## Luca Incurvati’s Conceptions of Set, 11

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. . .

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