Predicative Classes and Strict Potentialism Open Access

Abstract

1 Introduction

Within set theory, it is customary to distinguish between sets and classes. On pain of contradiction, there is (in ZF) no set of all sets, but, at least on some views, there is a class of all sets. It has a name, V. Others have objected to this talk of classes, or at least proper classes like V that are not sets. Boolos [1998], for example, rejected (proper) classes altogether:

Wait a minute! I thought that set theory was supposed to be a theory about all, “absolutely” all, the collections that there were and that “set” was synonymous with “collection”. If one admits that there are proper classes at all, oughtn’t one to take seriously the possibility of an iteratively generated hierarchy of collection-theoretic universes in which the sets which ZF recognizes play the role of ground-floor objects? I can’t believe that any such view of the nature of [membership] can possibly be correct. Are the reasons for which one believes in classes really strong enough to make one believe in the possibility of such a hierarchy?

Whether you agree with Boolos or not, he certainly asks a good question. If classes aren’t just sets that we “forgot to form”, what are they?

A promising line of response is to distinguish two different conceptions of “collection”, the combinatorial and the logical (see, for example, [Parsons, 1974] and [Maddy, 1983]). On the combinatorial conception, collections are determined solely by their members; combinatorial collections are rigid and complete. At least in part, this is expressed by an extensionality principle: collections are identical just in case they have the same members. Of course, the sets of ordinary ZFC set theory are combinatorial. By contrast, on the logical conception, a collection is characterized in terms of its membership criterion, say, as |$\{x:\varphi(x)\}$| for some suitable condition |$\varphi(x)$|⁠. Thus, in different circumstances, the members of a logical collection may be different. One can then take classes to be logical collections.

The difference between (combinatorial) sets and (logical) classes becomes particularly stark when we consider the domain of sets as potential, as generated in stages. Each set, once generated, is thoroughly extensional–it has the same members at every stage at which it exists. Classes, or at least proper classes like V or Ω, are not like this. They “grow”, or gain more members, as more sets are generated.¹ They cannot be completed. In this context, classes are intensional, more like properties. This suggests a response to Boolos’s complaint. It does not make sense to think of classes like V or Ω as sets, in order to start a longer iterative hierarchy. As incompletable collections, proper classes are inherently unsuited to generate more sets.

To spell out this (so far) entirely programmatic response, potentialists need an account of how logical classes are generated. This is where predicativism comes in. The familiar iterative conception of sets provides a natural and systematic account of the generation of combinatorial sets. Predicativism, we argue, provides a similarly natural and systematic account of the generation of logical classes. If we are right, then predicativism about classes will provide the desired non-ad hoc response to paradox and to Boolos’s “wait a minute” objection concerning (proper) classes. Each set is completed and therefore rigid, whereas classes are property-like and capable of “growth”.

Our main aim is thus to explore and develop a predicative approach to classes. In principle, the classes can be classes of just about anything: classes of natural numbers, classes of real numbers, classes of frogs, classes of tornadoes (ignoring vagueness in the last two examples). To stay far from paradox, we will not consider classes of classes, at least not here. Our primary example will be classes of sets.

This example leads to an important difference vis-à-vis the more familiar study of predicative classes of natural numbers. In both cases the domain of predicative classes is understood as merely potential. However, studies of predicative classes of natural numbers typically take an actualist view of the domain of numbers.² Each stage (or each world) has the same natural numbers, namely all of them. In principle, we could think of the sets too as existing all at once. But we suggest it is better to think of the sets too as generated in stages (following Linnebo [2013] or Hellman [1989], among others). This means that not only the domain of classes is merely potential but also the domain of objects from which the classes draw their members. And this, in turn, leads to classes that can “grow” as more objects are generated and thus come to satisfy the membership condition that defines the relevant class. Despite being inherently interesting, such “growing” classes have not received much attention.³

Additionally, our analysis of predicative classes of combinatorial sets has some surprising consequences concerning potentialism more generally. The dominant form of potentialism about sets has been a liberal one, which insists that each object is generated at some stage of an incompletable process. Our analysis reveals some new reasons to adopt a stricter form of potentialism that adds an analogous requirement concerning truths: loosely speaking, that each truth be “made true” at some stage of the incompletable process. The natural logic of this strict form of potentialism turns out to be semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic. Thus, when more classically minded potentialists about sets add a theory of predicative classes, some new and surprising reasons emerge to adopt a stricter form of potentialism, based on semi-intuitionistic logic.

The rest of the paper is organized as follows. The next three sections provide a potentialist approach to predicative classes of sets. First, §2 provides an overview of potentialist treatments of various types of mathematical objects, including a few choice-points along the way. Next, §3 turns to predicativist accounts of classes. We show that predicative classes are naturally understood in a potentialist manner. Then, §4 develops a potentialist account of predicative classes of sets, inspired by our earlier account of predicative classes of natural numbers [2023], but with some important differences to allow the classes to “grow”.

We turn, then, to the advertised reasons to adopt stricter form of potentialism. §5 analyzes a revenge problem concerning classes. If we have made sense of classical quantification over a potential hierarchy of predicative classes, this can be used to define yet more such classes–thus undermining our attempt to talk about all of them. Our resolution is to restrict ourselves to intuitionistic logic when defining classes: one should not invoke Excluded Middle, and other strictly classical theses and inferences in those contexts. This, we argue, is well-motivated in a way that is quite independent of the anti-realism often associated with intuitionistic logic. Then, §6 examines a different revenge problem, this time concerning sets. Since it appears possible to iterate the generation of classes to Ω and beyond, why shouldn’t the same–absurdly–go for sets? The best responses, we argue, involve a stricter form of potentialism, not only about classes (as required by the first revenge problem), but also about sets. Finally, §7 summarizes and concludes. There are also two appendices.

2 A brief guide for potentialism

The emphasis here is on the word “brief”. For more details, see [Linnebo, 2013; Linnebo and Shapiro, 2019], or [Linnebo and Shapiro, 2023, §3].

2.1 A Modal Analysis

Aristotle famously rejected the notion of the actual infinite–a complete, existing collection with infinitely many members. This attitude toward the infinite was echoed by the majority of mathematicians and philosophers at least until late in the nineteenth century. Aristotle did accept what is sometimes called potential infinity. Subsequent mathematicians followed this, and, indeed, made brilliant use of potential infinity.

But what is potential infinity? The notion can be motivated by considering procedures that can be repeated indefinitely. A nice example is provided by Aristotle’s claim, against the atomists, that matter is infinitely divisible. Consider a body w of water. However many times one has divided the water, it is always possible to divide it again (or so it is assumed).

By endorsing (1) and (3), one is asserting that the divisions of the body of water are (merely) potentially infinite.

A note on methodology: it is often useful to invoke the heuristic of possible worlds when discussing the modality in question. Here we insist that this is only heuristic, as a manner-of-speaking. Our official theory is formulated in the modal language, with (one or both of) the modal operators as logically primitive. The operators are not explained in terms of anything else.

2.2 The Modal Logic of Potentiality

To invoke the heuristic, the idea is that a possible world has access to other possible worlds that contain objects that have been constructed or generated from those in the first world. From the perspective of the earlier world, the “new” objects in the second exist only potentially.

An Aristotelian would insist that every possible world is finite, in the sense that it contains only finitely many objects. As noted, we make no such assumption here. An actual infinity–or, to be precise, the possibility of an actual infinity–is realized at a possible world if it contains infinitely many objects of the given kind.

For present purposes, we can think of a possible world as determined completely by the mathematical objects–regions, numbers, sets, classes, etc—it contains. This motivates the following principle:

Partial ordering: The accessibility relation |$\leq$| is a partial order. That is, it is reflexive, transitive, and anti-symmetric.

So the underlying logic is at least S4. So far, then, we have S4 plus (CBF).

The constructions we envisage here all have this property. So we adopt the following principle:

Convergence: The accessibility relation |$\leq$| is convergent.

The modal propositional logic that results from adding (G) to an axiomatization of S4 is known as S4.2.

2.3 Reasoning About the Potentially Infinite

The language of contemporary mathematics is strictly non-modal. The potentialist thus needs a translation to serve as a bridge connecting the non-modal language in which mathematics is ordinarily formulated with the modal language in which our analysis of potentiality is developed. Suppose we adopt a translation * from a non-modal language |$\mathcal{L}$| to a corresponding modal language |$\mathcal{L}^\Diamond$|⁠. This translation, together with the modal logic, will tell us which entailment relations obtain in the non-modal language |$\mathcal{L}$|⁠. With a little more detail, in order to determine whether |$\varphi_1,\ldots,\varphi_n$| entail ψ, in the non-modal system, we need to (i) apply the translation and (ii) ask whether |$\varphi^\ast_1,\ldots,\varphi^\ast_n$| entail |$\psi^\ast$| in the modal system.

The heart of potentialism, as we see it, is the idea that the existential quantifier of ordinary non-modal mathematics has an implicit modal aspect. Consider the statement that a given number has a successor. For a potentialist about the natural numbers, this is a proposition that each number potentially has a successor–that it is possible to generate a successor. This suggests that the right translation of |$\exists$| is |$\Diamond\exists$|⁠. Dually, when a potentialist says that a given property holds of all objects (of a certain sort), she means that the property holds of all objects (of that sort) whenever they are generated. This suggests that |$\forall$| be translated as |$\Box\forall$|⁠.

Let us call the strings |$\Diamond\exists$| and |$\Box\forall$|modalized quantifiers. Our proposal is thus that each quantifier of the non-modal language is translated as the corresponding modalized quantifier. Each connective is translated as itself. Let us call the result the potentialist translation, and let |$\varphi^\Diamond$| represent the translation of |$\varphi$|⁠.

Intuitively, a formula is stable just in case it never “changes its mind”, in the sense that, if the formula is true (or false) of certain objects at some world, it remains true (or false) of these objects at all “later” worlds as well. For example, set-theoretic membership |$\in$| is stable. If a is (or is not) an element of b, it will remain so no matter what further objects are generated.

We now state two key results, which answer the question about the correct logic for those kinds of potentiality that enjoy the above convergence property. For the first, let |$\vdash$| be the relation of classical deducibility in a non-modal first-order language |$\mathcal{L}$|⁠. Let |$\mathcal{L}^\Diamond$| be the corresponding modal language, plus the decidability of all atomic formulas of |$\mathcal{L}$|⁠. Let |$\vdash^{\Diamond}$| be deducibility, in this corresponding language, by |$\vdash$|⁠, S4.2, and axioms asserting the stability of all atomic predicates of |$\mathcal{L}$|⁠.

2.4 Different Orientations Towards the Infinite

We can distinguish several opposing orientations to infinite collections. First, actualism simply accepts actual infinities of the given kind. So actualists find no use for modal notions in explicating the relevant type of collection. The non-modal language of ordinary mathematics is already fully explicit. Most actualists accept classical logic when reasoning about these infinite collections. Potentialism is opposed to actualism. According to it, at least some of the collections with which mathematics is concerned are generated successively, and at least some of these generative processes cannot be completed.

There is room for disagreement among potentialists concerning which processes can be completed. The traditional Aristotelian form of potentialism takes a very restrictive view, insisting that at any stage, there are never more than finitely many objects, but that we always (ie, necessarily) have the ability to go on and generate more. A potentialism about set theory (such as those of [Linnebo, 2013] and [Hellman, 1989]) would take a generative process that is indexed by a set-theoretic ordinal to be completable, but insist that it is impossible to complete the entire process of forming sets. The so-called “relative predicativism” of Feferman [1964; 2005] and others lies in between the traditional Aristotelian orientation and potentialized set theory. The view accepts the natural numbers as a complete, actual infinity, but insists that sets of natural numbers are defined in stages, and there is no stage at which all sets of natural numbers exist together, so to speak.

Potentialists can also differ from each other with respect to a more qualitative matter. It concerns the truths of mathematics. As above, on any form of potentialism, these are understood as modal truths about certain generative processes. But how should these modal truths be understood? Liberal potentialists adopt bivalence for the modal language.⁵ Their approach to modal theorizing in mathematics is thus much like a realist approach to modal theorizing in general: there are objective truths about the relevant modal aspects of reality, and this objectivity warrants the use of some classical form of modal logic. Strict potentialists disagree. They insist not only that every object (of the relevant kind) be generated at some stage of a process, but also that every truth be “made true” at some stage. Linnebo and Shapiro [2019] argue that strict potentialists should adopt a modal logic whose underlying logic is intuitionistic (or intermediate between classical and intuitionistic logic).⁶

3 Predicativism as a Form of potentialism

It is illuminating to analyze predicativism as a form of potentialism. We will now explain how this is done and highlight some of the choices to which this approach gives rise.

3.1 From the Vicious Circle Principle to Definitional Stability

As noted above, we think of classes as predicative. But what is that? A definition is said to be impredicative if it defines an entity by using quantifiers whose range includes the entity to be defined; otherwise, the definition is predicative. Bertrand Russell [1908] once argued that impredicative definitions are circular, and thus illegitimate, via what he called the Vicious Circle Principle:

No totality can contain members defined in terms of itself. [Russell, 1908, p. 237]

Or later:

…whatever in any way concerns all or any or some of a class must not be itself one of the members of a class. [Russell, 1973, p. 198]

One now classic paper [Feferman, 1964] highlights another aspect of predicativism, namely the view that some totalities are inherently potential:

|$\dots$| we can never speak sensibly (in the predicative conception) of the “totality” of all sets as a “completed totality” but only as a potential totality whose full content is never fully grasped but only realized in stages.

As the context makes clear, the sets in question are what we are calling classes of natural numbers. As noted above, Feferman, and those who follow his program, take the natural numbers to be a completed, actual infinity.

A similar view of sets is found in the work of Henri Poincaré, the father of predicativism. He gives two prima facie different diagnoses of the logical and set-theoretic paradoxes. He blames the paradoxes on the use of viciously circular definitions (see, for example, [Poincaré, 1906, §XI]) on the Cantorian assumption that there are completed infinite collections:

There is no actual (given complete) infinity. The Cantorians have forgotten that, and they have fallen into contradiction. [1906, §XI]

Poincaré thus insists that there are no completed infinite collections. In later work, he insists on a potentialist conception:

[W]hen we speak of an infinite collection, we mean a collection to which we can add new elements unceasingly (similar to a subscription list which would never end, waiting for new subscribers). [Poincaré, 1909, p. 463] ([1913, p. 47])

This much, of course, is similar to Aristotle’s conception of infinity as merely potential.

Poincaré notes some challenges to which this view gives rise. Consider attempts to “classify” the elements of one of these potentially infinite collections. In some cases, this poses no problems. For example, the classification as being among the first 1000 subscribers to a certain newspaper is undisturbed by later additions to the subscription list. In other cases, however, “the principle of the classification rests on some relation of the elements to be classified with the entire collection” (pp. 46–47). An example is the classification as being the youngest subscriber to the newspaper. Although a fifteen-year old may be thus classified today, tomorrow that honor might go to a new subscriber who is only twelve years old. Poincaré concludes:

From this we draw a distinction between two types of classifications applicable to the elements of infinite collections: the predicative classifications, which cannot be disordered by the introduction of new elements; the non predicative classifications in which the introduction of new elements necessitates constant modification.

That is, a classification of the elements of a potentially infinite collection is deemed predicative or not in accordance with whether or not the classification can be “disordered” (bouleversé) by later additions to the collection. So for Poincaré, the stability of classifications is the very heart of predicativity.⁷

In words: a condition is stable just in case, whenever an object a is generated, a is immediately “classified” as either satisfying the condition or not, and this classification will never be “disordered” or overturned. We will later consider various explications of the “stages” and of what it is for a stage to “say” that something is the case.

The easiest and most natural way to ensure that a classification is stable is to restrict all of its quantifiers to elements of the potentially infinite collection that have already been generated. For plainly, if a classification disregards any objects not yet generated, it will not be “disordered” by the generation of any new objects. We have thus connected the two aspects of predicativism. Where potentialism poses a threat to the stability of definitions of classes, the Vicious Circle Principle–understood as the requirement that the definition of a new object may only quantify over objects already available–provides a natural and effective way to defuse that threat.

This connection tells against Kurt Gödel’s influential claim that the Vicious Circle Principle is justified only if one is a constructivist: “it seems that the vicious circle principle […] applies only if the entities are constructed by ourselves” [Gödel, 1944, p. 136]. On our account, the motivation derives from potentialism, not from a metaphysical view that mathematical objects are literally our constructions.⁸ When it comes to motivating the Vicious Circle Principle, what matters isn’t constructivism versus realism but potentialism versus actualism. Potentialists of all stripes–whether anti-realist constructivists or robust realists–have a reason to invoke predicativity restrictions to ensure the stability of their definitions.

3.2 Stability in a Potentialist Setting

What classes are permissible from a predicativist point of view? Let us first clarify the question. As in [Linnebo and Shapiro, 2023], we use a higher-order (or two-sorted) language. The first-order variables range over the objects that are eligible to figure as members of classes. In [Linnebo and Shapiro, 2023], these objects are natural numbers. Here they can be just about anything, but our primary example is sets.

On the potentialist approach, by contrast, we use modality to do the work previously done by Russell’s and Whitehead’s order indices. Instead of quantifying solely over classes of certain orders to define a class of a yet higher order, the idea will now be to quantify solely over available classes to define a new class that is merely potential relative to the classes over which its definition quantifies. This results in a more streamlined theory, with greater expressive power. In the ramified theory, every class variable has an order index that restricts which classes it is permitted to range over. By contrast, our plain class variables range over any classes available at the relevant world. We thus avoid “hardwiring” restrictions on the ranges of the class variables by means of syntactic order indices. We simply use plain class variables, whose ranges we explicitly restrict when and as needed.

For which formulas |$\varphi(x)$| is this principle permissible?

That is, for (M-Comp) to be permissible, the condition |$\varphi(x)$| needs to be stable.¹¹

It is important to notice that the requirement that every class be stable still allows classes to “grow” as the generation of the merely potential domain unfolds. The stability requirement applies only to objects that have been generated. It leaves open the possibility that as further objects are generated, some of these will “become” elements of the class. To illustrate, consider classes of natural numbers. Suppose that natural numbers too are successively generated, as Poincaré (and Aristotelians) would have it. Consider the formulas ‘n = n’ and ‘n is prime’, which seem to be predicative. If these formulas are allowed to define classes of numbers, then the resulting classes would “grow” as new numbers are generated. To use familiar terminology, a collection is rigid if it cannot exhibit this kind of growth.¹² While the requirement that classes be stable is obligatory, predicativists have a choice whether or not to require additionally that classes be rigid.¹³ Our primary interest will be in classes permitted to “grow”.

We turn finally to the question of sufficient conditions for (M-Comp) to be permissible. As observed, one sufficient condition is to adhere strictly to the Vicious Circle Principle, that is, to allow class comprehension on any condition all of whose quantifiers are restricted to objects that are available at the relevant stage of the generation. This restriction guarantees stability. A condition that does not even “look at” future objects cannot be “disordered” by their introduction.

We can sometimes be more relaxed, though, and allow class comprehension on conditions that contain unrestricted modalized first-order quantifiers. To do so is, in effect, to allow the defining condition to quantify over “future” objects of the base type. By contrast, it would be circular to generalize over “future” classes in an account of how classes are generated. This asymmetric treatment of first- and second-order quantification can be justified in cases where the generation of the base objects can be independently characterized, without any entanglement with the generation of classes. The generation of natural numbers, for example, can be characterized independently of the generation of classes of numbers. The case of sets is harder and more interesting. One might have thought that the generation of sets too can be characterized independently of that of classes. But we will later (in §6) find that the generation of sets and of classes are in fact entangled and thus that the envisaged relaxation of the Vicious Circle Principle is problematic.

4 Towards an account of predicative classes of sets

We have seen how predicativism can be understood and developed as a form of potentialism. Let us now examine, in more detail, the generation of predicative classes of sets. We begin with a potentialist set theory.

Various potentialist set theories might work. One that fits the bill is [Linnebo, 2013], but there are others too (e.g., [Studd, 2013]).

4.1 Three Phases of Predicativism About Classes

Following [Linnebo and Shapiro, 2023], there are three major phases to the development of a predicativist theory of classes (no matter what they are classes of). The first phase is where we think of a single class definable at a given stage or in a given world, in terms of what objects exist at that world. The defined class exists in some later, accessible world.

The second phase is where one starts with a given world w and talks about all of the classes that are definable in w. That is, we envision another world |$w^{\prime}$| that has all of the classes definable in terms of what exists at w. The move from w to |$w^\prime$| corresponds to the ascent from one order to the next in Russell’s and Whitehead’s ramified hierarchy of orders. Thus, a minimal move in the ramified hierarchy is non-minimal in the modal analysis, which is more fine-grained. Where the ramified hierarchy adds all classes definable relative to a stage simultaneously, the modal analysis allows predicative classes to be added one by one. This added freedom is particularly relevant for austere Aristotelian-style potentialists, who insist that each individual world be finite.

The third phase concerns iterations of Phase 2. Clearly, the second phase can be iterated to obtain an ω-sequence of worlds, each of which adds all classes definable relative to the previous one. Additionally, we wish to iterate applications of the second phase into the transfinite. This is the most challenging aspect of the project. If we try to iterate the procedure through all ordinals (or all countable ordinals, or all recursive ordinals, or …), we will end up with something more like (a part of) Gödel’s constructible hierarchy, and not a predicativist framework. The underlying predicativist theme is that if, in a given world, we can prove (predicatively) that a certain relation is a well-ordering, then we can iterate the construction through that well-ordering.

An (alleged) conceptual instability of predicativism comes to the fore in this last phase, since it does not seem possible to give a predicatively acceptable account of a predicatively acceptable proof, or a predicatively acceptable well-ordering, or, indeed, of a well-ordering. We will address this.

4.2 Using a Single Class to Represent a Collection of Classes

We find it convenient to be able to talk about some collections of classes, even though there is no formal type for this. We assume that there is a pairing operation on the items in the range of the first-order variables. In the case of natural numbers, one such operation is: |$\langle x, y\rangle$| is |$2^x3^y$|⁠. In the case of sets, we use the usual operation: |$\langle x, y\rangle$| is |$\{\{x\},\{x,y\}\}$|⁠.

One complication is that some of the worlds may not be closed under pairing. This will happen if we think of the natural numbers as themselves potential. If each world is finite, then there will be no worlds closed under pairing. With sets, it depends on the exact development. In an account based on [Hellman, 1989], each world is an inaccessible rank, and so is closed under pairing. Something similar holds of at least one of the options developed in [Linnebo, 2013], but on another option, the worlds are not (automatically) closed under pairing. That need not detain us here, as we can restrict attention to those pairs that exist in a given world.

Let’s abbreviate this as |${\rm{R\tiny {GD-FC}}}(X)$|⁠. So every class such that |${\rm{R\tiny {GD-FC}}}(X)$| rigidly represents a collection of classes, namely those that figure as sections of X. Note that although the collection of represented classes is rigid, each of the represented classes need not be–they are still allowed to “grow”.

4.3 Phase 1

This turns out to be a false start, however, since the formula is actually impredicative! To invoke the usual heuristic, in (8) the class Y is defined by a formula |$\varphi(x)$|that quantifies over all classes at the world where Y is introduced, not at the initial world at which we wanted to make the definition.

This alternative is adequate because it allows any rigid collection of classes, available at some initial world, to serve to define another class that is available only at a “later” world. This definition is predicatively acceptable in the sense that it quantifies only over classes available at the initial world.

Note that we are assuming, for the time being, that the formula defining a predicative class is allowed to contain modalized first-order quantifiers and thus to generalize over “future” sets. But the defining formula is not allowed to quantify over “future” classes, only available ones. As mentioned at the end of Section 3.2, this asymmetric treatment of first- and second-order quantification presupposes that the generation of sets (or whatever other base objects we consider) can be characterized independently of that of classes. We argue in Section 6 that this presupposition fails when the base objects are sets. This will give us a reason to restrict (P1) further.

4.4 Phase 2

Note that, to ensure that the classes we define are stable, the defining formula |$(\varphi^{\Diamond}) ^{\lt X}$| is fully modalized, before it is relativized to the classes represented by X. Otherwise we could use the condition “x is a set of highest rank generated so far”, to define a class, which would fail to be stable. Since any fully modalized formula is stable (see [Linnebo, 2013] or [Linnebo and Shapiro, 2019]), we know that Y will be stable.

4.5 Phase 3

As a brief warm-up, we begin with finite jumps. For each natural number n (ie, each finite von Neumann ordinal), we define the nth predicative jump of a given class of numbers. (Recall that Y_i is the ith section of Y, ie, |$Y_i =\{n\|\ \langle i, n\rangle\epsilon Y\}$|⁠.) Let |$JSeq(X,Y,n)$| abbreviate the formalization of the following:

|$Y_0 = X$| and for each i < n, |$J(Y_i, Y_{i+1})$|⁠.

The purpose of Phase 3 is to state the possibility of iterations of the predicative jump along any (predicatively definable) well-ordering. Let WO(R) abbreviate the formalization of the following:

• R is stable;

• R is a class of ordered pairs;

• necessarily for any class X, if X is possibly non-empty, and if every X is possibly in the field of R, then possibly, there is a set x such that x is necessarily the R-least member of X.

It is tempting to adopt an axiom asserting the possible existence of transfinite iterations of the predicative jump along any well-ordering R. Let |$JSeq^\prime(X,Y,R))$| abbreviate the formalization of the following:for every set a, if necessarily a is the R-smallest set, then Y _a = X; if b is necessarily the R-successor of a, then |$J(Y_a,Y_b)$|⁠; and if, necessarily, c is an R-limit, then |$(Y_a)_d=Y_d$| for each d that precedes c in the ordering R.

It is not clear that this is predicatively acceptable.¹⁶ On the face of it, the notion of a well-ordering is impredicative: a relation is a well-ordering if and only if every non-empty class of its field has a least element in that ordering. But, for the predicativist, there can be no complete totality of such classes. In the context of predicativity relative to the natural numbers, Feferman therefore objects to the analogue of the naïve Phase 3 principle formulated above. He complains that this attempt to implement Phase 3

ignore[s] one crucial point if predicativity is only to take the natural numbers for granted as a completed totality: namely, that they involve in an essential way […] the impredicative notion of being a well-ordering relation. [2005, p. 606]

We will shortly consider some ways to develop a version of Phase 3 that avoids any reliance on a completed totality of classes.

There appears to be an important difference, however, between Feferman’s example of predicative classes of numbers and our present example of predicative classes of sets. In our case, a relation R is understood as a class of ordered pairs. To define what it is for such a relation to be a well-ordering, it suffices to quantify over non-empty sets that possibly overlap the relation R.¹⁷ Since our predicativist about classes (so far) accepts classical reasoning about sets, the statement that there is no such set is unproblematic. Thus, we appear to have a response to Feferman’s naïvité worry that has no analogue in the case of classes of numbers. When the notion of R being a well-ordering is given the mentioned set-theoretic formulation, this particular worry dissipates.

It does not follow, of course, that the Phase 3 principle is acceptable. All we have suggested is that one particular worry about an illicit reliance on modalized quantification over classes not yet generated can be addressed. As we will see, there is a different, more compelling reason to restrict our use of modalized quantification over classes, along the lines suggested by Feferman. Thus, our observation in the previous paragraph serves only to delay the imposition of the restrictions that Feferman advocates.

5 A revenge problem concerning classes

The advertised reason takes the form of a revenge problem, which we now describe. Then we discuss two responses to the problem.

5.1 The problem

Predicativism, as we understand it, is about a bottom-up generation of classes, understood as logical rather than combinatorial collections. The central question, then, is what it takes to generate such a class. Since a logical class is completely characterized by its membership condition, all it takes to generate a class is to formulate such a condition, in a clear and unambiguous manner, drawing only on expressive resources to which we are entitled.¹⁸

Suppose we have made sense of modalized quantification over a potential hierarchy of sets and classes in an acceptable bottom-up manner. Suppose further, along with liberal potentialists, that a condition of this form, say |$\varphi^\Diamond(x)$|⁠, is bivalent, in the sense that each instance is either true or false. Then we have all it takes to generate the corresponding class. Of course, to comply with the bottom-up character of the generation, there is no guarantee that this class will be in the potential hierarchy over which the membership condition generalized. But no matter; if need be, we let the class lie beyond this hierarchy.

A comparison with the ramified hierarchy of orders will be instructive. Suppose we have in a predicatively acceptable way made sense of (classical) quantification over various domains of classes, where these domains are indexed by some well-ordered collection of order indices. Then we can introduce a “super”-order–say, using index ‘s’ — above all of the ones recognized thus far, which collects together all of the mentioned domains. We can now define yet further classes–of order s + 1 — by quantifying over classes of order s. As usual, this is just the beginning.

In this way we use the extended modality to do the work that was done in the ramified hierarchy by introducing a “super”-order, located above all of the orders previously accepted.

The revenge problem poses a challenge to the combination of liberal potentialism and predicativism about classes that we are currently exploring. This view assumes that we can make good sense of modalized quantification over classes. It also assumes that the modal logic is classical. Given these two assumptions, it would be dogmatic to deny that it is possible to define yet more predicative classes. For we can “close off” the totality of classes over which we are modally quantifying and, much as in the ramified hierarchy, define yet more classes “on top”. If we permit such extensions, however, we would undermine our attempt to use modal operators to generalize over the entire potential hierarchy of sets and classes.

We will discuss two responses to the revenge problem. Both responses are inspired by Feferman’s suggestions about how we can develop Phase 3 without any illicit reliance on a completed totality of classes. The first suggestion is to eschew all quantification over the totality of predicative classes, whether modalized or not, instead getting by with the generality afforded by the use of free variables. This corresponds to rejecting the assumption that we can make sense of modalized quantification over classes. The second suggestion is to invoke an unorthodox conception of generality, which is available even when the domain is indefinite, but which supports only a semi-intuitionistic logic. This corresponds to rejecting the assumption that the modal logic is classical.

5.2 Response 1: Only Free-Variable Generality Concerning Classes

David Hilbert made an important distinction between generalizations effected by quantifiers and by free variables. A quantifier requires a well-defined domain. A free variable, by contrast, does not require any domain. Hilbert writes of the free-variable expression of the law of commutativity, “|$a + b = b+a$|“, that it

is in no wise an immediate communication of something signified but is rather a certain formal structure whose relation to the old finitary statements

$$\begin {align} 2\,+\,3 & = 3\,+\,2 \\ 5\,+\,7 & = 7\,+\,5\end{align}$$

consists in the fact that, when a and b are replaced in the formula by the numerical symbols 2, 3, 5, 7, the individual finitary statements are thereby obtained, ie, by a proof procedure, albeit a very simple one. [Hilbert, 1926, p. 196]

Thus, the idea is, a formula with free variables does not express a statement but can nevertheless be endorsed when we have a “proof procedure” which, when applied to any instance of the generalization, yields a proof of that instance.

Suppose we accept only free-variable generalizations over the totality of classes. Since, according to Hilbert, formulas with free variables do not express statements at all, there is no longer any pressure to take such formulas to define classes.

One might worry, though, that this response will throw the baby out with the bathwater. A formula with free variables enables us to simulate, or get the effect of, the universal closure of the entire formula. However, the generalizations simulated by using free variables cannot be negated or freely combined in other truth-functional ways. Consider a formula |$\varphi(x)$|⁠, which simulates the generalization |$\forall x\,\varphi(x)$|⁠. Suppose we wish to simulate the negation of this generalization. But the negated formula, |$\neg\varphi(x)$| simulates a universally generalized negation, |$\forall x\neg\varphi(x)$|⁠, not the desired negated universal generalization, |$\neg\forall x\varphi(x)$|⁠. More generally, free-variable-based generality enables us to simulate absolutely general |$\Pi_1$|-sentences, but not |$\Sigma_1$|-sentences or beyond.

This limited expressive power poses a danger. Even Phases 1 and 2 of our analysis of predicativity appear to involve complex generalizations over the totality of predicative classes. For example, Phase 2 says that for every class there exists a predicative jump. But on a closer examination, this generalization turns out to involve only free-variable-based generality over such classes, not quantification. For the Phase 2 principle is fully captured by an operation |$X\mapsto X^+$| such that |$\Diamond J(X, X^+)$|⁠.

Phase 3 is a different story. It involves the conditional WO|$(R)\to\ldots$|⁠, which is |$\Sigma^1_1$| and thus cannot be captured by free-variable-based generality. Predicativists who allow only free-variable generalizations over the totality of classes are therefore forced to adopt a rule-based version of Phase 3: |$\Diamond J(X, X^+)$|⁠.

Suppose WO(R) can be proved using only “predicatively acceptable means”. Then it is possible to iterate J along R.

In the case of classes of natural numbers, it is natural to adopt the Feferman–Schütte characterization of what “predicatively acceptable” means.

Let us explore what happens in our present case of predicative classes of sets. We can show, by predicatively acceptable means, that various relations are well-orderings. Consider, for example, the standard ordering < on the von Neumann ordinals. Let X be a class of ordinals. Then, reasoning with X as a free variable, the predicativist can show that X has a <-least element. So, it appears, we can iterate the predicative jump Ω many times. In fact, that is only the beginning. It is easy to define a class well-ordering R of order type |$\Omega\times\Omega$| and to prove it to be a well-ordering, using just free-variable reasoning. (See  Appendix A for a proof.) So it appears we can iterate the predicative jump (at least) |$\Omega\times\Omega$| many times.

How far can we go? As is well known, in the case of predicative classes of natural numbers, Feferman and Schütte provide an answer by pinpointing an ordinal |$\Gamma_0$| that provides a least upper bound on how far we can go. It is therefore tempting to think that in our case, we want a class-theoretic ordinal that stands to Ω the way |$\Gamma_0$| stands to ω.

It is not clear, though, that this thought makes sense. In the case of classes of natural numbers, there is an external perspective, namely transfinite set theory, from which point of view we can describe and study |$\Gamma_0$|⁠. Is there an analogous external perspective on our hierarchy of sets and classes thereof? Perhaps the best we can do is to choose some |$V_\kappa$| to serve as a model of V, and then identify the (set-theoretic) cardinal that stands to κ the way |$\Gamma_0$| stands to ω. A truly external perspective would require something like the impredicative class theory MK or some higher-order equivalent thereof. But a committed predicativist would almost certainly reject such a theory. We come up, then, against the familiar complaint that to draw the limit of predicativity (even relative to the cumulative hierarchy of sets), we need to go beyond what is predicatively justifiable (in this relative sense). See, for example [Hellman, 2004].

The predicativist will have to respond that to accept predicative classes of sets, there is no need to pinpoint an exact limit of such predicativity. The best we can do is to approximate the limit from below. We need not decide whether this response is tenable. For even if it is, the present form of predicativism about classes faces a puzzling phenomenon. It appears possible to iterate certain operations beyond Ω — that is, beyond absolute infinity! In Section 6 we show that such iterations are problematic for the present project.

5.3 Response 2: Semi-Intuitionistic Logic and an Indefinite Domain of Classes

Feferman has a second suggestion concerning how to avoid an illicit reliance on a definite totality of predicative classes. This is encapsulated in the following “slogan” (cf also [Dummett, 1991, p. 319]):

What’s definite is the domain of classical logic, what’s not is that of intuitionistic logic”. [Feferman, 2011, p. 23]

He elaborates as follows:

In the case of predicativity, this would lead us to consider systems in which quantification over the natural numbers is governed by classical logic while only intuitionistic logic may be used to treat quantification over sets of natural numbers or sets more generally. (ibid.)¹⁹

The suggestion, then, is that so long as quantification over classes is governed only by intuitionistic logic, the axiomatic version of the phase 3 principle, (P3), is acceptable after all.

Before proceeding to the technical details, we wish to observe that this suggestion, just like Feferman’s first one, enables a response to the revenge problem concerning classes. In essence, when quantification over all classes is intuitionistic, bivalence can no longer be assumed, and we thus get a principled reason to deny that some conditions |$\varphi^\Diamond(x)$| define classes, namely, those conditions for which universally generalized LEM is not valid. An analogous view can be developed concerning our primary example of classes of sets, as opposed to Feferman’s classes of numbers.

There is an important difference, however, concerning the objects that can figure as members of the classes. While it is natural to regard the domain of natural numbers as definite, it is not unreasonable to deny that the domain of all sets is definite (as we do in Section 6). We therefore need to consider two different analogues of Feferman’s semi-intuitionistic theory of classes of numbers. Both analogues regard the domain of classes as indefinite. But they differ on whether or not the domain of sets is definite.

It is important to notice that even the principle of bounded excluded middle entails that identity, as well as set and class membership, are decidable. This observation has some interesting consequences. First, the decidability of identity and set membership entails that well-orderings such as Ω, |$\Omega +\Omega$|⁠, and |$\Omega\times\Omega$|⁠, defined above, are decidable.²⁰ This will be important shortly. Second, the decidability of class membership gives rise to a necessary condition on acceptable class comprehension. Suppose that class comprehension is permitted on a condition |$\varphi(x)$| and that membership in the defined class is decidable. Then the condition |$\varphi(x)$| too must be decidable. Thus, class comprehension can at most be permitted on decidable formulas.

This necessary condition on class comprehension raises an important question: do the class-existence principles associated with our three phases of predicativity satisfy the condition? Suppose first that the domain of all sets is regarded as definite. Then the answer is affirmative. In phases 1 and 2, all quantification over classes is bounded by rigid collections, which define sets and are therefore ensured to behave classically. In phase 3, we quantify over a well-ordering R of sets along which we iterate the predicative jump. Since we are assuming that the domain of sets is definite, the quantification will behave classically. Overall, then, we see that the necessary condition derived from Feferman’s new semi-intuitionistic approach to predicativity is compatible with our investigation thus far.

The situation changes dramatically when we deny that the domain of all sets is definite–as we will do in Section 6. Then we are only entitled to bounded (not set-theoretic) Excluded Middle and Omniscience. This, in turn, means that all three phases need to be restricted. Instead of quantifying freely over sets when defining predicative classes, theorists who choose this option need to ensure that the formulas they use to define classes obey LEM. A safe way to do so is to use only bounded quantification over sets. We will have more to say about this option in Section 6.

5.4 The Need for Strict Potentialism about Classes

Recall from Section 2.4 that strict potentialism goes beyond the liberal variant by adding the requirement that every truth be “made true” or accounted for at some stage or other of the generative process. We contend that both responses to the revenge problem concerning classes are versions of strict potentialism.

Let us explain. A universal generalization is traditionally seen as “made true” by each and every instance. This instantial conception of generality is problematic when the domain is indefinite. If there is no stage at which all of the instances of a universal generalization are available, there can be no stage at which the generalization is “made true” in an instantial manner. How, then, can we avoid instantial quantification over the potential hierarchy of all predicative classes?

In Section 5.2 we explored the option of understanding all such generality as achieved by the use of free variables rather than quantification proper. As Hilbert realized, a free-variable-based generalization can be fully accounted for at some stage of the process where we have a procedure for producing a proof of any instance of the generalization, whether this instance is currently available or not.

Another option is to is invoke some non-instantial conception of generality, which allows a universal generalization to be “made true” in a way that avoids the need to invoke each and every instance. In this way, we ensure that a universal generalization can be “made true” at some stage of the generative process even though not all of its instances are available at that stage. Such a non-instantial conception of generality can be employed even when the domain is indefinite, as explored in Section 5.3.

Several non-instantial conceptions of generality exist–and have in common that they validate intuitionistic logic, not classical. The most famous example is the BHK interpretation of the logical operators, familiar from the literature on intuitionism. The idea is that p is a proof of |$\forall x\,\varphi(x)$| iff, for any a, p(a) is a proof of |$\varphi(a)$|⁠. Thus, p provides a general method for proving any given instance, which can be available even though each instance is not. The BHK conception of universal generality is not suitable for present purposes, however. We want every set–including huge transfinite ones–to be domains of classical logic. This would require us to consider equally huge infinitary proofs.

A better option for our purposes is the non-instantial conception of generality developed in [Linnebo, 2022]. Instead of appealing to the notion of a proof, this conception is based on an abstract truthmaker semantics, where various states of the world cooperate to verify universal generalizations and other statements. Some generalizations are made true in a partially instantial manner. For example, it is true that everyone in the room is hungry because a₁ is hungry, …, a_n is hungry, and because everyone in the room is among these n individuals. We obtain a truthmaker for the generalization by fusing truthmakers for these n + 1 claims. Any set of states has a fusion. Other universal generalizations are made true in a wholly non-instantial manner. For example, the fact that every object has a singleton set can be explained, without recourse to any individual instance, in terms of wholly general principles about set formation–which is good, since the required instances do not form a set.

The picture that emerges is that the only robust responses to the revenge problem concerning classes involve embracing a strict form of potentialism about classes. One option is to require that all generalizations over the totality of classes be free-variable-based. Another is to require that such generalizations be understood in a non-instantial way and therefore let the logic be semi-intuitionistic, not classical.

It remains an open question, though, whether we should be strict or merely liberal potentialists about sets. This question sets the agenda for our final substantive section.

6 A revenge problem concerning sets

We just suggested that the revenge problem concerning classes requires us to be strict potentialists about classes. Suppose we accept this conclusion but try to hold on to liberal potentialism concerning sets. This hybrid view may seem unmotivated. If every truth about classes must be “made true” at some stage or other of the generative process, why should not the same go for sets? Instead of pursuing this loose thought, we will discuss another revenge problem, which provides a more clear-cut reason to accept strict potentialism about sets as well.

6.1 The Problem

The problem is easy is describe. We have seen that there are decidable well-orderings of length Ω and far beyond. (Indeed, this holds irrespective of whether we are liberal or strict potentialists about sets; cf Section 5.3 and  Appendix A below.) This leads to trouble, given two natural thoughts. First, it seems natural to iterate the formation of classes along these well-orderings. This would be permitted even by the rule-based version of our Phase 3 principle and parallels what is permitted in the case of the natural numbers. Second, if class formation can be iterated that far, why not also set formation? We see no principled reason why the iteration of set formation should not be allowed to proceed just as far as that of classes. But if set formation is allowed to continue beyond Ω, we get a Burali-Forti contradiction–since Ω is, by definition, the height of the hierarchy of sets.

To be entirely clear: none of our commitments formally entail that set formation can be iterated along any decidable well-ordering. So although the revenge problem is serious, there is no formal inconsistency. What we have is a serious conceptual-philosophical challenge.

We will consider two attempted responses. One seeks to hold on to liberal potentialism about sets and tries to justify a differential treatment of sets and classes. Another is based on strict potentialism about sets and gives a reason why neither iteration can be allowed to proceed to or beyond Ω. Since we find the first response problematic and the second convincing, we conclude that there is reason to be strict potentialists about sets as well as about classes.

6.2 Response 1: Each Set Is Completed, Whereas Classes Need Not Be

Both forms of potentialism, liberal and strict, take each set to be completed, but regard the hierarchy of sets, V, and the sequence of von Neumann ordinals, On, as incompletable. To form a set, we therefore need to specify a complete collection of elements of the would-be set. Since V and On are incompletable, these collections are not suitable for set formation. Classes are very different in this regard. To form a class, all we need is a sharp and stable membership condition, formulated in a way that draws only on available resources, without presupposing any objects not yet generated. This ensures that classes do not have to be completed (or even completable); for example, V and On have sharp and unproblematic membership conditions (namely, being a set and being a von Neumann ordinal, respectively) but are incompletable.

We can now state the first attempted response to the revenge problem concerning sets.

The profound difference just observed between sets and classes justifies a differential treatment of these two types of collection. It is true that class formation can be iterated further than set formation. Since classes do not have to be completable, we can iterate class formation along incompletable well-orderings such as |$\Omega+\Omega$| or |$\Omega\times\Omega$|⁠. Since each set, by contrast, has to be completed, it would be unacceptable to iterate set formation along an incompletable well-ordering; for example, there cannot be a set of all von Neumann ordinals.

The problem is that this response just shifts the bump in the carpet. We are told that set formation cannot be iterated to Ω or beyond because Ω is incompletable. Where does incompletability set in? Why at Ω and not higher up in the sequence of class-theoretic well-order types that we have available? The problem is particularly pressing for liberal potentialists about sets. On their view, there is no deep difference between Ω and the various class-theoretic well-order types beyond Ω; these are all perfectly good and classically behaved well-orderings. Why, then, should precisely Ω mark the point at which incompletability sets in?

We submit that an Ω-stage for either iteration would undermine the entire program sketched here. Again, we take V to be the growing (and therefore incompletable) class of all sets, and Ω to be the growing (and incompletable) class of all von Neumann ordinals. Then we can use our Phase 1 principle to define a rigid class of all von Neumann ordinals at that stage. And so we do have something very set-like that is not, and supposedly cannot be, a set. This is the very thing that Boolos objected to in his famous “Wait a minute!” response noted in the opening of this article. We have a class that is in every respect like a set–it is rigid–yet it is not, we are told, a set.

It is worse. We have not given a full-blown theory of how sets are generated, but on most potentialist accounts (such as those of Linnebo [2013] or Hellman [1989]) if s is a stage and we can characterize some sets at that stage, then there is a later stage that contains a set whose members are exactly those sets. So if we do manage an Ω stage, then both V and Ω will be sets at a future stage. The members of this set V are all sets that ever will be generated. We then encounter Cantor’s paradox and the Burali-Forti paradox. We end up with von Neumann ordinals that are not in Ω and sets that are not in V.

To leave things here, with this observation that an Ω-stage would undermine our project is, to quote Michael Dummett in a closely related context, “to wield the big stick, but not to offer an explanation” [1991, p. 316]. Let us try to do better.

6.3 Response 2: Become Strict Potentialists About Sets Too

We move on to a second and far superior response to the present revenge problem. This response embraces strict potentialism about sets as well as about classes and uses this to block the problem. Let us explain.

According to strict potentialism, part of what it means for each set to be completed is that quantification restricted to a set behaves classically (in the precise sense articulated in Section 5.3, namely Bounded Excluded Middle and Bounded Omniscience). But according to strict potentialism about sets, quantification over Ω obeys only intuitionistic logic, not classical. Since any set has to obey BOM, we thus have a principled reason to deny that there can be a von Neumann ordinal corresponding to Ω. Although Ω and V exist as classes, quantification over them is not classical, which explains why there are no corresponding sets. This blocks the generation of sets beyond Ω.

Of course, strict potentialism about sets needs to be properly developed. We describe one way to do so in  Appendix B. But the explanation just given is robust, as it draws only on general aspects of the view.

Our response to the revenge argument concerning sets has consequences for the theory as classes as well: it blocks the generation of classes beyond Ω.²¹ To define a class, we recall, we need a sharply defined membership condition. As Weyl [1919, p. 85] ([1994, p. 109]) put it, the condition needs to have an “intrinsically clear sense”; that is, generalized Excluded Middle needs to hold for the condition. Since quantification over Ω (or V) does not obey classical logic, generalized Excluded Middle cannot be assumed for any condition defined in terms of such quantification. Now, recall from Section 4.5 our characterization of what it is for a class Y to encode the iteration of the predicative jump J along a well-ordering R, starting with a class X:

for every set a, if necessarily a is the R-smallest set, then Y _a = X; if b is necessarily the R-successor of a, then |$J(Y_a,Y_b)$|⁠; and if, necessarily, c is an R-limit, then |$(Y_a)_d=Y_d$| for each d that precedes c in the ordering R.

Note that this characterization uses unbounded quantification over V. In particular, the characterization of Y’s sections at step Ω or beyond would involve such quantification and thus cannot be assumed to obey Excluded Middle. This means that the existence of these sections, and a fortiori of Y in its entirely, is not predicatively acceptable. For as observed in Section 5.3, predicative class comprehension can only be permitted on decidable formulas. This prevents us from iterating the formation of classes to or beyond Ω.

Instead, strict potentialists about both classes and sets must restrict the P3 principle to iteration along any set well-ordering: for we need decidability of quantification over the well-ordering. This completes our journey of scaling back the permissible lengths of iterations of the predicative jump. At the end of Section 4.5, it briefly looked as if iteration along any well-ordering might be permissible. Then, in response to the revenge problem concerning classes, we learnt that the most we can permit is iteration along any predicatively acceptable well-ordering. Now, in response to the revenge problem concerning sets, we have found a need to scale back yet further, permitting only iteration of the predicative jump along any set well-ordering.

An attractive picture emerges of Ω as an absolute infinity. That is, Ω is an upper bound on the length of any iterative generation of determinate mathematical objects such as sets or classes. This upper bound can be approximated from below but never reached or surpassed.

7 Concluding summary

It is time to sum up and conclude. Our two most important and novel conclusions are based on the two revenge problems. Both point in the direction of strict potentialism.

First: A predicativist account of classes naturally leads in the direction of strict potentialism about classes. We hinted at this conclusion in [Linnebo and Shapiro, 2023] too (see, especially, pp. 6, 13), but are far clearer and more explicit here. In particular, we articulated the revenge problem concerning classes and argued that both of the responses available–letting all generalizations over classes be achieved by means of free variables or some form of non-instantial generality–involve a form of strict potentialism about classes.

Second: The possibility of extending a potentialist account of sets with a potentialist account of predicative classes reveals a new reason to embrace strict potentialism about sets as well as about classes. The reason is that liberal potentialism about sets gives rise to a revenge problem. It shows that it is possible to iterate the generation of classes to and beyond Ω, and once this is permitted, there is no good reason not to make the same allowance for sets. This would mean, absurdly, that there are sets that the liberal potentialists “forgot to form”. To avoid this absurd conclusion, we have a good reason to be strict potentialists about sets as well.

Three further conclusions echo ones made in [Linnebo and Shapiro, 2023] but are here transferred to and confirmed for classes of sets, as opposed to classes of natural numbers.

Third: There is a close connection between predicativism, in the usual sense associated with the Vicious Circle Principle, and potential infinity: where the latter threatens definitional stability, the former suffices to defuse the threat.

Fourth: It is natural and illuminating to analyze predicativism in a modal framework that explicates potentialist ideas.

Fifth: As suggested in Feferman’s later work, predicativity admits of a particularly nice analysis using a semi-intuitionistic logic.

Overall, we have argued that potentialists about sets should accept predicative classes as well, and that when they do so, reasons emerge to adopt strict potentialism about sets as well as classes. This conclusion is independent of the details of the potentialist set theory with which we begin. More work is needed, however, to identify and investigate the theories with which we end up.

We end with a remark on the revisionism involved in our proposal. While strict potentialism involves a step away from a classical foundation for mathematics, it is important to realize that departure is modest and represents no loss for the ordinary mathematicians. Every set-sized domain behaves just as in classical mathematics. Thus, by asserting the possible existence of sufficiently large transfinite sets (say, |$V_{\omega+6}$| or even |$V_\kappa$| for κ a strong inaccessible), we obtain domains that are safe for classical set-theoretic reasoning. Thus, all of ordinary classical mathematics can proceed just as before. It is only the foundation of mathematics as a whole that takes on a different character–but in return becomes capable of withstanding the revenge problems.

Funding

ØL gratefully acknowledges support from the European Union (ERC Advanced Grant, C-FORS, project number 101054836).

Footnotes

Acknowledgments

We are grateful Neil Barton, Laura Crosilla, Salvatore Florio for comments and questions on earlier drafts. Thanks also to two anonymous referees and to audiences in Minneapolis, Oslo, and Oxford.

References

APPENDICES

A Large well-orderings

The defining condition for Ω is “x is a von Neumann ordinal” or perhaps “x is transitive and every member of x is transitive”. This is bounded, and so LEM can be used with respect to members of Ω. We can show that each member of Ω is well-ordered under membership.

This defines a class. The quantifiers here are all bounded, although x occurs free and Ω is perhaps a class constant. We claim that WO|$(\Omega^*)$|⁠.

Clearly WO|$(\Omega^*)$| is stable and it is a class of ordered pairs–we don’t need LEM to show either of those. Its field is Ω.

Note that on the present conception, a set can be formed only if all if its members are available. Suppose |$k\epsilon X$| and |$k\epsilon\Omega$| at some stage. We also have to suppose that X is a subclass of Ω. Every member of k is available at that stage. Suppose also (going to a later stage if necessary) that k + 1 is also available. Using separation, let b be the set of members of k that are also in X. This is a subset of k + 1, and so it has a least member c. This is also the least member of X.

Now let |$\Omega^{**}$| be the class of pairs of pairs |$\langle\langle a,b\rangle,\langle c,d\rangle\rangle$| defined by:

|$a,b,c,d$| are all von Neumann ordinals, and either |$a\in c$| or both a = c and |$b\in d$|

We show that this is a well-ordering of “length” |$\Omega\times\Omega$|⁠.

B Strict potentialist set theory

Several such set theories have been developed, but we have used that of [Linnebo, 2013] as our primary example. This theory is a form of liberal potentialism about sets. Here we have argued that the addition of predicative classes of sets yields a reason to adopt instead a strict potentialist theory of sets. So we now wish to describe what we take to a plausible candidate for such a theory.

Some constraints on the desired strict potentialist theory of sets have emerged from our discussion.

(C1) As explained in Sect. 5.3, we want LEM for all atomic predicates, including both set-theoretic membership (⁠|$\in$|⁠) and plural membership (symbolized, as customary in plural logic, by |$\prec$|⁠).

(C2) As explained in Sect. 5.3, we want Bounded Omniscience for quantification restricted to a set and to a plurality.

(C3) We want AC for sets, because sets are understood as arbitrary, combinatorial collections.

With these constraints in mind, we will now present a slightly streamlined and improved version of the liberal potentialist set theory of [Linnebo, 2013] and explain what needs to be modified in order to turn it into a strict potentialist set theory.

B.1 The Plural Logic

As discussed in Sect. 2.4, the propositional modal logic of liberal potentialism is classical S4.2, whereas that of strict potentialism is intuitionistic S4.2; let’s call it iS4.2. See [Linnebo and Shapiro, 2019] for details and references.

Next, we add the usual axioms and inference rules for the singular and plural quantifiers. There is also an extensionality principle for pluralities, to the effect that any two coextensive pluralities are indiscernible (with respect to non-intensional contexts). All of these principles remain unchanged in the strict potentialist setting.

Notice that these principles ensure that classical logic prevails in any domain based on a plurality. This logic intermediate between intuitionistic and classical is known as semi-intuitionistic; cf, e.g., [Feferman, 2010].

The modifications just described have several important consequences. Most importantly, they constitute an important first step towards heeding constraints (C1) and (C2). For as Proposition 1 will show, these modifications will allow us to derive the decidability of set membership and bounded omniscience for sets. We note, further, that the decidability of |$\prec$| and the plural omniscience principle (P-BOM) entail the decidability of plural inclusion |$\preccurlyeq$|⁠.

Since pluralities are arbitrary combinatorial collections, we add (to both the liberal and strict theories) a plural Choice principle:

Assume |$\forall x\prec xx\exists y\,\psi(x,y)$| and |$\psi(x,y)\wedge\psi(x^\prime,y)\to x=x^\prime$|⁠. Then there are yy such that |$\forall x\prec xx\exists! y\prec yy\,\psi(x,y)$|⁠.

This takes care of constraint (C3). We note that our restricted plural comprehension scheme is essential here in order to avoid Diaconescu’s famous result that the axiom of choice and unrestricted Separation entail LEM and thus full classical logic [Diaconescu, 1975].²²

To sum up, the move from liberal to strict potentialism requires a move from classical plural logic to a semi-intuitionistic plural logic. This means, in particular, that the plural comprehension scheme must be restricted so as to require that the separation condition be ID.

B.2 The Modal Logic of Plurals

This principle can be regarded as a generalization of the modal axiom G (§2.2). While G says that any two branches along the accessibility relation converge in a single “world”, (P-Repl) says that any plurality-indexed family of branches can be brought together. We contend that this principle is warranted in the strict theory as well as in the liberal one.

To summarize, the principles concerning the interaction of plurals and modals remain unchanged when we move from liberal to strict potentialism.

B.3 The Relation Between Pluralities and Sets

We also add the decidability of this predicate. Finally, we add an axiom scheme of |$\in$|-induction:

Suppose that every urelement is |$\varphi$| and that necessarily, for any xx each of which is |$\varphi$|⁠, {xx} too is |$\varphi$|⁠. Then necessarily every set is |$\varphi$|⁠.

This principle captures the idea that the only way to generate a set is by collapsing a plurality.

The following proposition, whose proof is routine, ensures that the semi-intuitionistic behavior of pluralities described in Sect. B.1, together with the principles adopted in this subsection, ensure that sets too behave in a semi-intuitionistic manner.

To summarize: none of these principles concerning the relation between pluralities and sets require any modification.

B.4 The Resulting Non-Modal Set Theory

For ease of comparison, let us formulate classical ZFC with the axiom scheme of |$\in$|-induction rather than the classically equivalent axiom of Foundation.

It turns out, then, that strict potentialism about sets validates a theory very close to Feferman’s [2010] Semi-Constructive set theory, SCS, except that (i) this latter contains Markov’s principle, and (ii) our theory has ID-Separation rather than just bounded Separation.²⁴

To sum up, the changes that result from going from liberal to strict potentialism are surprisingly modest. If we choose to axiomatize classical ZFC using |$\in$|-induction, then the only changes are that the logic goes semi-intuitionistic and that Separation must be restricted to ID-Separation.