, first-order, theory that is often presented with a Limitation of Size axiom, namely NBG (see §5.1 here)?
But here we will have to leave those questions hanging.