2.1 Part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets

We start Part 1 with the following story:

The Basic Story. Sets are arranged in stages. Every set is found at some stage. At any stage $\mathbf {s}$ : for any sets found before $\mathbf {s}$ , we find a set whose members are exactly those sets. We find nothing else at $\mathbf {s}$ .

We formalise this by working against the background of second-order logic, and taking two kinds of first-order variables. Variables x, y, z etc. range over sets, and $\mathbf {q}$ , $\mathbf {r}$ , $\mathbf {s}$ range over stages. We also, in addition to set-membership, have three primitive binary predicates (i) ‘ $\in $ ’ for set-membership, (ii) ‘ $<$ ’ for one stage being before another, and (ii) ‘ $\preceq $ ’ is a relation for a set x being ‘found’ at a stage $\mathbf {s}$ .

Since the list of axioms needed is rather short, it’s perhaps worth just stating the axioms of his stage theory or $\mathsf {ST}$ :

• Extensionality: As usual, the axiom that sets with the same members are identical.

• Order: $<$ is a transitive relation on the stages.

• Staging: Every set is a member of some stage.

• Priority: For any set, its members are found before it is.

• Specification: For any property F, if all Fs are found before a stage $\mathbf {s}$ , then the set of Fs is found at $\mathbf {s}$ .

Button then lays down a set theory (i.e., a theory just with ‘ $\in $ ’) called Level Theory (or $\mathsf {LT}$ ). This consists of the familiar axioms of Extensionality and Separation, with a Stratification principle that every set exists at some level, where a level is effectively a set-theoretic proxy of a stage.

On to some important theorems. Let’s start with Theorem 4.1:

$$\begin{align*}\mathsf{ST} \vdash \phi \text{ iff }\mathsf{LT} \vdash \phi\text{ for any }\mathsf{LT}\text{ sentence }\phi.\end{align*}$$

This vindicates the idea that $\mathsf {LT}$ is the ‘right’ set-theoretic correlate of the multiple-primitive $\mathsf {ST}$ ; they prove exactly the same things about the sets.

An interesting payoff of this theorem is that it immediately implies that any cumulative hierarchy is well-ordered. The Basic Story might seem to leave open room to have different kinds ordering on the stages, beyond their length (e.g., partial, dense, or ill-founded orderings). No such orderings are possible.

A final theme of Part 1 concerns different kinds of categoricity. A theory is categorical when it has exactly one model up to isomorphism, and readers may be familiar with the quasi-categoricity result that any two models of second-order $\mathsf {ZFC}$ are either isomorphic or one is isomorphic to an initial segment of the other. Button shows that any set-sized model of Level Theory is isomorphic to a $V_\alpha $ and vice versa (Theorem 6.1). Moreover, he shows an internal deductive full categoricity result. Those wishing to learn about these different kinds of categoricity might consider reading Part B of his Philosophy and Model Theory (co-authored with Sean Walsh).

The axioms of $\mathsf {ST}$ and $\mathsf {LT}$ are rather minimalist; since (almost) any $V_\alpha $ provides a model for $\mathsf {LT}$ , both theories are compatible with the existence of only finitely many sets. As it turns out, there are various level-theoretic bolt-ons that one can consider—roughly corresponding to Infinity, Powerset, and Replacement—which yield the expected interpretability results with subtheories of $\mathsf {ZF}$ .

The conclusion of the paper provides a beautifully concise history of the subject, tracing how the results given emerged from previous work. In particular, the discussion of the Scott and Montague approach (contained within a manuscript that was sadly never published due to Montague’s untimely death) is a historical gem that might be missed on a casual read through. This is an area that merits further study, and Button has done the community a service in drawing attention to it.

2.2 Part 2: Axiomatizing the bare idea of a potential hierarchy

The next two parts can be, in some respects, viewed as variations on the theme of the first. Each takes a slight tweak of the Basic Story, and then shows equivalences with Level Theory and categoricity results.

The second story is tensed:

The Tensed Story. Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.

The setting for formalising the Tensed Story is negative-free second-order logic with an axiom assuming that time is past-directed. There are then three modal operators: asserts that it was the case that $\phi $ , asserts that it will be the case that $\phi $ , and there is an unlimited operator that may be glossed as ‘sometimes $\phi $ ’.

Button then provides a Potentialist Set Theory ( $\mathsf {PST}$ ) using these resources. I won’t state the axioms here; they are simply modal reworkings of the axioms of the Stage Theory of Part 1. Interestingly in this context it is only well-foundedness that can be proved. Linearity (and thus well-ordering) must be axiomatically thrown in.

One can then, similarly to Part 1, consider ‘equivalence’ theses between versions of $\mathsf {LT}$ and $\mathsf {PST}$ . One important distinction here concerns mutual interpretability and definitional equivalence. Mutual interpretability occurs when two theories can each provide an interpretation of another, whereas definitional equivalence (roughly speaking) adds the additional requirement that composing interpretations gets you back where you started. It may be helpful to consider some more mathematically familiar examples; many extensions of $\mathsf {ZFC}$ are mutually interpretable ( $\mathsf {ZFC}$ and $\mathsf {ZFC}$ + $V=L$ for example). However we know from the work of Albert Visser and Ali Enayat (among others) that two non-identical deductively-closed extensions of $\mathsf {ZFC}$ are never definitionally equivalent ( $\mathsf {ZFC}$ is a so-called ‘tight’ theory).

The equivalence results Button provides do not quite achieve definitional equivalence, but they do come desperately close. Button terms his kind of equivalences ‘near-synonymies’. He provides a helpful classification of the different synonymies available. Importantly, he shows that whether one is a necessitist or contingentist about second-order entities (i.e., whether or not one takes the second-order entities to exist out of necessity or not) has a significant role to play—a deductive near-synonymy is not available for full $\mathsf {LT}$ and the second-order contingentist, and a semantic version is only available when the full semantics are employed.

Button then goes on to assess the significance of the near-synonymies. In the case of definitional equivalence, many are happy with the idea that two definitionally equivalent theories ‘say the same thing’ or ‘carve up the same facts’ in different languages. This suggests the Equivalence Thesis that Actualism and Potentialism are different but equivalent ways of expressing the same facts. Button avoids overstating his case, and ultimately concludes that whether or not the Equivalence Thesis is plausible will depend on whether the potentialist can supply a sufficiently clear conception of the the mathematical modality in play. Whilst I would have liked Button to stick his neck out a little more on whether he thinks the Equivalence Thesis hold, I think it was an appropriate choice given the context of an article in the Bulletin.

Similar to Part 1, comparison with other modal approaches—in particular those of Parsons, Linnebo, and Studd—is handled in the conclusion. Whilst not as gripping as the conclusion to Part 1, it is nonetheless instructive for those wishing to learn about some subtleties of setting up modal set theories.

2.3 Part 3: A boolean algebra of sets arranged in well-ordered levels

Part 3 aims to show just how far the Basic Story can be modified whilst retaining categoricity and synonymy results. We are presented with:

The Complemented Story. Sets are arranged in stages. Every set is found at some stage. At any stage $\mathbf {s}$ : for any sets found before $\mathbf {s}$ we find both.

• (Lo): a set whose members are exactly those sets, and

• (Hi): a set whose non-members are exactly those sets.

We find nothing else at $\mathbf {s}$ .

To axiomatise this idea, we have the same primitives as in Part 1 (i.e., $\in $ , $<$ , and $\prec $ ). Extensionality, Order, and Staging are the same as in the Basic Story. Some small tweaks are required to handle complements, however. Call a set ‘low’ if it found using clause $Lo$ and ‘high’ if it found using clause $Hi$ . Priority and Specification then receive relativised versions for the high and low sets. An axiom Cases is also required to ensure that every set is either high or low.

As in Part 1, we define a Boolean Level Theory ( $\mathsf {BLT}$ ) with just the primitive $\in $ . There’s a couple of interesting differences to $\mathsf {LT}$ , notably (1) there is an axiom asserting the existence of absolute complements, and (2) we need a new notion of level (called a Boolean level, or just bevel). Again we have a result (Theorem 3.3) that $\mathsf {BST}$ proves $\phi $ iff $\mathsf {BLT}$ proves $\phi $ , and some examination of what can be proved with extensions of $\mathsf {BLT}$ (e.g., using $\mathsf {BLT}$ analogs of the level-theoretic bolt-ons of Part 1).

We then see that Boolean Level Theory admits of categoricity theorems (Theorems 5.1, 5.2, and 5.3). Moreover it turns out that $\mathsf {ZF}$ and and extension of $\mathsf {BLT}$ are definitionally equivalent (Theorem 7.1). This is despite the fact that $\mathsf {BLT}$ proves that every set has a complement.

The paper closes with some interesting remarks about Conway games and the surreals, showing that one can prove that they form a totally ordered field within $\mathsf {BLT}$ (Theorem 8.5). The appendices contain routine checking of the details, a choice which has rendered the main body of the text very readable.