Among the newly published books in the CUP bookshop today, here’s one that could well be of interest to some readers of this blog, namely Sharon Berry’s *A Logical Foundation for Potentialist Set Theory. *

It is, as is now the default for new CUP monographs, published at a ludicrous price. However, it is good to report that there is a late draft downloadable from Berry’s website here (though a Word document, sad to relate, so some of the symbolism is a bit gruesome).

The headline news is that Berry advocates a version of potentialist set theory — as she nicely puts it, “the key idea … is that, rather than taking set theory to be the study of a single hierarchy of sets which stops at some particular point …, we should instead interpret set theorists as making modal claims about what hierarchy-of-sets-like structures are possible and how such structures could (in some sense) be extended.” The virtue of this idea is supposed to be that we can avoid problems that arise from assuming that the height of the universe is fixed (giving us baffling questions along the lines of “fixed *how*?”, “why stop *there*?*”*). There are already potentialist set theories out there in the literature; but Berry gives a new version which depends on deploying a certain natural generalization of the logical possibility operator. Then,

I show that, working in this framework, we can justify mathematicians’ use of the ZFC axioms from general modal principles which (unlike those used in prior potentialist justifications for use of the ZFC axioms) all seem clearly true. This provides an appealing answer to classic questions about how anyone (realist or potentialist) can satisfyingly justify use of the axiom of replacement.

The main work is done in the first two Parts of the book, a bit over a hundred pages. This is indeed interesting stuff. And there is a shorter but attractive exposition of some of the key ideas available here, in a paper jointly by Sharon Berry and Peter Gerdes. (See also and compare Tim Button on potentialism in the second paper of his trilogy linked here.)

The third Part of Berry’s book, another hundred pages, looks beyond set theory, and “turn[s] to larger philosophical questions”. That’s perhaps a mistake, though; from what I have read it might have been better to take some of the Part II topics in a slightly more relaxed and expansive way, increasingly accessibility, and then kept only the most immediately set-theory relevant sections on Part III.

Rowsety MoidI don’t understand the ‘height’ issue. Who assumes that the height of the universe is fixed and thus faces the baffling questions? (And if the answer is ‘no one’, then why is this a problem that needs a solution?)

Peter SmithAs Sharon Berry puts it, “the height of the hierarchy of sets is left vague or mysterious”. Or in so far as gestures are made, they tend to have a modal flavour (however high you go, you can go higher). So it is natural to explore the options for cashing out this idea in some reasonably well understood modal framework — and (rightly or wrongly) Berry thinks that there is a real pay-off from making the effort. Namely, we get a better justification for the standard axioms and for Replacement in particular [NB, the book isn’t intended to be revisionary about set theory]. Or that’s the sales claim!

Rowsety MoidDeveloping a better justification for Replacement makes sense to me. It’s the ‘height’ issue I don’t understand, if it’s about assuming or (using terms from the linked paper) positing / commitment to / designating an arbitrary height where ‘the actualist hierarchy of sets just happens to stop’. Because who does that? I’m expecting some names and the heights they assume. Else why think it’s a real problem?

Leaving the height ‘vague or mysterious’ seems quite different, though perhaps the idea is that, behind the vagueness and mystery, there is still some arbitrarily chosen height.

Perhaps that is indeed the idea, because the Berry and Gerdes paper says on p 5 (emphasis added):

Who posits that? Name names, Berry and Gerdes!

Sharon BerryHi Rowsety and Peter Smith, and thanks very much for your interest!

My longtime policy is not to interact with namespace social media.

However, I feel obliged to mention that the ancient coauthored draft linked above was written on a bet to see whether I could motivate interest in my book project (as it was then) without mentioning the conditional logical possibility operator*. My dear husband Peter Gerdes accused me of blocking interest in my papers by using that logical novelty gratuitously (and being confusingly fretful about heading off philosophical misunderstanding). So I agreed to create an article following all his writing suggestions. Peter Gerdes has his own views on the philosophy of mathematics (in the neighborhood of Putnamian internal realism as interpreted by Tim Button) and shouldn’t be held accountable for any flaws in the content of that paper.

Also if you (Rowsety) or anyone later reading this has questions/comments, I would love to hear and discuss your thoughts! Please drop me a line at seberry@invariant.org if you’re interested (but no pressure).

The book got heinously long due to a silly misunderstanding about word count for equations and my general cowardice. However, here are some key sections that I think/hope readers of this blog might be interested in checking out (in the final version or the penultimate draft linked above):

• Chapter 3 says a bit about what I take the problems for extant versions of the “Putnamian” (aka minimalist) potentialism to be.

• Chapter 4 introduces and explains the conditional logical possibility operator, and broadly indicates how we might use it to reformulate potentials set theory in a way that solves these problems.

• Chapter 5 (sections 5.6 and 5.6) of the book raise a bunch of worries about Linnebo and Studd’s “Parsonian” (aka dependence theoretic) potentialism.

• Chapter 2 Section 5 (perhaps hubristically) states a view about Dummett’s motivation for accepting something like indefinite extensibility vs. mine.

• Chapter 8 Section 12 gives the key/most controversial modal axiom for my justification of Replacement (and Chapter 7 explains the crucial notion of content restriction used in stating most of my axioms)

*https://philarchive.org/rec/BERMSS-2]