1 Introduction

Admittedly, the present state of affairs where we run up against the paradoxes is intolerable.
Just think, the definitions and deductive methods which everyone learns, teaches, and uses in
mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking
is defective, where are we to find truth and certitude?
– Hilbert (Bencerraf and Putnam, 1983)

Philosophers have often used first-order logic to analyze mathematical and scientific
claims. However, we seem to grasp a notion of logical possibility prior to, and
independent from, our grasp of mathematical objects like set-theoretic models.
Powerful reasons to accept this notion as an additional logical primitive have
emerged (Boolos 1985; Etchemendy 1990; Gómez-Torrente 2000; Hanson 2006;
Field 2008).
In this book, I’ll make a case that philosophical analyses using (a natural general-
ization of) this notion of logical possibility can illuminate the philosophy of mathem-
atics, metaphysics and philosophy of language. Much of this case will focus on pure
mathematics and the philosophy of set theory. For example, I will show that formulat-
ing set theory in terms of logical possibility (along Potentialist lines suggested by
Putnam and Hellman in response to the Burali-Forti paradox) yields a new and more
appealing justification for one of the standard ZFC (Zermelo–Frankel with choice)
axioms of set theory. This brings us closer to realizing the traditional hope of justifying
mainstream mathematics from principles that seem clearly true.
However, we will see that using a primitive logical possibility operator can also help
us develop a modestly neo-Carnapian philosophy of language. And philosophical
analyses of scientific theories using the logical possibility operator can illuminatingly
“factor” scientific claims into a logico-mathematical component and a remainder, in
a way that reveals hidden heterogeneity in the role of mathematics in the sciences and
clarifies debates over Quinean and post-Quinean indispensability arguments.

1.1 Mathematics as a Touchstone and the Centrality of Set Theory

Mathematical proofs provide a touchstone of clarity and convincingness which

serves as an inspiration to philosophy and other disciplines. While it is possible

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 2 A Logical Foundation for Potentialist Set Theory

to doubt the results of mainstream mathematical arguments (philosophers are

capable of doubting anything), there’s something striking about just how convin-
cing mathematical proofs often are. Consider the standard argument that there are
infinitely many primes. Even philosophers who deny that there are numbers (and
hence think the argument as usually stated is unsound) are strongly tempted to
say that we know something like the premises and that these proofs provide some
kind of valuable amplification of this knowledge. The premises we use in
informal mathematical reasoning have a combination of prima facie obviousness,
power and generality, which makes them exemplary tools for expanding our
knowledge and resolving disputes in cases where people’s initial hunches dis-
agree. It’s no surprise that Leibniz1 wished philosophers could resolve their
disputes like mathematicians by saying “let us calculate” (or at least, “let us
each look for a proof”).
In many ways, set theory lies at the heart of modern mathematics, and it does
powerful mathematical (not just philosophical) work as a foundation for the
whole. So, one might hope that the set-theoretic foundations for mathematics
would share the clarity and convincingness we hope for from mathematical
arguments.
However, certain problems in the philosophical foundations of set theory raise
worries. These concerns are more mathematical and specific to set theory than standard
philosophical worries about, e.g., whether there are any abstract objects. And these
concerns are more threatening to mathematical practice than philosophical doubts
typically are, insofar as they raise doubts about whether the standard ZFC axioms of
set theory are even logically consistent.
The development of set theory resolved a great many problems in analysis. It
also provided a formal framework to allow interactions between various areas of
mathematics – creating, as Hilbert (1926) famously observed, a kind of mathem-
atical paradise. However, contradiction, in the form of Russell’s paradox, threat-
ened Hilbert’s paradise.
This problem was almost, but not quite, solved by accepting the iterative hierarchy
conception (also called the cumulative hierarchy conception) of sets and the standard
ZFC first-order axioms for set theory. On the iterative hierarchy conception of set, we
think about the sets as being formed in layers, with the sets at each layer containing
only elements from prior layers. This lets us avoid the appearance that there should
be a set of all sets that aren’t members of themselves, and hence Russell’s paradox.
And it is hard to deny that the mathematical results which are currently stated in
terms of this conception of set theory reflect genuine and important knowledge of
some kind.
However, as we will see below, a few further problems arise, including a question of
how to justify our acceptance of all the ZFC axioms. So, we may ask, is the price of

1
See Chrisley and Begeer (2000: 14).

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 Introduction 3

remaining in Cantor’s set-theoretic paradise giving up the old ambition of founding

mathematics on intrinsically obvious seeming principles?
One of the main projects of this book will be to answer this question in the negative.
I will develop a unified understanding of set-theoretic talk, which vindicates our
intuitive expectations regarding set theory – and demonstrates that all theorems of
set theory (ZFC) are derivable from principles that seem clearly true (not to say
indubitable). The approach I’ll develop differs from standard Actualist set theory, in
a few important ways.

1.2 Actualism and Its Discontents

On standard Actualist approaches to set theory, set theory studies abstract mathemat-
ical objects called “sets,” which form an iterative hierarchy as evoked above. Apparent
existence claims made by set theorists (like “there is a set which has no elements”) are
made true by the existence of corresponding objects, just like ordinary existence claims
about cities or electrons or cars. Crudely speaking, three problems arise when we look
at set theory from this familiar point of view (each of which I’ll describe in much
greater detail in Section 2.2).
First, there’s a problem about our conception of the hierarchy of sets as
a mathematical structure. We don’t seem to have a precise conception of the intended
structure of the iterative hierarchy of sets, in the way that we do seem to have
a conception of the natural numbers. In particular, the height of the hierarchy of sets
is left vague or mysterious. As the Burali-Forti paradox (Burali-Forti 1897) drama-
tizes, a certain naive conception of the hierarchy of sets (as containing ordinals
corresponding to all ways some objects could be well ordered by some relation) is
incoherent. And once this naive paradoxical conception of the height of the hierarchy
of sets is rejected, we don’t seem to have a precise idea about the intended height of the
hierarchy of sets left over to replace it. It appears that, for any height that the hierarchy
of sets could achieve, there could be a strictly larger structure, which extra layers of
“sets” on top of the original hierarchy and fits with everything in our conception of the
sets equally well. But it seems arbitrary to suppose that the hierarchy of sets happens to
stop at any particular point.
Second, there’s a worry about generality and the role of set theory as a foundation
for all of mathematics. One might hope that set theory would be able to represent any
mathematical structure one might want to study. The idea that set theory has this kind
of generality is prima facie quite intuitive. But Actualist set theory is prima facie
unable to represent the study of mathematical structures that are “too large.” Thus,
Actualism makes it hard to capture the intuition that “any possible structure” should, in
some sense, be fair game for mathematical study, and hence treatment within set
theory.
Third, there’s a problem about intuitively justifying the standard ZFC axioms of set
theory. As noted above, mathematical proofs can usually be reconstructed so as to
derive their conclusion from premises that are prima facie extremely plausible (if not

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 4 A Logical Foundation for Potentialist Set Theory

indubitable or impossible to philosophically or empirically cast doubt upon). So, one

might hope that (once we understand set theory correctly), every claim provable from
the ZFC axioms for set theory can be shown to follow from principles that seem clearly
true in the way that foundational mathematical axioms often do.
However, philosophers have noted that the axiom of Replacement (one of the
standard ZFC axioms) doesn’t seem clearly true and deriving it from principles that
do seem clearly true has proved challenging. For example, Hilary Putnam (2000)
writes, “Quite frankly, I see no intuitive basis at all for . . . the axiom of Replacement.
Better put, I do not see that a notion of set on which that axiom is clearly true has ever
been explained.” Instead, philosophers of mathematics and mathematicians have made
do with less ambitious approaches to justification. For example, some mathematicians
have invoked external justifications, like the failure of mathematicians to discover any
contradictions during over a century of work with ZFC. Others have shown that the
axiom of Replacement follows from certain powerful and plausible (if not clearly
consistent or true) principles that also imply many of the other axioms of set theory.
But insofar as the latter powerful conceptions aren’t clearly consistent, showing this
doesn’t suffice to show Replacement follows from clearly true premises. Nor does
simply combining Replacement itself with the bare-bones conception of the intended
width of an iterative hierarchy of sets evoked above yield something that seems
obviously consistent. This state of affairs can feel unsatisfying.

1.3 Potentialism and the Justification Problem

In response to the first two problems above, philosophers like Putnam, Parsons,
Hellman and Linnebo (Putnam 1967; Parsons 1977; Hellman 1994; Linnebo 2018a)
have proposed that we should reject2 Actualism about set theory in favor of a different
approach: Potentialism. The key idea behind Potentialism is that, rather than taking set
theory to be the study of a single hierarchy of sets which stops at some particular point
(as the Actualist does), we should instead interpret set theorists as making modal
claims about what hierarchy-of-sets-like structures are possible and how such struc-
tures could (in some sense) be extended.
As we will see in more detail in Chapter 2, switching to a Potentialist understanding
of set theory solves the first problem for Actualism noted above. The Potentialist
avoids postulating an arbitrary (or indeterminate) height for the hierarchy of sets,3 and
Potentialism also plausibly solves the second problem above, by honoring the intuition
that any possible mathematical structure can be studied within set theory.

2
Strictly speaking Putnam proposes Actualism and Potentialism are (in some sense), two perspectives on
the same thing.
3
At least Potentialists like Hellman (1996), Linnebo (2018a) and Studd (2019) avoid positing such an
arbitrary stopping point for the sets. Putnam’s view, on which Actualist set theory and Potentialist set
theory are (somehow) two perspectives on the same thing, does not let us avoid this problem in any
obvious way.

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 Introduction 5

However, the problem of justifying the axiom of Replacement from premises that
seem clearly true remains. Contemporary Potentialists can, and generally do, prove
that (the Potentialist translations of) every theorem of ZFC can be derived from certain
intuitive assumptions about logical possibility, or some other such modal notion.
However, these proofs all use principles of modal logic that aren’t (and aren’t claimed
to) be clearly true in the way invoked by Putnam. The existing Potentialist literature
has shown that Potentialism is no worse off than Actualism with regard to the problem
of justifying Replacement that Putnam raises.4 However, neither Potentialists nor
Actualists have put forward a solution to this problem.

1.4 Outline

In this book, I will attempt to solve the above problem of justifying the axiom of
Replacement from principles that seem clearly true (or at least improve on existing
solution attempts) and clarify the foundations of set theory.
In Part I, I will argue that we should indeed be Potentialists about set theory for
essentially the reasons indicated above, and then review major existing formulations of
Potentialism about set theory and some problems for each. I’ll discuss and contrast two
major existing versions of Potentialist set theory: the Putnamian approach developed
by Putnam and Hellman which I will largely follow, and an alternative Parsonian
approach recently explored by Linnebo and Studd, which appeals to a notion of
interpretational possibility, rather than metaphysical or logical possibility.
I will develop and advocate a particular form of Potentialist set theory. Although
this approach largely blends and simplifies ideas from Putnam and Hellman, it has the
distinctive feature of replacing claims that “quantify-in” to the diamond of logical
possibility (and thereby talk about what’s logically possible for objects) with claims
about what’s logically possible given certain structural facts, expressed using a new
piece of logical vocabulary I’ll call the conditional logical possibility operator (◊... ).
Cashing out Potentialist set theory in these terms lets us avoid certain philosophical
controversies,5 as well as practically helping us state axioms that can be easily grasped
and recognized as saying something clearly true.
In Chapter 2 I will discuss Actualist approaches to set theory and expand on the
worries for them noted above. In Chapter 3 I’ll discuss how adopting some Potentialist
approach to set theory promises to solve these worries and review existing forms of the
Putnamian style of Potentialism. I will defend Hellman’s use of a notion of logical
possibility to cash out Potentialist set theory but note that controversies over quantified
modal logic raise some problems for using his version of Potentialism in our founda-
tional project.

4
Existing potentialists (Hellman 1994; Linnebo 2018a; Studd 2019) have generally adopted some version
of a Potentialist translation of Replacement as an axiom (schema). For while these Potentialist translations
are not clearly true, they are (we will see) as attractive as corresponding instances of the Replacement
schema understood actualistically.
5
See Section 3.3.1.

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 6 A Logical Foundation for Potentialist Set Theory

In Chapter 4 I’ll introduce my preferred style of Potentialist paraphrase and the key
notion of conditional (structure-preserving) logical possibility I’ll use to give these
paraphrases. Finally, in Chapter 5 I’ll contrast the above approach to Potentialist set
theory with those advocated by Linnebo and Studd, major proponents of an alternate
“Parsonian” school of Potentialist set theory.
In Part II I will turn to the core mathematical project of this book: justifying the ZFC
axioms. I’ll propose general purpose axioms for reasoning about conditional logical
possibility which (I claim!) seem clearly true in the way our foundational project
requires. Then I will show that these axioms justify our use of normal first-order
reasoning for set-theoretic claims (i.e., claims in the first-order language of set theory)
even when those claims are understood potentialistically. Specifically, if we let ϕ◊
stand for the Potentialist translation of a set-theoretic claim ϕ, let ‘FOL be provability in
first-order logic and ‘ be provability in the formal system proposed in this book, we
can show the following.
Theorem 1.1 (Logical Closure of Translation). Suppose Φ; Ψ are sentences in the
language of set theory and Φ ‘FOL Ψ then Φ◊ ‘ Ψ ◇.
With this theorem in mind, all that’s needed to justify normal mathematical practice is
to demonstrate that if ϕ is an axiom of ZFC then ‘ ϕ◊ holds. A key idea here will be to
use certain non-interference intuitions to justify the (Potentialist translation of) the
axiom of Replacement, rather than simply taking the latter as an axiom, as current
Potentialists tend to do. Putting these pieces together we can conclude that for all set
theoretic sentences ϕ:

ZFC ‘ ϕ then ‘ ϕ◊
FOL

That is, reasoning in ZFC as if one were talking about an Actualist hierarchy of sets is
harmless. If one can prove that ϕ in ZFC then the Potentialist translation of ϕ (written
ϕ◊ above) is (true and indeed) provable in my formal system.
Note that since I choose axioms of reasoning about conditional logical possibility which
are attractive for general use rather than ones that directly mirror Actualist ZFC set theory
(as other Potentialists have done in proving versions of the theorem above), it’s not at all
obvious whether the reverse direction of the above conditional, i.e., “If ‘ ϕ◊ then
ZFC ‘FOL ϕ” is true. In principle, there is some hope that the modal axioms
I propose (or, more plausibly, further principles about conditional logical possibility
that seem equally clearly true ) will let one vindicate new axioms for set theory, going
beyond the ZFC axioms.
Finally, in Part III of the book, I’ll turn to larger philosophical questions. In
Chapter 10, I consider two ways my story about set theory can fit into a larger
philosophical picture of mathematics and its applications: a Nominalist approach
and the weakly neo-Carnapian approach I ultimately favor.
In Chapters 11–14, I’ll discuss the Nominalist approach to non-set theoretic math-
ematical objects and Indispensability arguments. I’ll argue that adding some cheap

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 Introduction 7

tricks to the above paraphrase strategy lets the Nominalist answer certain classic
Quinean and Explanatory indispensability arguments. However, I’ll suggest that the
mathematical Nominalist may face serious and under-discussed worries about refer-
ence and grounding.
In Chapters 15 and 16, I’ll explain the weakly neo-Carnapian approach to non-set
theoretic mathematical objects I favor, and argue that adopting it helps solve or avoid
these reference and grounding problems and has certain other advantages (while retain-
ing many benefits of Nominalism). The resulting view is a kind of neo-Carnapianism
realism about mathematical objects, which drops Carnap’s radical anti-metaphysical
ambitions but keeps mathematicians’ freedom to talk in terms of arbitrary logically
coherent pure mathematical structures.
Finally, in Chapters 17–19, I’ll discuss how the overall picture of mathematics
developed in this book relates to traditional questions about Logicism, Structuralism
and human access to facts about objective proof-transcendent mathematical facts.

1.5 Caveats and Clarifications

Let me finish this introduction with some quick caveats about the nature and aim of my
project.
First, I don’t claim set theorists should literally rewrite set theory textbooks in
Potentialist terms. Mathematicians’ current practice of (making arguments which can
be reconstructed as) proving things in first-order logic from the ZFC axioms is fine.
And doing something like logical deduction from purely first-order axioms may be
unavoidably easier (for minds like ours) than thinking about the elaborate modal
claims that figure in Potentialist set theory. If one thinks about typical set theoretic
talk as abbreviating Potentialist claims, then the main result of Part II shows that it’s
unnecessary to unpack this abbreviation in most mathematical contexts.
However, I am suggesting Potentialism reflects what people should say when we think
about set theory in many philosophical and foundational mathematical contexts. Replacing
Actualist set-theoretic claims with their Potentialist paraphrases solves various intuitive
puzzles and makes sense of things that we normally want to say about set theory.6

6
In this proposal I somewhat mirror Hellman’s response (Hellman 1998) to Burgess and Rosen’s dilemma
(Burgess and Rosen 1997). Burgess and Rosen argue that nominalistic paraphrases must be intended as
either a hermeneutic theory of what scientists mean or a revolutionary theory of what they should say, but
typical Nominalist paraphrases don’t seem adequately supported by scientific motivations for either as
they wouldn’t be accepted to linguistics or physics journals.
One response to this would be to say that nominalistic paraphrases reflect what we should say in
philosophical contexts, and this differs from what we should or do say in any scientific context. Hellman
points out that one can appeal to useful divisions of labor within the sciences to motivate such a distinction.
For example, a physicist who hypothesizes that heat is molecular motion (and regiments physical theories
accordingly) isn’t thereby making a revolutionary proposal about what higher-level scientists (biologists
or ecologists) should say or a hermeneutic proposal about what they currently mean. So the untroubled
friend of metaphysics can think about ontology as its own discipline, with its own level of analysis and
corresponding explanatory work this analysis is invoked to perform. A Nominalist of this stripe might say:
metaphysics is to physics as physics is to biology and ecology. That’s why good proposals about what we

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 8 A Logical Foundation for Potentialist Set Theory

Arguably, this book’s project of developing Potentialist foundations for set theory is
analogous to the familiar project of providing a set-theoretic foundation for analysis.
Our naive reasoning about certain concepts (limits in one case, the height of an
iterative hierarchy of sets that “goes all the way up” in the other) turns out to lead to
paradox. So, it is desirable to find a different way of thinking about relevant mathem-
atical claims which will let us capture their intuitive significance and interest, while
blocking paradoxical inferences.
Second, the Potentialist understanding of pure set theory advocated in Parts I and II
of this book is compatible with a range of different views about how to understand
other areas of mathematics. I hope my version of Potentialism will be compelling even
to readers who find both Nominalism and the neo-Carnapian realism about mathemat-
ical objects (outside set theory) I advocate in Part III unacceptable.
Third, I aim to provide a foundation for Potentialist set theory which rests entirely
on intuitively compelling principles that are subject matter neutral and constrain the
behavior of all objects (c.f., Frege’s characterization of logic in Frege (1980)). Thus, in
a sense I’m defending a kind of Logicism about set theory. But I don’t mean to claim
that my foundational principles are analytic, cognitively trivial, or impossible for any
rational being to doubt. I merely claim they’re clearly true in the sense evoked by
Putnam above.7 I also don’t mean to suggest that facts about conditional logical
possibility discussed in this book constitute some kind of metaphysical free lunch.8
Fourth, we must distinguish the foundational project in this book from a less
ambitious justificatory project. Actualist philosophers have sometimes aimed to find
a unified conception of set theory from which all the various ZFC axioms clearly
follow – without worrying whether this conception itself is clearly coherent. This
project can be valuable in various ways, e.g., in showing the naturalness and appeal of
certain mathematical hypotheses (like proposed large cardinal axioms) which also
follow from the relevant conception. However, finding such a unified conception
doesn’t suffice for my foundational project. For if the unifying conception isn’t clearly
consistent then, surely, it isn’t clearly true (even on a view which allows

should start to say in philosophy journals can differ radically from what physics journals would or should
publish. Perhaps Sider’s distinction between metaphysical semantics and linguistic semantics discussed in
Section 11.4.2 suggests a similar line of response (Sider 2011).
However, the motivations I urge for Potentialist set theory are closer to those for foundational projects
within mathematics than the explicitly philosophical motivations Hellman and Sider reference. Thus,
I think the Potentialist paraphrases I advocate might be accepted by extreme naturalist readers, who would
reject the above suggestion that philosophy or metaphysics could provide a legitimate further level of
analysis beneath the sciences. Also note that the motivations for Potentialist set theory I press in this text
aren’t among the specific philosophical motivations for Nominalist formalizations of mathematics which
Burgess and Rosen (1997) criticize.
7
I take the axiom of choice to be prima facie clearly true, despite the fact that it can be doubted on grounds
like the Banach–Tarski paradox. But readers who find Choice less immediately appealing can read this as
a claim to justify Replacement from principles “as prima facie obvious as the other axioms of ZF set
theory” instead.
8
I take accepting a primitive modal notion of (conditional) logical possibility to be a significant, but
warranted, addition to our fundamental ideology.

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 Introduction 9

mathematicians to introduce arbitrary coherent structures). So, we haven’t succeeded

in justifying all theorems of ZFC set theory from premises that seem clearly true.
Finally, the set-theoretic foundational project of this book also differs from a more
ambitious project, along the following lines. Philosophers sometimes seek the most
metaphysically joint-carving successor to the naive concept of sets which generates
Russell’s paradox (something which might, e.g., be hoped to connect intimately with
the true answer to the liar paradox). So, for example, you could ask whether the
iterative hierarchy conception of sets is remotely on the right track, or whether the
“best” successor to naive set theory is something like Quine’s New Foundations
instead.
I think this more philosophically speculative project is legitimate, but not required
for the foundational justificatory project attempted in this book. I will try to show that
Potentialist translations which attractively explicate theorems of current mainstream
set theory follow from principles that seem clearly true. But I won’t take a position on
what the most metaphysically illuminating successor to naive set theory is.

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https://doi.org/10.1017/9781108992756.001 Published online by Cambridge University Press