Sharon Berry’s book, A Logical Foundation of Potentialist Set Theory, is the most recent addition to the debate on potentialist set theory. It tries not only to develop a novel modal set theory, but it also claims to justify it properly and show how it can be applied to other debates outside of the foundations of mathematics. A remarkable goal, and a worthy addition to the debate. In this review, after a brief overview of the book’s structure and main claims, I will comment on the many things that Berry’s book does right. Finally, since we philosophers are a polemic bunch, I will point out the (very few) points of Berry’s arguments that I think are doubtful.

A Logical Foundation for Potentialist Set Theory is divided into four parts, plus the Appendices. Part I functions as a general introduction for the whole book, explaining the limitations of the actualist position and the need for a potentialist set theory, a brief overview of Putnamian and Parsonian approaches, and finally an overview of the proposal of the book. Part II includes the main points of the arguments: Berry introduces her proposal in detail, explains how it works, and argues that all the axioms of $\mathsf {ZFC}$ can be deduced from her system. Finally, Part III applies her potentialist system to other debates: the platonism vs nominalism debate, the indispensability argument, if-thenism, problems around physical magnitude statements in the philosophy of language, philosophy of logic, neologicism, and finally pluralism in the philosophy of set theory. After this tour de force, the Appendices contain all the technical details to support Berry’s claim, and in particular, the fully detailed derivations of the results she argued for in Part II. Notice that in the printed books only Appendices A–D are present, while on Berry’s personal website (https://seberry.org/book-tab) the author added Appendices E–M, for a total of 73 new pages (and considering that the whole book is 233 this is a hefty addition)! In these online appendices, Berry adds several auxiliary definitions, a more fleshed out explanation of her modality, important results needed for the technical development of her system (Isomorphism Theorem, Translation Lemmata), and finally the fully detailed derivations of the $\mathsf {ZFC}$ axioms.

Berry’s starting point, common to all potentialist frameworks, is that the actualist position, i.e., the position that the set-theoretic universe $\mathsf {V}$ is a completed unity,Footnote ¹ is inherently vague and indefinite, and it cannot really justify the axioms of $\mathsf {ZFC}$ . On the other hand, a potentialist position can provide a better justification of them, and thus should be preferred (this idea is also at the core of Zermelo’s proof of the quasi-categoricity of $\mathsf {ZFC}_2$ ). Berry’s potentialism is Putnamian: the possibility is merely interpretational, that is, it assumes that the language that we use to interpret the objects existing might have been different (as opposed to Parsonian metaphysical potentialism, according to which there is the possibility of different objects existing). Moreover, she admits only height potentialism, i.e., the possibility of adding new ordinals to the set-theoretic hierarchy, and not new subsets (that would be width potentialism). The possibility that Berry introduces is a structure preserving logical conditional possibility: $\diamond _{R_0, \dots , R_n} \phi $ means that it is logically possible for $\phi $ to be true without any changes to the structural facts about relations $R_0, \dots , R_1$ . So, for example, $\diamond _{\in } \exists X [|\mathbb {N}| < |X| < |\mathbb {R}|]$ means that it is logically possible that there exists a set X with cardinality strictly in between the cardinality of the natural numbers and the real numbers, with no change in the membership relation (so the negation of the Continuum Hypothesis). This logical modality is a step weaker than the usual modalities used in the other potentialist frameworks, in the fact that it doesn’t allow to quantify in the scope of the modal operator. This avoids some conceptual problems, but it also makes the categoricity proof of arithmetic that Berry claims to have somewhat dubious (since at some points there is a need to quantify in the scope of the modality to make it work).Footnote ² Berry proposes a number of inference rules for this modal operator, that together with closure rules, content restriction rules, axioms, and translation functions (between the language of classical $\mathsf {ZFC}$ and her modal language $\mathcal {L}_{\diamond }$ ) form her logical system from which all axioms of $\mathsf {ZFC}$ can be formally deduced.Footnote ³

Berry’s system is perfectly tailored to do what she claims has to be done: it provides a justification of $\mathsf {ZFC}$ ’s axioms by deducing them from prior, more fundamental (in Berry’s argument purely logical) principles that only govern the modal operator. The idea is that we have an intuitive vision of how the set-theoretic universe $\mathsf {V}$ gets populated with sets (in a potentialist manner), and Berry’s modal operator provides a formalisation of this intuitive (potentialist) idea. Then, we can use the modal operator to deduce $\mathsf {ZFC}$ , putting everything on very sure foundations. In particular, this argument avoids the ad hoc objections against the axioms of replacement and separation (very briefly: these axioms are added as a restriction to Comprehension, to avoid the class paradoxes), since in Berry’s system they derive from prior, logical principles. Moreover, it allows Berry to propose a wider, very interesting philosophy of mathematics (this is done in Part III of the book, as I mentioned earlier). According to Berry, her kind of potentialism favours a weak Platonist approach (with some Carnapian conventionalist flavour) to mathematical objects that avoids the objections usually put forward against Platonism by nominalists (in particular, it solves Indispensability-like arguments) and structuralists (like Shapiro’s realist structuralism). Finally, it also integrates a (neo-)logicist reading of mathematics, and avoids the problem of epistemological access to mathematical truth without having to assume a pluralist position like Hamkins. All in all, a hefty list of advantages!

The only problem I have with this argument regards the claimed pure logicality of Berry’s system. She claims (in Ch. 4 and Part II) that her system is logical, and not mathematical, and for this reason it counts as a justification of $\mathsf {ZFC}$ that stands on very good grounds: according to Berry, no mathematical assumptions have been made, and all the axioms and inference rules are clearly true. My objection is essentially Quine’s objection to second-order logic: Berry’s system is set theory in sheep’s clothing. In particular, I don’t think that all her axioms are clearly true and logical: some of them are clearly mathematical in nature, and their clear truth should be argued. Since this is a very central objection, I will develop my argument in detail.

In chapter 8, Berry introduces the formal system for her conditional possibility. She introduces a number of axioms and inference rules that are generally expected: an introduction and elimination rule for $\diamond _{R_0, \dots , R_n}$ , a rule for ignoring it (that allows us to add new relations to content restrict our modality), two comprehension axioms (simple comprehension and modal comprehension), relabelling axiom, importing axiom, logical closure, and cutback axiom. To these axioms, she adds (and this is the fly in the ointment): Possible Infinity, Possible Powerset, Choice, and Possible Amalgamation. I will talk about Possible Amalgamation in a bit, for now let’s focus on the other axioms. Just by looking at the names in the two lists of axioms, it is clear that there is a stark difference between them: the axioms in the first list are clearly logical, and there is not much to object to their introduction (maybe a case can be made about the two comprehension axioms, but I think that they are OK). On the other hand, the axioms in the second list are clearly mathematical in their nature: the axiom of Possible Infinity states that it is possible that there is a successor of an object (with the usual properties of the successor relation), the Possible Powerset Axiom states that it is possible to add a layer of classes to the objects satisfying some predicate, and finally the Possible Choice Axiom is essentially the Axiom of Choice, with an initial $\diamond $ added (so it states that it is possible that there is a choice function). These axioms are all crucially needed to derive their corresponding set-theoretic axioms and some more (in particular, the axioms that we derive from them are Infinity, Powerset, Choice, Replacement, and Pairing). Up to this point there is nothing to object. But the assumption of Berry’s argument was that her derivation of the $\mathsf {ZFC}$ axioms started from purely logical principles. This is true for all the derivations that use only the axioms in the first list, but more doubtful for the ones that use axioms from the second list. The objection here is not that what these axioms state is not logically possible (it clearly is), but that this logical possibility is pure of any “contamination” with mathematics. These are mathematical assumptions, not logical ones. Indeed, it is equally logically possible that there is no successor of an object, no choice function, and no possibility of adding classes to the objects satisfying a predicate. This is essentially Quine’s objection to second-order logic: to make things work, we have to allow back in the logical system some mathematical facts that are not really logical. It is also the reason why Russell didn’t consider the reduction of arithmetic to logic in the Principia Mathematica complete: it needed the Axiom of Infinity, which he couldn’t consider a logical principle.

Of course Berry can reply to this objection by pointing out that $\mathsf {ZFC}$ is our foundation of mathematics, so when trying to derive its axioms from prior principles we need to use the utmost care in using the right one to derive precisely the axioms of $\mathsf {ZFC}$ . Moreover, since we are trying to derive a mathematical theory, objecting the fact that these axioms are mathematical in nature doesn’t seem to make sense. Of course we need to add an axiom of Possible Infinity, otherwise we are not doing mathematics! Finally, she can also point out that these are logical principles because they express logical possibility, and not because of their content (that’s actually mathematical). I believe that these replies can work, even though they cannot completely disperse the doubts, especially for the Axiom of Possible Choice. That is probably the strongest axiom, and carries the implicit assumption that we need to derive precisely $\mathsf {ZFC}$ , not $\mathsf {ZF}$ (that would also be a fine choice of foundational theory), or the possible fragments in between with weakening of $\mathsf {AC}$ . Admittedly, Berry also includes a weaker version of the Axiom of Possible Choice (the Axiom of Simplified Choice), so the possibility of deriving something weaker than $\mathsf {ZFC}$ is clearly on her mind. The last wrinkle here is that Berry needs (at least) the Axiom of Simplified Choice to derive the Axiom of Pairing, since in her system it’s not possible to “quantify in” using the logical conditional possibility. This technical matter limits her system regarding Choice; however, I don’t know at the moment whether it is possible to change the whole system in such a way that the derivation of the Axiom of Pairing avoids the use of Choice (I suspect that this has to be a thorough restructuring, probably too much if we want to preserve Berry’s goals). In summary, I think that this first objection needs to be answered, and having that in the text would have helped in assessing Berry’s assumptions and goals. Some of Berry’s possible replies can be found implicitly in the rest of the book, but having everything laid out could help avoid some doubts.

Another critical point is the Amalgamation Axiom, which Berry introduces in her system to derive the axioms of Infinity and Replacement. The idea behind this axiom is that it is possible that there exist disjoint structures that, at the same time, witness the conditional logical possibility of extending some core structure in different ways. This axiom has two crucial jobs: allowing us to expand our starting structure, thus making the set-theoretic universe $\mathsf {V}$ indeed potentially extendible, and allow us to deduce the Axiom of Replacement, the central axiom that gets us the potentialist flavour of the system, avoids the paradoxes, and the one axiom that seems the most difficult to justify in $\mathsf {ZFC}$ (as Berry states at the start of her book).

I think that there are two problems connected with the Axiom of Amalgamation. The first one concerns its relation with the rule of $\diamond $ -Elimination. This rule states that, if something that is content restrictedFootnote ⁴ to some relations is possible, it’s truth value is completely determined by facts about those relations, and thus we can eliminate the possibility operator and “actualise” it: $\diamond _{R_0, \dots , R_n} \phi \vdash \phi $ . I contend that this is problematic, since it gives us back a certain kind of actualism, and this seems to go counter Berry’s goal. It means that if something is possible, and all the elements (and properties) to make it possible are already realised, then it is not only possible, but actual. In turn, this gives us a very strong linearity to the system: not only there are no unrealised possibilities, but all the actualised possibilities must be mutually consistent. The reason is the Amalgamation Axiom: it states that there might be disjoint extensions of the core structure, but together with $\diamond $ -Elimination and the forced actualisation of all possibilities, this means that these disjoint structures must in some way co-exist. Consequently, they cannot be mutually inconsistent. An example can make this more understandable. Suppose that in our core structure $\mathsf {V}_{\alpha }$ we have the set of all natural numbers $\mathbb {N}$ . At the next stage $\mathsf {V}_{\alpha + 1}$ it is possible that $\mathbb {R}$ comes to exist (notice that, since $\mathbb {R}$ is the collection of all possible subsets of natural numbers, and it is content restricted to the relations present in $\mathsf {V}_{\alpha }$ , it will come to exists at $\mathsf {V}_{\alpha + 1}$ given $\diamond $ -Elimination). Using the Amalgamation Axiom, we discover that there are two disjoint structures that witness how $\mathsf {V}_{\alpha }$ can be expanded: $\mathsf {V}_{\alpha + 1}$ in which there is no subset of $\mathbb {R}$ with cardinality strictly bigger than the cardinality of $\mathbb {N}$ , and $\mathsf {V'}_{\alpha + 1}$ in which this set exists. In other words, in one extension $\mathsf {CH}$ is true, and in the other one it’s false. Such a setup is logically possible, but it is not possible in Berry’s system. This is because the possibilities expressed in both extensions are content restricted to what we can find in the core structure $\mathsf {V}_{\alpha }$ , and, by the application of the $\diamond $ -Elimination rule, they become actualised “by force”. But they cannot be if they are mutually inconsistent, on pain of an overall inconsistency in the system. The only way to avoid this problem is assuming the strong linearity of the system: the extensions are all mutually consistent, even in the case of the disjoint ones of the Amalgamation Axiom. This means that in the case of mutually inconsistent statements, that are nevertheless both consistent with $\mathsf {ZFC}$ (mainly independent statements like $\mathsf {CH}$ ), we need to assume at the outset which way we are settling them, to ensure that all the stages of the construction and all the possible extensions are mutually consistent. In the book, this discussion is not present. I don’t think it is central (for the reasons I will explain in a bit), but I believe it would have been great, especially to connect the debate around potentialism with current debates in the philosophy of set theory surrounding contemporary set-theoretic research (in particular the problem of independence and set-theoretic pluralism).

Berry doesn’t need to take this objection into consideration, and for a good reason. Linearity is not an unexpected consequence of the interplay of her axioms. On the contrary, it is backed in into her system by choice, and so the system works as intended. Indeed, Berry’s potentialist system’s main goal is to justify the axioms of $\mathsf {ZFC}$ using only a height modality. That is, in her system the only extensions of the core structure possible are height extensions, produced by adding new ordinals. This kind of extensions do not change the truth values of independent statements. On the other hand, to actually produce extensions that change the truth value of independent statements we need to add new subsets, thus producing width extensions. To do so we need a completely different set of modalities (one for height and one for width), and this is something that Berry is not trying to do. So Berry fits perfectly in the debate with her choice of linearity. As a further argument, Berry can point out that her goal is to build a system for $\mathsf {ZFC}$ alone, and not to chime in the problem of independence. Consequently, if the system is not built to produce extensions with $\mathsf {CH}$ or its negation, it is working as intended. The system allows us to build all the possible extensions that validate $\mathsf {ZFC}$ . And in doing so, this works perfectly.

It should be clear at this point that the two objections I have against Berry’s book are mainly nitpicking. They are philosophical objections, not at all flaws in her argument or in her book. The fact that I had to come up with objections to the actual content from my philosophical position should be taken as a further testament to the amazing work that Berry did in putting together her system.

In conclusion, I believe that Berry does an amazing job in presenting this novel position and developing it in detail. The rare slips in the presentation don’t change this fact, and I really hope that a second edition, which also integrates all the novel material in the online appendices, can be published someday. This book is a worthy addition to the debate on set-theoretic potentialism, and obviously to the library of any philosopher and logician interested in it.