# Philosophical Uses of Categoricity Arguments

Penelope Maddy
University of California, Irvine  Jouko Väänänen
University of Helsinki

###### Abstract

This book addresses the viability of categoricity arguments in philosophy by
focusing with some care on the nature of the specific conclusions that a
sampling of prominent figures have attempted to draw – the same theorem might
successfully support one such conclusion while failing to support another. It
begins with the historical cases of Dedekind, Zermelo, and Kreisel, casting
doubt on received readings of the latter two, at least, and highlighting the
success of all three in achieving what are argued to be their actual goals.
These earlier uses of categoricity arguments are then compared and contrasted
with more recent work of Parsons and the co-authors Button and Walsh.
Highlighting the roles of first- and second-order theorems, of external and
(two varieties of) internal theorems, the book eventually concludes that
categoricity arguments have been more effective in historical cases that
reflect philosophically on internal mathematical matters than in more recent
uses for questions of pre-theoretic metaphysics.

##  1 Introduction

Mathematicians and philosophers have appealed to categoricity arguments in a
surprisingly varied range of contexts. One familiar example calls on second-
order categoricity in an attempt to show that the Continuum Hypothesis,
despite its formal independence, has a determinate truth value, but this
doesn’t exhaust the uses of categoricity even in set theory, not to mention
its appearance in various roles in discussions of arithmetic. Here we compare
and contrast a sampling of these deployments to get a sense of when these
arguments tend to succeed and when they tend to fail. Our story begins with
two historical landmarks, Dedekind and Zermelo, on arithmetic and set theory,
respectively, and ends with two leading contemporary writers, Charles Parsons
and the co-authors Tim Button and Sean Walsh, again on arithmetic and set
theory, respectively. In between, we pause over the well-known contribution of
Georg Kreisel. In each case,111With the exception of Section 4, where no new
mathematics is involved. we ask: what does the author set out to accomplish,
philosophically?; what do they actually do (or what can be done),
mathematically?; and does what’s done (or can be done) accomplish what they
set out to do? We find this focus on context illuminating: these authors have
qualitatively different philosophical goals, and what works for one might not
work for another.

##  2 Dedekind in ‘Was sind und was sollen die Zahlen? (1888)

###  2.1 What does he set out to accomplish?

Dedekind’s motivation for his treatment of natural numbers in ‘Was sind und
was sollen die Zahlen’ (Dedekind (1888)) is explicit: ‘in science nothing
capable of proof ought to be believed without proof’ (p. 790).222Unless
otherwise indicated, all references in this section are to the English
translation of Dedekind (1888) in (Ewald, 2005, pp. 796-833). Ideally, such a
proof should not depend on the vagaries of geometric intuition or temporal
intuition or anything of that sort; rather, the ‘number-concept’ and the
resulting proofs should emerge as ‘an immediate product of the pure laws of
thought’ (pp. 790-791). This is what Dedekind means when he says that
‘arithmetic [is] merely a part of logic’ (p. 790), so it’s important to note
that these ‘pure laws of thought’ include what we would regard as set
formation:

> It very frequently happens that different things, a, b, c, … for some reason
> can be considered from a common point of view, can be associated in the
> mind, and we say that they form a system. (§2, p. 797)

(Of course, a ‘system’ here is what we’d call a ‘set’.) In sum, then, Dedekind
is out to show that the concept of number can be characterized and the
theorems of arithmetic proved using only logical notions, in his generous
sense of ‘logical’. This would complete the project, initiated in Dedekind
(1872), of founding the calculus and higher analysis without appeal to
geometric or temporal intuition.333The weak set theory of Dedekind (1888) is
enough for the construction of reals via Dedekind cuts in Dedekind (1872).

###  2.2 What does he actually do?

Dedekind begins by articulating some of his logical laws of thought, in
particular, some pre-theoretic metaphysical truths about sets.444 Fortunately,
difficult questions about the status of these sets – e.g., are they understood
realistically or idealistically? – can be set aside for present purposes. The
most important of these, obviously, is the above-noted assumption that ‘very
frequently’ some things ‘can be considered from a common point of view’ and
thus collected into a set. (If this isn’t Unlimited Comprehension, Dedekind
tells us nothing about how it differs; only later does he recognize the danger
(see below).) From there, he develops an early theory of sets: extensionality
and the existence of singletons (§2), unions (§8) and intersections of
families (§17). He introduces functions (‘mappings’, §21) as entities distinct
from sets, assumes that identity functions exist, that the restriction of a
function to a subset is still a function, and that functions can be composed
(implicitly).

He then introduces the key idea of a ‘$\phi$-chain’ (§37):555In (Dedekind, ,
pp. 100-101), Dedekind explains the importance of this notion to his critic,
Keferstein, who thought it could be eliminated. Dedekind points out that a
simply infinite system, $(N,a,\phi)$, could be contained in a larger system,
$S$, with some ‘arbitrary additional elements $t$, to which the mapping
$\phi$’ could be applied and a subset, $T$, containing $t$ and closed under
$\phi$. The question is how ‘to cleanse our system $S...$ of such alien
intruders $t$ as disturb every vestige of order and to restrict it to $N$?’ We
might try saying ‘an element $n$ belongs to the sequence $N$ if and only if,
starting with the element [a] and counting on and on steadfastly $...$ through
a finite number of iterations of the mapping $\phi...$ I actually reach the
element $n...$ But this $...$ would $...$ contain the most pernicious and
obvious kind of vicious circle$...$ Thus, how can I, without presupposing any
arithmetic knowledge, give an unambiguous conceptual foundation to the
distinction between the elements $n$ and the elements $t$? Merely through
consideration of $\textit{chains}...$ and$...$ by means of these, completely!’
He notes that Frege’s ancestral ‘agrees in essence with my notion of chain
$...$ only, one must not be put off by his somewhat inconvenient terminology’.

######  Definition 1.

If $\phi$ is a function from S to S, and $K\subseteq S$, then K is a _$\phi$
-chain_ iff $\forall{x}\in{K}(\phi({x})\in{K})$.

This notion in hand, in Dededkind defines

######  Definition 2.

S is _simply infinite_ iff there is a one-to-one $\phi$ from S to S and an
$s\in{S}$ such that

  1. 1.

there is no ${x}\in{S}$ with $\phi({x})={s}$,

  2. 2.

$S=\bigcap\\{{X}\mid{s}\in{X}\mbox{ and }{X}\mbox{ is a
}\phi\mbox{-chain}\\}$.

By this point, he’s defined Dedekind-infinite (§64) and proved that there’s an
infinite set in the notorious §66: the collection of ‘all things which can be
objects of my thought’ is infinite because a function that sends each element
therein to the thought of that element in fact maps the set to a proper subset
of itself (at least one element is not a thought,‘e.g., my own ego’)! He then
shows that every infinite set contains a simply infinite set.

So, finally, where are the numbers? Here Dedekind abstracts:

> If in the consideration of a simply infinite system N ordered by a mapping
> $\phi$ we entirely neglect the special character of the elements, simply
> retaining their distinguishability and taking into account only the
> relations to one another in which they are placed by the ordering mapping
> $\phi$, then these elements are called natural ${numbers}$. (§73, p. 809)

The precise ontological status of these numbers won’t concern us here, but for
the record they are ‘a free creation of the human mind’ (§73, p.
809).666Exactly what Dedekind intends by ‘free creation’ is beside the point
for our purposes, but it is a fascinating question. See Reck (2003) and Sieg
Morris (2018) for quite different readings.

The so-called Peano Axioms are implicit in the definition of simply infinite:
if $S$ is simply infinite, with $\phi$ from $S$ to $S$, and $s\in S$, then
$A\subseteq S$, $s\in A$, and $A$ closed under $\phi$ implies $A=S$. With the
axioms in place, the usual theorems of arithmetic follow and ‘are always the
same in all ordered simply infinite systems, whatever names may happen to be
given to the individual elements’ (§73, p. 809).

This is certainly true – the consequences of PA hold in all relevant systems –
but it isn’t enough to guarantee that performing the process of abstraction
described in the previous paragraph on different simply infinite systems won’t
generate different natural numbers. Apparently sensitive to this loose end,
Dedekind refers us forward to his proofs that any simply infinite set is
isomorphic to the numbers (§132) and that any set isomorphic to the numbers is
simply infinite (§133). Thus, the simply infinite systems form an isomorphism
class (§34), so it doesn’t matter which one he happens to abstract from in
§73. Only then, in §134, does he conclude that ‘the definition of the notion
of numbers given in §73 is fully justified’ (§134, p. 823). This is the role
of his famous categoricity result in the overall project of showing that the
natural numbers can be characterized without appeal to intuition, in purely
logical terms.

So, how does the proof go? Averting our gaze from ‘free creations’, we sketch
a direct Dedekind-style proof that any two simply infinite sets are
isomorphic:

######  Theorem 1.

Suppose $N$ and $N^{\prime}$ are simply infinite, that is, that there are
$a\in N$ and $a^{\prime}\in N^{\prime}$, and one-one functions $\phi:N\to N$,
and $\phi^{\prime}:N^{\prime}\to N^{\prime}$ such that

$N=\bigcap\\{X:a\in X\mbox{ and $X$ is a }\phi\mbox{-chain}\\}$ (1)

and

$N^{\prime}=\bigcap\\{X:a^{\prime}\in X\mbox{ and $X$ is a
}\phi^{\prime}\mbox{-chain}\\},$ (2)

then $N\cong N^{\prime}$.

###### Proof.

Suppose $N$ and $N^{\prime}$ are as above. To construct an isomorphism between
$(N,\phi,a)$ and $(N^{\prime},\phi^{\prime},a^{\prime})$, proceed by induction
in $N$: for $b\in N$, define $\psi_{b}$, an isomorphism from $b$ and its
predecessors into $N^{\prime}$ such that $\psi_{b}(a)=a^{\prime}$ and
$\psi_{b}(\phi(x))=\phi^{\prime}(\psi_{b}(x))$ (§125). Then let
$\psi(b)=\psi_{b}(b)$ (§126), and show, by induction on $N^{\prime}$, that
$\psi$ is onto $N^{\prime}$. $\psi$ is the required isomorphism (§132).

∎

###  2.3 Does what he did accomplish what he set out to do?

It seems fair to say that Dedekind has achieved his goal of characterizing the
natural numbers without appeal to geometric or temporal intuition, but from
our contemporary perspective, it’s impossible to ignore the freedom with which
he allows ‘different things’ to be ‘considered from a common point of view
[and] associated in the mind’ so as to form a set (§2, p. 797). Dedekind
reports that the dangers of this free use of comprehension came clear to him
in 1903 and troubled him so much that he was at first reluctant to allow a
third edition of his little book:

> When I was asked $...$ to replace the second edition of this work (which was
> already out of print) by a third, I had misgivings about doing so, because
> in the meantime doubts had arisen about the reliability of important
> foundations of my conception. Even today I do not underestimate the
> importance, and to some extent the correctness, of these doubts. (Dedekind,
> 1888, p. 796)

Still, though he was ‘prevented by other tasks from completing such an
investigation’, Dedekind remains confident:

> My trust in the inner harmony of our logic is not $...$ shattered; I believe
> that a rigorous investigation of the power of the mind to create from
> determinate elements a new determinate, their system, that is necessarily
> different from each of these elements, will certainly lead to an
> unobjectionable formulation of the foundations of my work. (p. 796)

As we know, he turned out to be right about this.

Of course, any way of regularizing Dedekind’s reasoning to contemporary
standards will inevitably distort it in some way. Two methods appear in the
literature – one set-theoretic, one second-order – with the latter perhaps
more common. To assess the moral of Dedekind’s work for ourselves today, we
look at each of these in turn.

The first is fairly direct. Dedekind’s presentation is already formulated in
an early set-theoretic context; the only problem is his assumption of what
looks perilously close to Unlimited Comprehension. In his categoricity
theorem, this comes into play when he forms the family of all $\phi$-chains
containing $a$ ($N$ is its intersection). In fact, though, this isn’t
problematic, because all those $\phi$-chains are subsets of $N$, so the
relevant family can be generated by Separation from $\mathcal{P}(N)$. On
closer inspection, the relevant intersection can be formulated as …

$\begin{array}[]{lcl}\\{x\in N:\forall y((\forall z\in y(z\in N)\wedge a\in
y\wedge\forall z\in y(s(z)\in y))\to x\in y)\\}.\end{array}$

… so only $\Pi_{1}^{1}$-Separation is needed. Beyond that, we know today that
much weaker principles suffice: in Simpson Yokoyama (2013), Simpson and
Yokoyama show that $\mbox{WKL}_{0}$ (Weak König’s Lemma777Every infinite
binary subtree of $2^{<\omega}$ has an infinite branch.) suffices over
$\mbox{RCA}_{0}$ (Recursive Comprehension Axiom). This is interesting because
$\mbox{WKL}_{0}$ is weaker than Peano arithmetic itself (i.e.
$\mbox{ACA}_{0}$).

In the second, more common reading, Dedekind’s reasoning is cast, not set-
theoretically, but as proving the categoricity of second-order Peano
Arithmetic.888See, e.g., (Shapiro, 1991, pp. 82-83), (Button Walsh, 2018, p.
155). The conditions on simply infinite sets are then replaced by the second-
order Dedekind-Peano Axioms, and Theorem 1 becomes:

######  Theorem 2.

Suppose $(N,S,0)\models\mbox{PA}^{2}$ and
$(N^{\prime},S^{\prime},0^{\prime})\models\mbox{PA}^{2}$, where $N$ and
$N^{\prime}$ are full models.999 A second-order model is full if it interprets
the second-order variables as ranging over the full power set of the domain –
as opposed to a Henkin model, which only needs enough subsets to satisfy the
axioms of second-order logic (comprehension and choice). Then there is a
bijection $f:N\to N^{\prime}$ such that

$\begin{array}[]{l}f(0_{N})=0_{N^{\prime}}\\\
f(S_{N}(x))=S_{N^{\prime}}(f(x))\end{array}$

that is, $f:N\cong N^{\prime}$.

Dedekind’s proof easily translates to treat models of $\mbox{PA}^{2}$ instead
of simply infinite sets, but it still takes place within his set-theoretic
background theory.

Though the two readings don’t differ in their underlying mathematical content,
the contrast between them is worth noting. In Theorem 1, the conditions on a
‘simply infinite’ set are stated directly: for example, there is no $x\in N$
such that $S(x)=0$. In Theorem 2, those same conditions appear indirectly: in
the same example, N satisfies the axiom ‘there is no x such that $S(x)=0$’.
The latter involves semantic notions, while the former does not; as a
philosopher might put it, the former uses the set-theoretic conditions, while
the latter only mentions them.101010For those unfamiliar with this
terminology, the word ‘cat’ is used in the sentence: the cat is on the mat.
It’s mentioned in the sentence: the word ‘cat’ has three letters. As we’ll use
the terms, Theorem 1 is a weakly internal categoricity argument and Theorem 2
is external.111111Strongly internal or just internal will be introduced and
eventually formalized in Subsection 5.2. Despite the apparent preference in
the literature for the external reading, the weakly internal reading strikes
us as considerably closer to Dedekind’s own thought, but either way, the main
moral we wish to draw remains: Dedekind has successfully achieved his goal;
he’s shown that the natural numbers can be described without appeal to
geometric or temporal intuition.

##  3 Zermelo in ‘On boundary numbers and domains of sets’ (1930)

###  3.1 What does he set out to accomplish?

Perhaps the first thing to note about the aims of this well-known paper
(Zermelo (1930)) is that Zermelo isn’t out to do what many recent observers
have imagined; he isn’t out to introduce the iterative hierarchy, to argue
that the axioms are true therein, and to use this purported fact to motivate
or defend ZFC. In fact, the paper moves in the opposite direction, beginning
with a list of the axioms and arguing that any model thereof – any ‘normal
domain’ – can be stratified into well-ordered ranks, into $V_{\alpha}$’s. In a
subsequent report back to his funding agency on what he’s done so far, Zermelo
catalogs the benefits of this analysis and proposes, as a distinct
continuation, the project of constructing ‘a set-theoretic model’ (Zermelo,
19301, p. 441). A subsequent sketch (Zermelo (19302)) indicates what he has in
mind – laying out a pre-theoretic iterative picture and arguing that the
axioms are true there – but his stated goal even then is to establish the
consistency of the axioms, not their truth. So we need to look a bit more
closely at his intentions in ‘On boundary numbers’ (Zermelo (1930)).

The only aim mentioned in the opening paragraph of the paper is the resolution
of the paradoxes:

> it is this sharp distinction between the different models of the … axiom
> system that allows us to resolve the ‘ultrafinite antinomies’ (Zermelo,
> 1930, p. 29(401))121212The first page reference is to the original
> publication in Fundamenta Mathematica, the second to the English translation
> in Zermelo (2010).

This was also an explicit aim of his original axiomatization in Zermelo
(19082):131313Moore (Moore, 1982, pp.159–160) points out that Zermelo intended
his early axiomatization to bolster his proof of the well-ordering theorem by
clearly isolating the assumptions required. Ebbinghaus (Ebbinghaus, 2007, pp.
76–79) acknowledges this goal, but places Zermelo’s early efforts squarely in
the larger context of Hilbert’s foundational project, where axiomatization and
consistency proofs were central – obviously a reaction to the paradoxes. In
fact, Zermelo was inclined to withhold publication of his Zermelo (19082)
until he had the desired consistency proof, but Hilbert, out of concern for
his young colleague’s immediate career prospects, advised him not to wait. As
noted above, Zermelo continued to pursue a consistency proof in the 1930s.

> In solving the problem [of axiomatization] we must, on the one hand,
> restrict these principles sufficiently to exclude all contradictions and, on
> the other, take them sufficiently wide to retain all that is valuable in
> this theory. (Zermelo, 19082, p. 191(200))141414Page references are to the
> reprinting in Zermelo (2010), followed by the page reference in van
> Heijenoort (1967).

But he was clearly dissatisfied with the resolution given there:

> I have not yet been able to prove rigorously that my axioms are
> ‘consistent’, though this is certainly very essential; instead I have had to
> confine myself to pointing out now and then that the ‘antinomies’ discovered
> so far vanish one and all if the principles here proposed are adopted as a
> basis. (Zermelo, 19082, pp. 191(200-201))

So at least one goal of Zermelo (1930) is to provide a broader and more
explanatory treatment of the paradoxes.

It’s also worth noting the second clause in the first quotation above from
Zermelo (19082): ‘sufficiently wide to retain all that is valuable in this
theory’. Zermelo’s eye is always on the mathematical powers of his
assumptions; in his Zermelo (19081), for example – the ‘new proof’ of well-
ordering, companion to Zermelo (19082) – he defends the axiom of choice on the
basis of its theoretical benefits:

> Principles must be judged from the point of view of science [i.e.,
> mathematics], and not science from the point of view of principles fixed
> once and for all. (Zermelo, 19081, p. 135(189))

Zermelo takes it as obvious that there is much of mathematical value in
infinitary set theory and that axioms can be properly assessed in terms of the
mathematical achievements they enable. (These are so-called extrinsic
justifications, which Zermelo pioneered.151515For the record, extrinsic
considerations weigh the mathematical benefits of a claim, as opposed to its
intrinsic obviousness or connection to the concept of set. See, e.g., Maddy
(2011).) So we should also expect some discussion of the mathematical
consequences of the axioms he considers.

###  3.2 What does he actually do?

Like Dedekind, Zermelo works in a pre-theoretic background theory of
sets,161616Once again (cf. footnote 4), we set aside question about the
metaphysical nature of these sets. but unlike Dedekind, he doesn’t make this
explicit. (The content of that background theory is inferred from his
argumentation in what follows.) He begins, instead, by listing the axioms of
the theory he’ll be studying. Extensionality, Separation, Pairing, Power Set,
and Union carry over from Zermelo (19082). Infinity is omitted ‘because it
does not belong to “general” set theory’ (Zermelo, 1930, p. 30(403)), an
omission whose rationale emerges only later in the paper.171717See footnote
32. Choice is now regarded, not as a set-theoretic axiom, but as ‘a general
logical principle upon which our entire investigation is based’ (ibid., p. 31
(405)), giving us our first glimpse of what’s true in Zermelo’s implicit
background theory. Finally, the explicit list is supplemented by
Replacement,181818Zermelo notes the redundancy this introduces. attributed to
Fraenkel, and Foundation. Of course, the property in Separation and the
function in Replacement are to be understood as ‘unrestricted’ or ‘arbitrary’,
what we now think of as second-order, and Kanamori points out that Foundation,
too, must be second-order in the absence of Infinity.191919See (Kanamori, 2,
pp. 393-394). This system Zermelo calls $\mbox{ZF}^{\prime}$.

Little is said in defense of this selection of axioms. This is understandable
for the carry-over axioms, as they were treated in Zermelo (19081) and Zermelo
(19082), and perhaps it was fair to assume that the case for Replacement has
already been aired in the literature, for example, by Fraenkel and von
Neumann.202020Though see footnote 36. The one comment Zermelo does make
concerns the new axiom of Foundation:

> This last axiom, which excludes all ‘circular’ sets, all ‘self-membered’
> sets in particular, and all ‘rootless’ sets in general, has always been
> satisfied in all practical applications of set theory, and, hence, does not
> result in an essential restriction of the theory for the time
> being.212121It’s not clear to us what work Zermelo’s many quotation marks
> are intended to do here and elsewhere, so we pass them by without further
> comment. (Zermelo, 1930, p. 31(403))

So Foundation is benign, at least as far as currently understood, and perhaps
there is some intuitive appeal to ruling out ‘pathological’ sets, but the full
case in its defense only comes later (see Subsection 3.3).

Zermelo then introduces the central object of his study:

> We call ‘normal domain’ a domain of ‘sets’ and ‘urelements’ that satisfies
> our ‘$\mbox{ZF}^{\prime}$-system’ with regard to the ‘basic relation’ $a\in
> b$. (Zermelo, 1930, p. 31(405))

To the contemporary ear, this sounds like a model satisfying the theory, but
Zermelo isn’t working with a syntax/semantics distinction.222222Cf. Ebbinghaus
(2007), pp. 182-183, describing Zermelo’s discussion of ‘definite properties’
in 1929: ‘there is no sharp distinction between language and meaning. In fact,
Zermelo will never be clear on this point. His negligence will have a major
impact in later discussions with Gödel. The separation of syntax and semantics
or – more adequately – its methodological control became the watershed that
separated the area of the “classic” researchers such as Zermelo and Fraenkel
from the domain of the “new” foundations as developed by younger researchers,
among them Gödel, Skolem, and von Neumann’. To begin with, ‘$\in$’ isn’t to be
reinterpreted: the $\in$ of the normal domain is the $\in$ of the background
theory.232323From a modern point of view, we can see that if
$(M,R)\models\mbox{ZF}^{\prime}$, which includes second-order Foundation, then
it’s isomorphic to the $\in$-model generated by a Mostowski collapse. So the
fact that Zermelo doesn’t entertain such reinterpretation is ultimately less
significant than it might seem. The same is true for the second-order
quantifiers: $N\models$ Separation2 (where Separation2 is second-order
Separation) allows for sparse Henkin-style interpretations of the second-order
quantifier; Zermelo claims, instead, that for every property $P$ and every set
$x\in N$, there is a set $y\in N$ containing just the members of $x$ with $P$,
where the relevant properties include all subsets of $N$ as understood in the
background theory. Or, to take another example, for $N$ to satisfy Power Set
in Zermelo’s sense is for $N$ to include $\mathcal{P}(a)$ for every $a\in N$,
where that’s the real $\mathcal{P}(a)$, as judged from the perspective of the
background theory. Zermelo’s normal domains are what we’d call _full_ models
of the second-order axioms.242424Recall footnote 9.

With the axioms and the notion of normal domain in place, Zermelo begins his
study by introducing the von Neumann ordinals, starting not from the empty set
but from an arbitrary urelement: $u,\\{u\\},\\{u,\\{u\\}\\},...$. Working in
the background theory here, he’s implicitly relying on Pairing. Moments later,
Union is applied to define the successor of the $\alpha$th $u$-based ordinal,
$g_{\alpha}$, as $g_{\alpha}\cup\\{g_{\alpha}\\}$, but more importantly, this
happens in the course of a definition by recursion – revealing that versions
of all the $\mbox{ZF}^{\prime}$ axioms, including Replacement, are present in
the background theory (with the possible exception of Foundation, see
Subsection 3.3). The collection of all ordinals in a given normal domain, $N$,
is itself an ordinal, though not an ordinal in $N$. This ordinal is called the
‘characteristic’ or ‘boundary number’ of $N$, and Zermelo proves that it must
be a (strongly) inaccessible cardinal.

From here, the path to the theorem of interest is direct. It takes one lemma:

######  Lemma 1.

If $N$ is a normal domain, $N^{\prime}\subseteq N$ with the same urelements,
and

  1. 1.

$N^{\prime}$ is transitive, and

  2. 2.

$A\subseteq N^{\prime}$ and $A\in N\rightarrow A\in N^{\prime}$,

then $N^{\prime}=N$.

This is where Foundation (in $N$) makes its essential appearance (in a proof
by contradiction from the assumption that $N-N^{\prime}$ is
nonempty).252525This is also where Kanamori (Kanamori, 2, pp. 393-394) locates
the second-order axiom. Recall footnote 19.

Now suppose $N$ is a normal domain with boundary number $\kappa$. Then let

$\begin{array}[]{lcl}N_{0}&=&\mbox{ the urelements of $N$},\\\
N_{\alpha+1}&=&N_{\alpha}\cup\mathcal{P}(N_{\alpha}),\\\
N_{\nu}&=&\bigcup_{\beta<\nu}N_{\beta},\mbox{ for limit
$\nu<\kappa$}.\par\end{array}$

Given this familiar stratification on $N$ into ranks, all that’s left to show
is that it exhausts $N$:

######  Theorem 3.

$\bigcup_{\alpha<\kappa}N_{\alpha}=N$.

$\bigcup_{\alpha<\kappa}N_{\alpha}\subseteq N$, and
$\bigcup_{\alpha<\kappa}N_{\alpha}$ and $N$ have the same urelements by
definition, so the lemma applies. The proof naturally uses the fact that
$\kappa$ is inaccessible.

This sets up the central theorem:

######  Theorem 4.

If $N$ and $N^{\prime}$ are normal domains with the same boundary number and
equinumerous urelements, then $N\cong N^{\prime}$.

The proof begins from a bijection of the urelements of $N$ to the urelements
of $N^{\prime}$ and builds an isomorphism from $N$ to $N^{\prime}$ in stages
for each rank $N_{\alpha}$ of $N$ in a way that’s now familiar to observers of
set-theoretic categoricity arguments.

Several observations are in order about Zermelo’s line of thought here. To
begin with, though he himself uses the term, this isn’t really a categoricity
theorem in the sense in use today: ‘$\in$’ isn’t given different
interpretations in $N$ and $N^{\prime}$. In fact, if the urelements are
eliminated, the result reduces to the elementary fact, familiar from Set
Theory 101, that any $\in$-model of $\mbox{ZF}^{\hskip 1.0pt2}$ is a
$V_{\kappa}$, for some inaccessible $\kappa$. If you asked Zermelo, he’d no
doubt insist that this is proved in (what we regard as) a second-order
background theory, but all that actually matters is that $N$ and $N^{\prime}$
satisfy the second-order $\mbox{ZF}^{\prime}$ – ordinary, first-order ZFC is
enough in the background; in fact, ZFC restricted to $\Sigma_{2}$-separation
and $\Sigma_{2}$-replacement will do (see Barwise (1972)).

So, ironically, what’s made Zermelo’s proof a touchstone in the contemporary
literature on categoricity arguments is the presence of those urelements.
Because those of $N$ and $N^{\prime}$ are only equinumerous, not identical,
the isomorphism isn’t the identity – it has to be constructed rank by rank,
from the initial bijection. Though Zermelo doesn’t reinterpret ‘$\in$’, he
could have done so with essentially the same proof, which is why many
commentators262626See, e.g., (Hellman, 1989, pp. 66-67), (Button Walsh, 2018,
pp. 178-179) take him to have proved the following theorem about second-order
ZFC ($\mbox{ZFC}^{\hskip 1.0pt2}$):

######  Theorem 5.

If $N\models\mbox{ZFC}^{\hskip 1.0pt2}$ and
$N^{\prime}\models\mbox{ZFC}^{\hskip 1.0pt2}$, where $N$ and $N^{\prime}$ are
full models with the same height and equinumerous urelements, then $N\cong
N^{\prime}$.

Actually, as with Dedekind, it’s probably more faithful to see him as having
proved Theorem 4 in his background theory of sets. In any case, from there he
goes on to observe that if two normal domains have the same urelements, then
one is isomorphic to an initial segment of the other, and that normal domains
starting from the same single urelement (as $\emptyset$) stack neatly in the
order of their boundary numbers.

###  3.3 Does what he did accomplish what he set out to do?

To answer this question, we need to follow Zermelo’s line of thought a bit
further. Working in his background theory,272727Setting urelements aside. he’s
shown that there’s effectively one normal domain of each height, that each of
these breaks down into ranks – essentially the theorem $V=\bigcup_{\alpha\in
Ord}V_{\alpha}$,282828This notation just means: $\forall x\exists\alpha(x\in
V_{\alpha})$. – and that they form a hierarchy matching that of their boundary
numbers. These mathematical facts suggest an image of the universe of sets as
an endless series of neatly stratified domains. This image quickly grew into a
compelling intuitive picture of emphatic richness,292929This includes the
combinatorial ideas of Bernays (1935)., 303030Boolos (1971) is an outlier. His
austere version of the Iterative Conception doesn’t include replacement –
which many would regard as the first affirmation, after the axiom of infinity,
that the sequence of ordinals is ‘very long’ – or choice – which many would
regard as an affirmation that the power set is ‘very thick’ (see Bernays
(1935), p. 260). the Iterative Conception:

> We believe that the collection of all ordinals is very ‘long’ and that each
> power set (of an infinite set) is very ‘thick’. (Wang, 1974, p. 553)

This picture suggests new methods like reflection and maximality arguments,
new axioms like large cardinal and forcing axioms, along with all manner of
elaborations and analyses.313131In a striking example, (Bowler, 2019, §18.3),
argues that the iterative picture ‘isn’t just a nice extra feature that allows
set theory lecturers to draw pretty pictures on the board’, rather ‘it is
vital’ (p. 402): ‘Humans are adapted to build complex tools, which means
building up objects from component parts, themselves built from yet simpler
components’ (p. 403). Zermelo would insist on judging each resulting
mathematical proposal by its extrinsic mathematical merits, and there’s little
doubt that he’d regard the Iterative Conception in this heuristic use as an
impressive source of productive mathematical ideas.

All very good, one might say, but what if there are no boundary numbers?
Zermelo addresses this question directly in the final section of the paper,
beginning with the observation that the hereditarily finite sets form a normal
domain that even the intuitionist would accept.323232Now we see why Zermelo
leaves Infinity off the list $\mbox{ZF}^{\prime}$. The trouble with stopping
here, he goes on to note, is that it doesn’t permit ‘Cantorian set theory’,
and thus would not, in words going back to 1908, ‘retain all that is valuable
in this theory’ (Zermelo, 19082, p. 192(200)). This was Zermelo’s extrinsic
justification back then, and it remains so now: he includes the existence of
$\omega$, the boundary number of the normal domain of hereditarity finite
sets, and with it the larger normal domain whose boundary number is the first
uncountable inaccessible. Only a domain that satisfies
$\mbox{ZF}^{\prime}+$Infinity deserves to be called ‘Cantorian’.

Notice that Zermelo isn’t advocating that we add Infinity to
$\mbox{ZF}^{\prime}$; he’s advocating that we add it to the background theory,
to the theory that determines which normal domains there are and what they’re
like. In that background theory, he claims the right to Cantorian set theory,
rejecting the intuitionist’s effort to restrict set-theoretic inquiry, while
at the same time preserving the normal domain $V_{\omega}$, where
constructivist studies with their own extrinsic virtues can still be carried
out.333333We abuse notation by using $V_{\omega}$ for the hereditarily finite
sets: it suggests the existence of $\omega$, which, of course, is not present
in that normal domain. We hope that this is understood, that the usage doesn’t
mislead, here and elsewhere, where $V_{\kappa}$ is used for the normal domain
whose boundary number is $\kappa$. Thus he takes the non-categoricity of
$\mbox{ZF}^{\prime}$ as an advantage, not a disadvantage: ‘set theory as a
science must … be developed in greatest generality’ (Zermelo, 1930, p.
45(427-428)). The italicized ‘science’ is telling. Here, as in his earlier
writings, it signifies his commitment to what he sees as productive
mathematics. ‘The comparative investigation of individual models’ (ibid.) is
then one aspect of this productive project.

Under the banner of this generality, Zermelo also resists efforts to limit set
theory to ‘the lowest infinitistic domain’, to $V_{\kappa_{1}}$, for
$\kappa_{1}$ the first inaccessible. An axiom to that effect would secure
categoricity, but it would not ‘concern set theory as such’; it would ‘only
characterize the special model chosen by the respective author’ (Zermelo,
1930, p. 45(427)). The better course is to acknowledge that the ordinals of
$V_{\kappa_{1}}$ constitute a boundary number greater than $\omega$ – namely,
$\kappa_{1}$ – and that this uncountable boundary number itself exists as a
set in a normal domain larger than $V_{\kappa_{1}}$.

> It is of course not possible to ‘prove’, that is, to deduce from the general
> $\mbox{ZF}^{\prime}$-axioms, either its existence or non-existence, simply
> because, for instance, the boundary number $\omega$, even though it exists
> in the ‘Cantorian’ domain [$V_{\kappa_{1}}$], does not exist in the
> ‘finitistic’ domain [$V_{\omega}$], because, in other words, the question
> receives different answer in different ‘models’ of set theory, and is thus
> not decided merely by the axioms alone. (Zermelo, 1930, p. 45(427))

So, in the interests of productive science, this and further boundary numbers
should be posited in the background theory:343434Zermelo also provides a
somewhat inscrutable direct argument – what might be regarded as an intrinsic
argument (cf. footnote 15) – for the existence of inaccessibles based on the
assumption that ‘every categorically determined domain can also be conceived
of as a “set” … an element of a (suitably chosen) normal domain’ (Zermelo,
1930, p. 46 (429)). Given that the categorically determined domains are the
$V_{\kappa}$s, this comes to assuming that every $V_{\kappa}$ is a set in a
larger $V_{\kappa^{\prime}}$.

> We must postulate the existence of an unlimited sequence of boundary numbers
> as a new axiom for the ‘meta-theory of sets’ … To the unlimited series of
> Cantorian ordinal numbers there corresponds a likewise unlimited … series of
> essentially different set-theoretic models in each of which the entire
> classical theory finds its expression. (Zermelo, 1930, p. 47(429-431)).

The upshot of this line of thought is (1) an informal background theory that
essentially comes to $\mbox{ZFC}+\mbox{I}$ (where ‘I’ asserts the existence of
arbitrarily large inaccessibles), (2) the development in that background
theory of detailed development of full models of the second-order theory
$\mbox{ZF}^{\prime}$, and (3) the heuristic, intuitive picture of the
Iterative Conception.

To return, at last, to our leading question, has Zermelo’s categoricity
theorem (and the developments he bases on it) accomplished what he set out to
accomplish? His explicit goal was to resolve the paradoxes, which he now
dismisses in one characteristically combative sentence:

> The ‘ultrafinite antinomies of set theory’, to which scientific
> reactionaries … appeal in their fight against set theory with such eager
> passion, are only apparent ‘contradictions’, due only to a confusion between
> set theory itself, which is non-categorically determined by its axioms, and
> the individual models representing it: what in one model appears as
> ‘ultrafinite … ’, is already a fully valid ‘set’ … in the next higher model
> and, in turn, serves itself as the bed-stone in the construction of the new
> domain. (Zermelo, 1930, p. 47(429-431))

The class of all sets (or all ordinals) of one normal domain is just an
ordinary set in the next – fine – but this doesn’t solve the problem for the
background theory: why is there no set of all sets (or all ordinals) there?
Zermelo doesn’t answer this question explicitly, but implicitly, he has a
response of a piece with one offered by some observers to this day: there’s no
set of all sets because every set occurs in some normal domain, and no normal
domain contains all sets.353535He says something very like this in (Zermelo,
19301, p. 439): ‘Every normal domain is itself a “set” in all higher domains,
but there is no highest normal domain containing all all normal domains as
sets’. Today we might simply say there’s no set of all sets because every set
first appears at some stage of the iterative hierarchy, and there’s no stage
at which all sets are available to be collected. Of course, there’s no
contemporary consensus on the proper treatment of the paradoxes, but Zermelo’s
approach remains among the live options.

Still, in truth, it would be a grave mistake to count this approach to the
paradoxes as the sole payoff to Zermelo’s categoricity theorems. In that
retrospective report to the funding agency, Zermelo describes his goal more
broadly than in ‘On boundary numbers’ itself:

> I posed for myself the decisive preliminary question … How does a ‘domain’
> of ‘sets’ and ‘urelements’ have to be constituted so that it satisfies the
> ‘general’ axioms of set theory? Is our axiom system ‘categorical’ or are
> there a multitude of essentially different ‘set-theoretic’ models? (Zermelo,
> 19301, p. 437)

Giving answers to these questions is obviously a leading achievement of his
Zermelo (1930), but in Zermelo (19301), Zermelo goes further. We noted earlier
that the only defense of Foundation in Zermelo (1930) is the observation that
it ‘does not result in an essential restriction of the theory for the time
being’ (Zermelo, 1930, p. 31(403)); now, in Zermelo (19301), he explicitly
acknowledges its indispensable role in the results of that paper:363636Oddly,
even in Zermelo (19301), he doesn’t acknowledge the decisive role of
Replacement in his Zermelo (1930). He remarks that it was Hausdorff’s
introduction of cofinality ‘that made possible a fruitful application of the
new axiom’ (Zermelo, 19301, p. 435), omitting reference to von Neumann’s use
of Replacement for Transfinite Recursion, on which Zermelo (1930) obviously
draws (cf. (Kanamori, 1, p. 432)).

> In order to tackle [these goals] successfully, however, I first had to
> supplement the ‘Zermelo-Fraenkel axiom system’ by adding a further axiom,
> the ‘foundation axiom’ … By using the new axiom it was possible to carry out
> a decomposition into layers … of a ‘normal domain’ … and to answer the
> decisive main question in the ‘isomorphism theorems’. (Zermelo, 19301, p.
> 437)

The fact that Foundation enables the clarifying and fruitful breakdown of $V$
into the $V_{\alpha}$s constitutes a strong extrinsic argument in its favor,
both as an axiom of $\mbox{ZF}^{\prime}$ and as an assumption of the
background set theory.

Finally, beyond a viable candidate response to the paradoxes, beyond the
clarification and insights arising from the theorem $V=\bigcup_{\alpha\in
Ord}V_{\alpha}$ and its surroundings, beyond persuasive defense of the new
axioms of Foundation and Inaccessibles, ‘On boundary numbers’ introduced the
Iterative Conception, whose mathematical fruitfulness for the future of set
theory can hardly be overstated. In the end, then, Zermelo accomplished more
than he set out to do – and ultimately more than he could have realized at the
time – so this application of categoricity arguments must be counted as a
resounding success.

##  4 Kreisel in ‘Informal rigor and incompleteness proofs’ (1967) and ‘Two
notes on the foundations of set theory’(1969)

Kreisel’s philosophical discussions of categoricity arguments in Kreisel
(1969) and Kreisel (1967) have continued to influence contemporary thought on
the significance of such results, so we pause to ask what conclusions he
intends to draw and to what extent he succeeds. (Since he presents no new
mathematics, we dispense with our usual division into subsections.)

Looking back at ‘On boundary numbers’, Zermelo notes in passing one
consequence of its categoricity results:

> From this it already follows, among other things, that Cantor’s … conjecture
> … does not depend on the choice of the model, but that it is decided (as
> true or as false) once and for all by means of our axiom system. (Zermelo,
> 19301, p. 437)373737Zermelo is actually referring to GCH, but here we keep
> the focus on CH. The claim is untenable, if not false, for GCH.

Understood as a theorem in Zermelo’s background set theory, the claim is that
CH has the same truth value in every normal domain,383838Really, every normal
domain with the same number of urelements. Our focus here is on ‘unit’ normal
domains, that is, normal domains with one urelement playing the role of
$\emptyset$, so from now on, we ignore these niceties. but Kreisel
reformulates it as a fact of second-order logic:

$\mbox{ZFC}^{\hskip 1.0pt2}\models CH$ or $\mbox{ZFC}^{\hskip
1.0pt2}\models\neg CH$

What conclusion(s) does he draw from this?

Kreisel is often taken to have argued that this theorem of second-order logic
shows that CH has a determinate truth value (even though, sadly, we haven’t
been able to figure out what that is).393939See, e.g., (Button Walsh, 2018, p.
180). In response, observers point out that if we’re concerned that CH may not
have a determinate truth value because we worry that the power set operation
isn’t fully determinate, then we are, or should be, just as concerned about
the determinateness of the second-order quantifiers.404040Weston was first to
note this in print (see Weston (1976)). Button and Walsh (Button Walsh, 2018,
p. 158, note 17) list several others since. (Cf. Quine’s assessment that
higher-order logic is ‘set theory in sheep’s clothing’ (Quine, 1970, p. 66).)
In contrast, Zermelo’s set-theoretic proof that that all normal domains agree
on CH might as well be carried out in first-order ZFC. The resulting
assessment is that Kreisel was wrong and no progress on CH has been made.

The surprise is that Kreisel is well aware of this problem, as he remarks in a
long endnote:

> we still have the following simple-minded puzzle: is it not circular to use
> second order notions which involve the concept of set (or: subset) in
> axiomatizations of this concept? (Kreisel, 1969, p. 111, endnote 2)

This raises serious questions about what conclusions, if any, Kreisel is
inclined to draw about the determinateness of CH – there’s much to ponder in
this endnote (see below) – but it’s reasonable to assume that his central
arguments appear, not there, but in the main text.

In fact, both articles are entirely explicit about his goal in appealing to
the categoricity of $\mbox{ZFC}^{\hskip 1.0pt2}$. Writing in the late 1960s,
Kreisel was keen to isolate and articulate the significance of Cohen’s recent
independence results, as were many others in the set-theoretic and larger
intellectual communities. (Kreisel also intended to improve ‘logical hygiene’
(Kreisel, 1969, p. 109) by rebutting various popular misreadings.) Kreisel has
a lot to say about this, some of it admittedly inconclusive, but he emphasizes
one clear point in light of Zermelo’s result, namely, the importance of

> Distinctions formulated in terms of higher order consequence. In contrast to
> the example of CH …, Fraenkel’s replacement axiom is not decided by
> Zermelo’s Axioms (because [Zermelo’s axioms plus Infinity are] satisfied by
> [$V_{\omega+\omega}$] and Fraenkel’s axiom is not); in particular it is
> independent of Zermelo’s second order axioms while by Cohen’s proof, CH is
> only independent of the first order schema (associated with the axioms) of
> Zermelo-Fraenkel. (Kreisel, 1967, pp. 150-151)

Returning to this point from Kreisel (1967) in Kreisel (1969), he puts it this
way:

> The continuum hypothesis $CH$ is (provably) not independent of the full
> (second order) version of Zermelo’s axioms; we know this much without
> knowing which way $CH$ is decided; the example is not empty since, for
> instance, the replacement axiom is second-order independent. Needless to
> say, from the point of view of our present knowledge of the hierarchy of
> sets, it is of great interest to establish what can and what cannot be
> decided from the first order schemas or, as we might put it, from our
> present analysis of the full axioms. (Kreisel, 1969, p. 107)

In addition to its inherent importance, this demonstration of the distinction
between first- and second-order independence also facilitates one of Kreisel’s
exercises in ‘logical hygiene’: the popular analogy with the independence of
the parallel postulate is misleading because the parallel postulate is
independent of second-order geometry; the parallel postulate ‘corresponds to
Fraenkel’s axiom, not CH’ (Kreisel, 1967, p. 151).414141See also (Kreisel,
1969, p. 111, endnote 2.).

But now, what about the determinateness of CH? In the endnote quoted above,
he’s considering the _standard argument_ so often attributed to him, an
argument to the conclusion – ‘subset of the reals’ is determinate – from the
premise – a second-order quantifier over the reals is determinate. If the
familiar reading of this paper were correct, we’d expect Kreisel to launch a
direct defense against the circularity objection, but he doesn’t. What he says
is:

> Sure, it would be circular if one were looking for a reduction, a definition
> of this concept [of set (or: subset)] in, say, more elementary terms.
> (Kreisel, 1969, p. 111, endnote 2)

Well, yes, something like that is precisely what we’re looking for in the
standard argument: we’re supposed to have confidence that the second-order
quantifier is determinate – it’s just logic, don’t you see? – and that
confidence is supposed to be transferred to the notion of subset.424242For
comparison, we might be suspicious of the semantic notion of first-order
consequence and find that suspicion relieved by the discovery that it’s
equivalent with a syntactic one (Kreisel’s own example). A few lines later, he
continues – ‘to suspect a vicious circle we should have to have independent
reasons, such as the ambiguities mentioned at the beginning’ (ibid.) of the
paper, namely concerns about ‘the basic notion of set’ (Kreisel, 1969, p. 94).
Well, yes, again, the standard objection points out that someone who begins
with worries about the concept of set won’t be helped by the standard
argument.

So it seems that such defense as Kreisel offers for the determinateness of CH
isn’t the standard argument, but a direct case against a few of the reasons
one might have to worry about the notion of subset, and hence about CH, in the
first place. The two reasons he considers are based on predicativism and
finitism, respectively, and he argues that they’re both ‘quite inconclusive’
(Kreisel, 1969, p. 96). But surely there are other, more compelling grounds on
which to doubt the determinacy of the notion of subset.

On this score, Kreisel himself doesn’t seem entirely confident. Speaking just
epistemologically, about our ability to determine the status of CH, he writes:

> There are a lot of subsets of $\omega$, there are a lot of 1-1 mappings from
> $\omega$ to an initial segment of the ordinals; we certainly do not know …
> any listing of the subsets of $\omega$ by means of ordinals. So why should
> we expect to know the answer to this particular ‘simple’ matter? (Kreisel,
> 1969, p. 109)
>
> Doesn’t one simply have an overwhelmingly strong impression that though we
> don’t know how to decide CH, any decision would involve considerations of a
> quite different character from those which have led to existing axioms?
> (Ibid., p. 108)

He observes that these new considerations needn’t be “arbitrary” or even non-
mathematical’ (ibid.).

He then moves from an epistemic worry that we don’t know how to find the truth
value of CH to a worry at issue in the standard argument, the worry that its
truth value might be metaphysically indeterminate. The response he considers
is based on what he calls ‘realism’,434343He leaves many specifics of what he
means by this far from univocal term to his reader’s imagination. which he
criticizes on exactly this point earlier in the paper:

> there is or, at least, seems to be a genuine defect in the realist position
> generally and, in particular, in the unanalyzed use of the power set
> operation. (Kreisel, 1969, p. 97)

Given that he links the determinateness of power set with realism and that he
takes this aspect of realism to be problematic, it’s unlikely that vanquishing
predicativism and finitism has left him immune from other worries about the
notion of subset – and we’ve seen that he doesn’t expect such worries to
evaporate in the face of the standard argument.

To better understand Kreisel’s attitude here, consider another criticism he
lodges against realism, this one more methodological than epistemological or
metaphysical:

> By the second order decidability of CH, [realism] demonstratively does not
> provide a framework within which one can discuss the impression [that
> considerations of a quite different character would be needed to settle it].
> ((Kreisel, 1969, p. 108), emphasis in the original)

It isn’t immediately obvious why he thinks realism can’t provide a framework
for discussing the need for new methods, but there are clues. After dismissing
the objections from predicativism and finitism, Kreisel tries out a realistic
posture himself and examines ‘the cumulative type structure’. From that
perspective, he finds that ‘we have very good evidence for some of the axioms
of infinity’, for example, ‘when one recognizes that [a candidate] is a
consequence of a second-order reflection principle then one has found good
evidence’ (Kreisel, 1969, pp. 98-99). Why is this evidence good? Because ‘the
validity of the reflection principle becomes apparent on closer analysis of
the hierarchy’, because ‘in terms of knowledge of the unbounded hierarchy,
reflection principles are quite evident’ (ibid.) – and he goes on to give an
intrinsic argument based on the unattainability of $V$. So perhaps he thinks,
as it not uncommon, that the only justificatory methods available to the
realist are intrinsic, and given that he also thinks progress on CH will
require methods of ‘a quite different character from those which have led to
existing axioms’ (ibid., p. 108), it would follow that realism hasn’t the
means to address the fundamental challenge posed by the phenomenon of its
independence.

As an alternative to realism, he also considers what he calls
‘formalism’.444444As with ‘realism’ (see footnote 43), the particulars of this
position, it’s relation to other views of the same name, are left unexplored.
Presumably a formalist eschews metaphysics, so isn’t bound by the idea that
only intrinsic considerations can be trusted to track the contours of some
abstract subject matter. This leaves open the possibility of new methods, but
unfortunately, ‘the formalist position does not normally attempt an analysis
of the choice of system at all’ (Kreisel, 1969, p. 108). Notice that this
isn’t a principled argument that a formalist couldn’t take this on, just a
complaint that his contemporary formalists don’t – the possibility remains
open of a version of formalism that does. Such a formalism might appeal, for
example, to mathematical effectiveness in its justifications, to extrinsic
justifications – following Zermelo.454545The first author, predictably, thinks
of the Arealism of her Maddy (2011) or the Enhanced If-thenism of her Maddy
(20221). Kreisel comes close to this theme in a discussion of ‘pragmatism’: he
grants it a ‘superficial plausibility’, but complains that it discourages
‘work on … intuitive notions’ (Kreisel, 1967, pp. 140-141). But this needn’t
be true; e.g., we’ve already noted the immense heuristic value of Zermelo’s
Iterative Conception (cf., e.g., (Maddy, 2011, p. 136)). Kreisel also worries
that a philosopher with such views will be left with nothing left to do, but
the first author obviously disagrees. In fact, Kreisel sees encouraging
exploration of new methods as an ‘attempt to use [independence results] for
some objectively significant purpose’ (Kreisel, 1969, p. 110).464646Kreisel
(1969) closes with an implicit appeal for more such exploration: ‘Here it must
be admitted that, if CH, resp. $\neg$ CH is to be distinguished from other
axioms of set theory by the kind of consideration needed to decide it, the
crudity of current discussions is certainly not surprising. For though
foundations have made striking progress in the precise analysis of notions,
the analysis of reasons has been much less successful; and what has been done
is little known’ (Kreisel, 1969, p. 110).

Finally, one last observation from the fascinating endnote 2 of Kreisel
(1969). We’ve seen that Kreisel regards the standard argument from the
determinateness of second-order logic to that of CH as potentially viciously
circular, but he also locates a benign circle in the vicinity. We might, he
suggests, simply ‘use a concept in order to state the facts about it’
(Kreisel, 1969, p. 111, endnote 2) and illuminate or develop a concept in this
way. Consider, for example, the use of logical particles to advance an account
of the satisfaction conditions for sentences containing the logical particles.
He doesn’t elaborate, but the idea must be that we better understand the
subset operation and the second-order quantifier when we appreciate their
interconnection in the case of CH.

In sum, then, Kreisel sees Zermelo’s categoricity arguments as accomplishing
at least two things. Since his overarching goal is to probe the significance
of the independence results, his main focus is on the first of these:
revealing that CH presents a new kind of independence – first-order but not
second-order – which differentiates it from familiar cases like Replacement,
large cardinals, or the parallel postulate. Along the way, in his analysis of
the standard argument for the determinacy of CH, he observes that the
categoricity results also provide a sort of elucidation or illumination of the
concept of subset, by bringing out its interconnections with second-order
quantification. As for the standard argument as intended, he doesn’t appear to
take the threat of circularity to be fully dispelled, but the determinateness
of CH wasn’t his target in the first place. At his actual goal – elucidating
the independence phenomenon – he succeeds.

##  5 Parsons in ‘The uniqueness of the natural numbers’ (1990) and
‘Mathematical induction’ (2008)

###  5.1 What does he set out to accomplish?

In Chapter 8 of his book, Mathematical Thought and its Objects Parsons (2008)
– especially ‘The problem of the uniqueness of the number structure’ (§48) and
‘Uniqueness and communication’ (§49) – Charles Parsons develops and refines
the line of thought in his 1990 paper, ‘The uniqueness of the natural numbers’
Parsons (1990). His topic isn’t the uniqueness of individual natural numbers,
like 3; as a structuralist, Parsons holds that various objects, perhaps
various sets, for example, could play the role of 3 in the natural number
structure. Rather, his concern is whether

> the natural numbers are at least determinate up to isomorphism: If two
> structures answer equally well to our conception of the sequence of natural
> numbers, they are isomorphic. (Parsons, 2008, §48, p. 272)

A claim like this would be vacuous if there were no structure answering to our
conception, but Parsons argues, like Dedekind before him, that there is at
least one such (see below). As he understands it, ‘the initial, intuitive
ground’ (Parsons, 2008, §48, p. 272) for concern is that our conception might
be vague somehow, raising the possibility that it could be made precise in
different ways.

The conception Parsons has in mind is given by collection of rules stating:

$\begin{array}[]{c}N(0)\\\ \\\ N(x)\rightarrow N(S(x))\\\ \\\ S(0)\neq 0\\\
\\\ x=yS(x)=S(y)\\\ \\\ P(t)\lx@proof@logical@and
P(0)P(S(x))P(x)N(t)\end{array}$

for any predicate $P$ and term $t$.474747See (Parsons, 2008, §§31, 47) An
‘inescapable vagueness’ (Parsons, 2008, §47, p. 270) arises because this
characterization involves ‘induction as an inference that could be made with
any well-defined predicate, without the prospect of specifying exactly what
the range of such predicates is’ (ibid., §48, pp. 272-273).484848Parsons often
discusses a purported threat of vagueness rooted in the existence of non-
standard models, but ultimately concludes that it is ‘quite unconvincing’
(Parsons, 2008, §48, p. 279). We happily leave this aside, along with the
difficult question of how a structure answering to a conception – e.g., the
Hilbertian stroke types described below – relates to a model of set-theoretic
model theory. The ‘modelism’ of Button Walsh (2018) appears to solve this
problem by fiat: ‘mathematical structures, as discussed informally by
mathematicians, are best explicated by … a class of isomorphic models’ (p.
38). We call this understanding of a predicate variable ‘open-schematic’.

The potential problem comes into sharper focus when we attend to Parsons’s
example of a structure that answers to our conception.494949See (Parsons,
2008, §§28-29). Here and elsewhere, we use Parsons’s own locution – ‘answers
to our conception’ – because we’re unsure what would count as a fair
paraphrase. More generally, our goal in this subsection is to describe
Parsons’s view, not to defend or endorse it. He starts from Hilbert Hilbert
(1926):

> Following Hilbert we begin by considering the ‘syntax’ of a ‘language’ with
> a single basic symbol ‘$|$’ (stroke), whose well-formed expressions are just
> arbitrary strings containing just this symbol, i.e., $|$, $||$, $|||$, …
> (Parsons, 2008, §28, p. 159)

On Parsons’s account, we perceive these concrete tokens of strings. The string
types he calls ‘quasi-concrete’: they are abstract and acausal, but also
typically intuited in the process of perceiving their tokens (assuming the
perceiver enjoys the modest conceptual resources required).505050Again
following Hilbert, Parsons (Parsons, 2008, §28, p. 161) says that ‘intuition
of a type [is] founded on perception of a token’. Only typically, though,
because concrete strings can also be imagined and their types intuited in that
way.

But there are finite limits on the length and number of concrete inscriptions
and even on the human capacity to imagine, so how do we get from here to a
potentially infinite sequence of types? How do we come to know that ‘each
string of strokes can be extended by one more’ (Parsons, 2008, §29, p. 173)?
Parsons’s answer involves imagining ‘an arbitrary string of strokes’ (ibid.).
As with Locke’s idea of a general triangle – neither equilateral, nor
isosceles, nor scalene –

> imagining an arbitrary string involves imagining a string of strokes without
> imagining its internal structure clearly enough to imagine a string of $n$
> strokes for some particular $n$. (Parsons, 2008, §29, p. 173)

He compares this with imagining a ‘crowd at a baseball game without imagining
a crowd consisting of 34,793 spectators’ (ibid., p. 173-174). This will form
Parsons’s basis:

> To see the possibility of adding one more, it is only the general structure
> that we use, and not the specific fact that what we have before us was
> obtained by iterated additions of one more. (Ibid., p. 175)

Building on this idea of a general structure, Parsons offers two ways of
justifying the claim that one more can always be added – involving our
perception of figure/ground or of temporal succession515151 The first is
derived from our perceptual ability to ‘shift [our] attention so that what was
previously ground is now figure’ (Parsons, 2008, §29, p. 177) and the other
(developing Brouwer’s ‘temporal two-ity’) is derived from the fact that ‘we
experience the world as temporal, and have the conviction that we can continue
into the proximate future, in which the immediate past is retained’ (ibid.).
Whatever Parsons may have intended, he offers no reason to suppose these are
anything other than ordinary cognitive mechanisms studied in empirical
psychology. – with the upshot that ‘our insight into [the necessity that every
string can be extended] is insight into experienced space and time’ (ibid., p.
178). The Kantian echo is obvious, but notice that the faculties involved in
Parsons’s Hilbertian intuitive picture525252We call this an ‘intuitive
picture’ rather than a ‘concept’ or ‘structure’ to indicate that it’s not an
epistemic or metaphysical but a psychological pheonemonon. (as we’ll call it)
aren’t aspects of transcendental psychology or philosophical constructs like
‘rational intuition’535353Parsons does discuss a form of rational intuition in
Chapter 9 of Parsons (2008), but in explicit contrast to the intuition of
Chapter 5, the notion engaged here. (Recall footnote 51.) but aspects of
ordinary human perception and cognition.

Perhaps it’s unsurprising that a concept instantiated by such a structure,
based in human imagination, would be considered potentially vague in one
troublesome sense or other. Like the imagined baseball crowd, our imaginings
are indeterminate in indefinite range of aspects. (Try imagining a perfect
summer day or the Mona Lisa, then ask yourself how many questions you can
honestly answer about the features of the resulting ‘image’.) As for Parsons’s
imagined Hilbertian series of strings of strokes, ‘what if the picture began
to flicker in the far distance?’ (Wittgenstein, Remarks on the foundations of
mathematics, (Wittgenstein, 1978, V.10, p. 268)).

What Parsons wants is assurance that nothing like this happens, that our
concept of natural number specifies a unique and determinate structure.
Following Michael Dummett, he poses the problem in terms of communication: how
can I be sure that your numbers are like mine?545454 The only argument for
this move – from a question about the determinateness of a structure to a
question about communication – that we could find in the two Dummett essays
that Parsons cites (M. Dummett (1963, 1967)) appears in the course of his
evaluation of a version of Platonism, in particular, one based on an analogy
between intuition of abstract structures and ordinary perception. If our
intuition of the structure of natural numbers couldn’t be communicated, ‘this
would reduce the intuitive observation of abstract structures to something
private and incommunicable – the analog not of observation of the physical
world in the normal sense, but of the experience of sense data as conceived by
those philosophers who hold these to be private and incommunicable’ (M.
Dummett, 1967, p. 210). But Parsons’s concern isn’t with Platonism, much less
this particular variety thereof; he’s posed his challenge in terms of concepts
– e.g., is your understanding of the concept of natural number the same as
mine? – not in terms of reference, as some do – are you talking about the same
numbers as I am? (See, e.g., the ‘objects-modelism’ of (Button Walsh, 2018,
pp. 144-145).) This focus on concepts rather than ontology explains the focus
on actual human faculties noted above. Maybe I’m working with one
precification of our shared vague concept and you’re working with another. Or,
as Parsons prefers to pose the question, consider two speakers of a first-
order language of arithmetic, Kurt and Michael, both of whom embrace the Peano
axioms, and then, ‘we might ask how one could come to know that his “numbers”
are isomorphic to the other’s’ (Parsons, 2008, §49, p. 283). This is the
question Parsons hopes to answer by appealing to a categoricity argument.
Exactly how a positive answer would bear on the original question of the
determinateness of our concept is a topic for Subsection 5.3.555555Recall
footnote 54.

###  5.2 What does he actually do (or can be done)?

Given this goal, one’s first thought is Dedekind’s theorem (Subsection 2.2),
and that’s precisely where Parsons’s discussion begins.565656See (Parsons,
2008, §§10-11). The first version of Dedekind’s result, Theorem 1, is a weakly
internal argument (the description of simply finite systems is used not
mentioned575757Recall footnote 10.) based on what might as well be a weak
first-order set theory. Parsons would presumably agree to the conclusion
Dedekind draws – that the structure of the natural numbers can be described
without appeal to temporal or geometric intuition – though his Hilbertian
instance of that structure depends on both perceptual and imaginative
intuitions.585858Both Dedekind and Parsons appeal to roughly psychological
instantiations of their structures – Dedekind’s thoughts of thoughts of
thoughts and Parsons’s Hilbertian perceived and imagined strokes – to
establish that their structural descriptions aren’t vacuous. They differ when
Dedekind insists on abstracting (creating) the numbers themselves; Parsons is
content to rest with the structure. Still, to Parsons’s mind, the role of set
theory is problematic.

Early on in Parsons (2008), he writes that ‘we do not think of elementary
number theory as involving commitment to properties, sets, classes, or Fregean
concepts’ (Parsons, 2008, §5, p. 20), a sentiment repeated in the later
discussion of uniqueness:

> It is a commonplace in the foundations of mathematics that the idea of
> natural number is more elementary than that of set. (Parsons, 2008, §48, p.
> 274)

Presumably, this means that Kurt could believe the Peano axioms while innocent
of the concept of set, so this version of Dedekind’s argument is of no use to
him.595959In response, it might be argued that the concept of finite set is
essentially bound up with the concept of natural number, so that Kurt
necessarily believes some set theory along with his assumptions about numbers.
What Parsons wants, then, is a categoricity theorem for arithmetic that relies
on no resources beyond what’s available to Kurt himself, no resources beyond
those of number theory itself – what might be called a pure categoricity
argument.606060We allude here to related themes of purity of method in the
philosophy of mathematics. For an overview, see, e.g., Detlefsen Arana (2011).

The external version of Dedekind’s result, Theorem 2, also depends on set
theory and thus faces the same impurity objection that Parsons raises to
Theorem 1. Still, the introduction of second-order semantics reminds us that

$N\models\mbox{PA}^{2}\land N^{\prime}\models\mbox{PA}^{2}\rightarrow N\cong
N^{\prime}$

is, in fact, a truth of second-order logic alone. In this form, it’s obviously
subject to a number-theoretic version of the challenge raised in Section 4,
the one Kreisel hasn’t fully overcome: if you’re worried about the
determinateness of the range of predicates allowed in the open-schematic
induction axiom, as Parsons is, then you’re also nervous about the
determinateness of second-order quantification. More in line with Parsons’s
thinking,616161See Parsons (2008), pp. 45-46. the second-order logical truth
can be recast without semantics, as a weakly internal theorem in syntactic
second-order logic. Let $\mbox{PA}^{2}(X,Y,Z)$ be the second-order Peano
axioms formulated for a unary predicate, a unary function symbol, and a
constant, and let $\mbox{\rm ISO}((X,Y,Z),(X^{\prime},Y^{\prime},Z^{\prime}))$
be the second-order sentence claiming that there’s an isomorphism between
$(X,Y,Z)$ and $(X^{\prime},Y^{\prime},Z^{\prime})$. Then626262$\vdash_{2}$ is
deriviability in the usual axiom system of second-order logic: first-order
axioms, introduction and elimination rules for the second-order variables,
plus all instances of comprehension and choice (see e.g., (Shapiro, 1991, pp.
66-67), (Button Walsh, 2018, p. 34).)) Button and Walsh, ibid., pp. 240-241,
recommend that Parsons content himself with this result. The text indicates
why Parsons rules this out.

######  Theorem 6.

$\vdash_{2}\forall
N_{1},S_{1},0_{1},N_{2},S_{2},0_{2}((\mbox{PA}^{2}(N_{1},S_{1},0_{1})\land\mbox{PA}^{2}(N_{2},S_{2},0_{2}))\rightarrow\mbox{\rm
ISO}((N_{1},S_{1},0_{1}),(N_{2},S_{2},0_{2})))$.

The proof involves only $\Pi_{1}^{1}$-comprehension, but Parsons echoes the
same concern about even that much: there’s ‘a question whether the second-
order quantifiers … can have a definite sense’ (Parsons (2008), §47, p. 270).
In particular, even $\Pi_{1}^{1}$-comprehension is committed to a well-
determined range of predicates of natural numbers.636363Presumably he’d raise
an analogous objection to the set-quantifiers in the weak set-theoretic
versions of Dedekind’s argument, even apart from the impurity problem.

So, given Kurt’s epistemic situation, Parsons wants a pure theorem. Given his
objections to second-order logic, either semantic or syntactic, it must also
be first-order. Finally, in light of these two desiderata, any appeal to
semantic notions is problematic, so such an argument must be weakly internal.
(Notice that the preference for internalness isn’t for its own sake but as a
means toward achieving the other two desiderata.) To sum up: the goal is a
pure weakly internal argument – let’s call this strongly internal or just
internal646464For the record, the term ‘internal categoricity’ was first used
in Walmsley (2002). After Parsons (1990), the idea occurred more or less
explicitly throughout the 1990s and beyond, for example, in Shapiro (1991),
Lavine (1994), McGee (1997), and Lavine (1999), as well as Väänänen Wang
(2015) and Väänänen (2021).  – that’s also first-order. This is the bar
Parsons sets himself.

Before asking how Parsons undertakes to do this and how his sketch can be
developed, we pause to note some fundamental differences between Dedekind’s
project and Parsons’s – in the role of a background metaphysics and in
motivation. Both begin with a kind of pre-theoretic metaphysics: Dedekind his
sets and Parsons his structures. (We haven’t and won’t attempt to pinpoint
their precise ontological status – realistic, idealistic, or otherwise.)
Dedekind begins by codifying a few relevant features of his sets and using
those assumptions to study something else (the concept of natural number). In
contrast, Parsons never produces an explicit theory of structures – they
remain pre-theoretic, doing no explicit mathematical work – but they are,
nevertheless, the ultimate object of study (in particular, the natural number
structure). On motivation, Dedekind has no doubts about the determinateness of
his concept of natural number, he’s only interested in what it takes to
characterize the relevant structure, while for Parsons, the whole inquiry
arises from his worry that the concept might be vague or otherwise
indeterminate. So, despite the fact that both are focused on the concept of
natural number, despite the fact that Parsons appropriates the mathematical
core of Dedekind’s theorem, their philosophical perspectives and their goals
are quite different.

So, how does Parsons hope to pull this off; how is Kurt to prove that his
numbers are isomorphic to Michael’s using only first-order number-theoretic
tools? We know the first-order Peano axioms aren’t enough, but we’ve ruled out
a move to second-order. Perhaps ironically, Parsons hopes to locate a happy
medium between too weak and too strong by exploiting the open-schematic
character of the induction rule in his characterization of the concept of
natural number, the very feature that led to his worry over vagueness in the
first place.

The idea is that open-schematic induction takes us beyond the usual first-
order axiom, which allows only formulas from a fixed vocabulary, but not far
enough to generate a second-order quantifier:

> In … referring to arbitrary predicates, the statement of the rule makes no
> assumption about what counts as a predicate. … The rule is not a
> generalization over a given domain of entities and could not be, because it
> is not determined what predicates will or can be constructed and understood.
> … Replacing the rule by an axiom, that is, a single statement expressed by
> quantification over sets … is not suitable for the explanation of the number
> concept by rules that we have been engaged in. ((Parsons, 2008, §47, pp.
> 269-270))
>
> This understanding of induction implies that the applicability of the rule
> is not limited to predicates defined in some particular first-order language
> such as that of first-order arithmetic. But we must not take it as implying
> the unavoidability or even the legitimacy of for second-order logic. …
> second-order logic is not forced on us. (Parsons, 2008, §47, p. 270)

Not all observers agree on this last point. For example, see Shapiro – ‘the
effect is the same as allowing only initial universal quantifiers over second-
order variables’ (Shapiro, 1991, p. 246) – or Field – ‘so we have an analog,
in the schematic first-order case, of assertions of $\Pi_{1}^{1}$ second-order
sentences’ (Field, 1998, p. 354) – or Button and Walsh – ‘this is to fall back
on (a syntactic fragment of) full second-order logic … our grasp on the idea
of totally open-ended induction is exactly as precarious as our grasp on full
second-order quantification’ (Button Walsh, 2018, pp. 240, 163).

This is a debate we hope not to engage. If these observers are wrong and open-
schematic induction isn’t just a re-description of a weak form of second-order
logic, then it’s hard to see how a precise mathematical treatment could be
given; it would require a formal account of ‘any well-defined predicate’ that
avoids ‘specifying exactly what the range of such predicates is’ (Parsons,
2008, §48, pp. 272-273). We prefer a different route in pursuit of Parsons’s
goal: investigating what unproblematically first-order resources Kurt would
need to argue for the desired isomorphism.

So, suppose Kurt speaks of his numbers, his successor function, and his first
number, Michael speaks of his, and they believe many of the same arithmetic
truths. How can Kurt be sure they’re talking about the same structure? To pose
the question more precisely, let $N_{1}$ and $N_{2}$ be unary predicate
symbols, $S_{1}$ and $S_{2}$ unary function symbols, and $0_{1}$ and $0_{2}$
constant symbols.656565As it happens, $+$ and $\cdot$ are definable from $0$
and $S$ in second-order PA, but not in first-order PA, so it would make more
sense to include them here. We remain faithful to Parsons’s presentation for
now, but switch to including $+$ and $\cdot$ below, when the question is how
to replicate something like second-order internal categoricity in a first-
order setting. Then let $T_{1}$ be the first order Peano axioms for
$(N_{1},0_{1},S_{1})$ with induction formulated as a schema in the vocabulary
$\\{N_{1},0_{1},S_{1}\\}\cup\\{N_{2},0_{2},S_{2}\\}$, that is,

$\begin{array}[]{l}(\psi(0_{1},\vec{y})\wedge\forall
x((N_{1}(x)\wedge\psi(x,\vec{y}))\to\psi(S_{1}(x),\vec{y})))\to\\\
\qquad\forall x(N_{1}(x)\to\psi(x,\vec{y})),\end{array}$ (3)

where $\psi(x,\vec{y})$ is any first-order formula in the vocabulary
$\\{N_{1},0_{1},S_{1}\\}\cup\\{N_{2},0_{2},S_{2}\\}$. Let $T_{2}$ be the
corresponding theory for ($N_{2},0_{2},S_{2}$).

Kurt’s native language is built on the vocabulary $\\{N_{1},S_{1},0_{1}\\}$,
but his theory of natural numbers is $T_{1}$, which extends his schematic
induction principle to allow vocabulary from Michael’s language, built on
$\\{N_{2},S_{2},0_{2}\\}$. This means that the formula $\psi$ in the induction
schema of $T_{1}$ can include vocabulary beyond the native language of
($N_{1},0_{1},S_{1}$) into the language of ($N_{2},S_{2},0_{2}$) and that its
quantifiers range beyond $N_{1}$ to include $N_{2}$. This is stronger than
what we ordinarily mean by first-order Peano arithmetic; by including more
instances of induction, it moves in the direction of second-order PA.
(Obviously, this is also true of Michael, mutatis mutandis.) In addition,
Kurt’s understanding of Michael entails that he recognizes $T_{2}$ as holding
in that added ontology, so his working theory includes $T_{1}\cup T_{2}$.

With this added strength, Kurt, arguing in $T_{1}$, can use his induction
schema to prove first-order statements about relationships between the two
vocabularies. This raises the possibility that Kurt’s $T_{1}\cup T_{2}$ might
be enough to produce the isomorphism he’s hoping for, an isomorphism $\phi$
between ($N_{1},S_{1},0_{1}$) and ($N_{2},S_{2},0_{2}$).666666In embryonic
form, this idea is due to Corcoran (1980). So, is $T_{1}\cup T_{2}$ enough to
prove categoricity?As it happens, the answer is no.676767Button and Walsh make
a similar observation in their Proposition 10.4 (Button Walsh (2018), §10.C).

######  Remark 1.

$T_{1}\cup T_{2}$ cannot produce such a $\phi$.

###### Proof.

Our goal is to construct a model $M^{\prime}$ of $T_{1}\cup T_{2}$ such that
structures $(N_{1}^{M^{\prime}},0_{1}^{M^{\prime}},S_{1}^{M^{\prime}})$ and
$(N_{2}^{M^{\prime}},0_{2}^{M^{\prime}},S_{2}^{M^{\prime}})$ are non-
isomorphic – so that Parsons’s desired $\phi$ cannot exist. First, let $M$ be
a model of the vocabulary of $T_{1}\cup T_{2}$ such that $M$ is the disjoint
sum of two copies of the standard model $(\mathbb{N},0,S)$. Thus
$(N_{1}^{M},0_{1}^{M},S_{1}^{M})\cong(\mathbb{N},0,S)$ and
$(N_{2}^{M},0_{2}^{M},S_{2}^{M})\cong(\mathbb{N},0,S)$. Then let
$(\mathbb{N}^{*},0^{*},S^{*})$ be a countable non-standard model elementarily
equivalent to (but again disjoint from) $(\mathbb{N},0,S)$ and let
$M^{\prime}$ be the disjoint sum of $(\mathbb{N},0,S)$ and
$(\mathbb{N}^{*},0^{*},S^{*})$, making it a model of the vocabulary of
$T_{1}\cup T_{2}$. A simple Ehrenfeucht-Fraïssé-game argument, as in Feferman
(1972), shows that $M\equiv M^{\prime}$. Since $(\mathbb{N},0,S)$ satisfies
even the second-order Induction Axiom, $M$ certainly satisfies $T_{1}$.
Respectively, $M$ satisfies $T_{2}$. Thus $M\models T_{1}\cup T_{2}$ and then
also $M^{\prime}\models T_{1}\cup T_{2}$. But
$(N_{1}^{M^{\prime}},0_{1}^{M^{\prime}},S_{1}^{M^{\prime}})$ and
$(N_{2}^{M^{\prime}},0_{2}^{M^{\prime}},S_{2}^{M^{\prime}})$ are non-
isomorphic, as the former is standard and the latter is non-standard. ∎

Though this argument demonstrates that $T_{1}\cup T_{2}$ isn’t enough to
generate the desired $\phi$, it also gives a hint of what more is needed. The
two non-isomorphic countable models in the proof are $n$-equivalent for all
$n$. For countable structures, isomorphism is equivalent to the existence of a
winning strategy for the so-called isomorphism player in the Ehrenfeucht-
Fraïssé game of length $\omega$. So the point of the above counter-example is
that the models imitate being isomorphic, in the sense that the isomorphism
player has a winning strategy, up to $n$ moves of the Ehrenfeucht-Fraïssé
game, for every $n$ separately, but if infinitely many moves are allowed, the
non-isomorphism player has a winning strategy. The finite strategies
corresponding to different finite lengths of the game do not cohere. We need a
global principle that knits the different finite pieces together, essentially
a winning strategy in the infinite game. The strength gained by replacing the
usual induction schemas in two copies of first-order PA with the more generous
induction principles of $T_{1}$ and $T_{2}$ isn’t enough, but there are
enhancements that are up to the task.

Parsons’s preferred enhancement is primitive recursion.686868See (Parsons,
2008, §49, p. 281): ‘in keeping with Skolem’s recursive arithmetic, we can
introduce by primitive recursion a functor’. In particular, he introduces his
$\phi$ by recursion on $N_{1}$:

$\left\\{\begin{array}[]{l}\phi(0_{1})=0_{2}\\\
\phi(S_{1}(x))=S_{2}(\phi(x)).\end{array}\right.$

He then reprises Dedekind’s proof from Subsection 2.2; to show $\phi$ is one-
to-one, he applies $N_{1}$-induction to $\forall m(\phi(n)=\phi(m)\rightarrow
n=m)$; to show $\phi$ is onto $N_{2}$, he applies $N_{2}$-induction to
$\exists m(\phi(m)=n)$. (He notes that both instances of induction are first-
order.) Clearly this $\phi$ would be the required isomorphism.

Unfortunately, Parsons’s brief presentation doesn’t extend beyond his gesture
toward ‘Skolem’s recursive arithmetic’, so the challenge is to formulate a
first-order internal theorem that captures the required recursion. The
standard method involves some version of $\Pi^{1}_{1}$-comprehension and a
second-order induction axiom – and, as we know, Parsons’s vision requires that
second-order logic, however weak, be abandoned altogether. What he wants is a
first-order extension of $T_{1}\cup T_{2}$ that’s capable of generating the
required isomorphism between Kurt’s numbers and Michael’s. Precisely what it
takes to do this is a difficult technical question because of the way $\phi$
crosses from one vocabulary to another – existing results in reverse
mathematics deal with one version of arithmetic at a time. We explore what’s
known of the mathematical situation, drawing on and extending the second
author’s work in Väänänen Wang (2015) (jointly with Wang) and Väänänen (2021).
First-order formulations introduce some non-trivial complications, so we begin
with the second-order case as a warm up.

Consider, then, a second-order theory $T_{1}^{2}$ in the vocabulary
$\\{R_{1},\ldots,R_{n}\\}$, where $R_{i}$ is $r_{i}$-ary for $1\leq i\leq
n$.696969We assume for simplicity that the vocabulary is relational and
finite. Neither constraint is essential. When we refer to constants, such as
$0$ in $\mbox{PA}^{2}$, we think of them here as unary relations which are
singletons. Suppose $\\{R^{\prime}_{1},\ldots,R^{\prime}_{n}\\}$ are new
relation symbols such that the arity of $R^{\prime}_{i}$ is $r_{i}$ for $1\leq
i\leq n$, and $T_{2}^{2}$ is the theory $T_{1}^{2}$ with every occurrence of
$R_{i}$ replaced by $R^{\prime}_{i}$, $1\leq i\leq n$. Let $F$ be a new unary
function symbol, $U$ and $U^{\prime}$ be new unary predicate symbols, and
$\mbox{\rm ISO}(F,U,U^{\prime})$ the first-order formula which says that $F$
is a bijection between $U$ and $U^{\prime}$ such that
$R_{i}(a_{1},\ldots,a_{r_{i}})\leftrightarrow
R^{\prime}_{i}(F(a_{1}),\ldots,F(a_{r_{i}}))$ for all $a_{1},\ldots,a_{n}\in
U$ and all $i$ with $1\leq i\leq n$. If $\phi$ is any second-order formula, we
let $\phi^{(U)}$ denote the formula obtained from $\phi$ by relativizing all
first- and second-order quantifiers to $U$. Finally, for any theory $T$,
$T^{(U)}$ denotes $\\{\phi^{(U)}:\phi\in T\\}$. We then formally define:

######  Definition 3.

The second order theory $T_{1}^{2}$ is _internally categorical_ if

$T_{1}^{2(U)}\cup T_{2}^{2(U^{\prime})}\vdash_{2}\exists F\hskip
2.0pt\mbox{\rm ISO}(F,U,U^{\prime}).$ (4)

$T^{2(U)}_{1}\cup T^{2(U)}_{2}$ appears on the left side of the turnstile – as
opposed to the right side, which may seem more natural for categoricity claims
– simply to allow for infinite theories.

Intuitively, $T_{1}^{2}$ stipulates some axioms for a relational structure
with the relations $\\{R_{1},\ldots,R_{n}\\}$, and it does so without semantic
notions. Similarly, $T_{2}^{2}$ stipulates the same axioms for the relational
structure with $\\{R^{\prime}_{1},\ldots,R^{\prime}_{n}\\}$ as the relations.
The sentence $\mbox{\rm ISO}(F,U,U^{\prime})$ says that $F$ is an isomorphism
between the two relational structures when their domains are restricted to $U$
and $U^{\prime}$, respectively. The categoricity of $T_{1}^{2}$ is obviously
equivalent to the statement that the second-order sentence $\exists F\hskip
2.0pt\mbox{\rm ISO}(F,U,U^{\prime})$ is true in any full model of
$T^{2(U)}_{1}\cup T^{2(U)}_{2}$. But (4) says that the categoricity of
$T_{1}^{2}$ is not just a set-theoretical fact but is, in fact, provable in
second-order logic. Semantically, this means that every Henkin model of
$T^{2(U)}_{1}\cup T^{2(U^{\prime})}_{2}$ satisfies $\exists F\hskip
2.0pt\mbox{\rm ISO}(F,U,U^{\prime})$. In other words, if a Henkin model
recognizes two models of $T_{1}^{2}$, it also recognizes an isomorphism
between those two models.

Notice, then, that internal categoricity is more general than our familiar
notion of categoricity:

######  Theorem 7.

Suppose $T_{1}^{2}$ is internally categorical. Then $T_{1}^{2}$ is
categorical, i.e. if $M\models T_{1}^{2}$ and $M^{\prime}\models T_{1}^{2}$,
where $M$ and $M^{\prime}$ are full models, then $M\cong M^{\prime}$.

###### Proof.

W.l.o.g., suppose the domains of $M$ and $M^{\prime}$, which we denote also
with $M$ and $M^{\prime}$, are disjoint. Let $M^{*}$ be the unique model with
$M\cup M^{\prime}$ as domain, $U^{M^{*}}=M$,
${U^{\prime}}^{M^{*}}=M^{\prime}$, ${R}_{i}^{M^{*}}=R_{i}^{M}$, and
${R^{\prime}}_{i}^{M^{*}}={R}_{i}^{M^{\prime}}$ for $1\leq i\leq n$. Clearly,
$M^{*}\models T_{1}^{2(U)}\cup T_{2}^{2(U^{\prime})}$. Hence by internal
categoricity,

$(U^{M^{*}},R^{M^{*}}_{1},\ldots,R^{M^{*}}_{n})\cong({U^{\prime}}^{M^{*}},{R^{\prime}}^{M^{*}}_{1},\ldots,{R^{\prime}}^{M^{*}}_{n}).$

Now $M\cong M^{\prime}$ follows. ∎

In particular, when $T_{1}^{2}$ is $\mbox{PA}^{2}$, we have

######  Theorem 8 (e.g. Hellman (1989)).

$\mbox{PA}^{2}$ is internally categorical.

Dedekind’s original result follows immediately:

######  Corollary 1 (Dedekind).

$\mbox{PA}^{2}$ is categorical.

This brings the story of internal categoricity for second-order theories full
circle.

Despite the purity of Theorem 8, its second-order character obviously rules it
out for Parsons’s purposes, so let’s now see to what extent this general
approach can be adapted to a first-order context. For various reasons that
aren’t directly relevant here,707070For theories with first-order schemas
arising from second-order $\Pi^{1}_{1}$-axioms, it’s clear how to define
internal categoricity, but for first-order theories in general the question is
open. it’s not easy to define first-order internal categoricity for general
theories $T$. Instead, since our particular interest is in theories of
arithmetic, we explore possibilities for a first-order analog to Definition 3
that’s designed specifically for the case of the Peano axioms. To that end,
suppose the vocabulary of PA is $\\{+,\cdot,0,1\\}$.717171Recall footnote 65.
Let $N_{1}$ and $N_{2}$ be unary predicate symbols, $+_{1}$ and $+_{2}$,
$\cdot_{1}$ and $\cdot_{2}$ binary function symbols, and $0_{1}$ and $0_{2}$,
$1_{1}$ and $1_{2}$ constant symbols. Then let $\mbox{PA}_{1}(N_{1})$ be the
first-order theory PA written in the vocabulary
$\\{+_{1},\cdot_{1},0_{1},1_{1}\\}$, with the functions $+_{1},\cdot_{1}$
mapping $N_{1}\times N_{1}$ to $N_{1}$, the constants $0_{1},1_{1}$ in
$N_{1}$, but with adjustments reminiscent of $T_{1}$: the Induction Schema
allows formulas from the larger vocabulary
$\\{+,\cdot,0,1\\}\cup\\{+_{1},\cdot_{1},0_{1},1_{1}\\}\cup\\{+_{2},\cdot_{2},0_{2},1_{2}\\}$
and (first-order) quantifiers range over the whole domain, including
$N_{1}\cup N_{2}$. Likewise $\mbox{PA}_{2}(N_{2})$.

The first, obvious obstacle to a first-order project is that the basic claim
of categoricity – ‘there’s an isomorphism between $N_{1}$ and $N_{2}$’– can’t
be formulated in our first-order language. It might be possible, however, to
devise a first-order formula, $\phi(x,y)$, that determines a functional
relation of the desired form. Devising an appropriate $\phi$ involves coding.
Let $\psi(x,u,v)$ say that $x$, using $+$ and $\cdot$, codes an initial
segment $I_{1}$ of $N_{1}$ ending with $u$, an initial segment $I_{2}$ of
$N_{2}$ ending with $v$, and a function $f:I_{1}\rightarrow I_{2}$ such that
$f(0_{1})=0_{2}$, $f(z+_{1}1_{1})=f(z)+_{2}1_{2}$ for all $z\in I_{1}$
preceding $u$ in $I_{1}$, and $f(u)=v$. Then let $\phi(u,v)$ be $\exists
x\psi(x,u,v)$. Saying that $\phi$ is an isomorphism is straightforward: let
$\mbox{\rm ISO}_{\phi}(N_{1},N_{2})$ be the first-order formula which says
that $\phi$ is a bijection between $N_{1}$ and $N_{2}$, and for all $x,y\in
N_{1}$:

$\begin{array}[]{l}F(0_{1})=0_{2}\wedge\\\ F(1_{1})=1_{2}\wedge\\\
F(x+_{1}y)=F(x)+_{2}F(y)\wedge\\\
F(x\cdot_{1}y)=F(x)\cdot_{2}F(y),\end{array}$ (5)

where $F(x)$ abbreviates the unique $y\in N_{2}$ such that $\phi(x,y)$.

With this machinery in place, a version of categoricity can be proved from a
collection of first-order arithmetic assumptions. With $\phi$ as above:

######  Theorem 9 (Väänänen (2021)).

First-order PA is _internally categorical_ in the sense that

$\mbox{PA}\cup\mbox{PA}_{1}(N_{1})\cup\mbox{PA}_{2}(N_{2})\vdash\mbox{\rm
ISO}_{\phi}(N_{1},N_{2}).$ (6)

###### Proof.

By induction on $+_{1}$, we show that $F$ is a function $N_{1}\to N_{2}$ and
that it satisfies the conditions (5). We then use induction on $+_{2}$ to
prove that $F$ is onto. These proofs exploit the fact that induction holds for
first-order formulas that have any of the symbols
$+,\cdot,0,1,+_{1},\cdot_{1},0_{1},1_{1},+_{2},\cdot_{2},0_{2},1_{2}$.
Induction on $+$ is used to establish the necessary properties of the coding.
∎

Intuitively speaking, from the perspective of PA, the rest of the left-hand
side of the turnstyle in (6) tells us that there are two models
$(N_{1},+_{1},\cdot_{1},0_{1},1_{1})$ and
$(N_{2},+_{2},\cdot_{2},0_{2},1_{2})$ with $N_{1},N_{2}\subseteq N$, both of
which satisfy the axioms of first-order PA. From these assumptions, Theorem 9
tells us that $\phi$ is an isomorphism between them. As with second-order
internal categoricity (Theorem 8), this is true even if the model of PA we’re
working in is non-standard or uncountable. It’s easy to see that the internal
categoricity (and thereby the categoricity) of $\mbox{PA}^{2}$ follows from
the internal categoricity of the first-order PA in the sense of Theorem 9.

Though it comes close, the sense of ‘internal categoricity’ in Theorem 9
doesn’t look quite like that of Definition 3, largely because of that extra PA
on the left-hand side. But note that, in Definition 3, the comprehension
axioms are built into $\vdash_{2}$. This is where the hidden power of second-
order comprehension comes into dramatic focus: it’s simply assumed that the
range of the second-order quantifier includes an item for every formula in the
combined vocabularies of $T^{2(U)}_{1}\cup T^{2(U)}_{2}$; this is what
generates the required bijection, the required link between the two models. In
the first-order case, where no such implicit background is in place, the extra
PA is required to play this role. (And, for the record, Dedekind’s weak set
theory does the same in his Theorem 1.)

Another option is to consider two copies of PA with the same domain:
$\mbox{PA}_{1}^{*}$ says that the axioms of first-order PA hold of
$+_{1},\cdot_{1},0_{1},1_{1}$, $\mbox{PA}_{2}^{*}$ says the same for
$+_{2},\cdot_{2},0_{2},1_{2}$, where both allow induction for first-order
formulas in the joint vocabulary
$\\{+_{1},\cdot_{1},0_{1},1_{1}\\}\cup\\{+_{2},\cdot_{2},0_{2},1_{2}\\}$. If
$\mbox{AUT}_{\phi}$ says that $\phi$ defines a permutation of the domain and
that it satisfies the preservation clauses in the definition of $\mbox{\rm
ISO}_{\phi}(N_{1},N_{2})$, then we get this result:

######  Theorem 10 (Väänänen (2019, 2021)).

First-order Peano arithmetic PA is _internally categorical_ in the sense that

$\mbox{PA}^{*}_{1}\cup\mbox{PA}^{*}_{2}\vdash\mbox{AUT}_{\phi}.$ (7)

This mimics Definition 3 more closely: there’s no extra PA overseeing the two
copies $\mbox{PA}^{*}_{1}$ and $\mbox{PA}^{*}_{2}$. Still, the assumption that
the two copies have the same domain (which, incidentally, needn’t be
countable) makes Theorem 10 less compelling than Theorem 9. Intuitively
speaking, we’re trying to establish that two models of first-order arithmetic
are isomorphic, and (knowing that this is impossible without some extra
assumption) we make the task easier by assuming that we are ‘half way there’
in the sense that at least the two models have the same domain.

###  5.3 Does what he did (or can be done) accomplish what he set out to do?

Does any of this help Kurt determine whether or not Michael’s numbers are
isomorphic to his own? In Theorem 9, he begins from PA – not
$\mbox{PA}_{1}(N_{1})$, his own idiosyncratic version – and views both
$\mbox{PA}_{1}(N_{1})$ and Michael’s version from that broader point of view.
It’s hard to see how this could honestly be described as Kurt proving,
exclusively from his own perspective, that his numbers are isomorphic to
Michael’s.727272There’s also the murkier question of whether Kurt – whose
thinking, we assume, is relentlessly first-order – could replicate the line of
thought that came before Theorem 9. It might be argued that understanding the
significance of $\phi$ depends on seeing it as a first-order replacement for
the second-order $\exists f$, which would be reason to doubt that a
relentlessly first-order Kurt would see its significance, let alone be
inspired to think it up. (Cf. (Kreisel, 1967, p. 148).) For that matter, how
would a relentlessly first-order Kurt pose the question of categoricity to
himself in the first place?  Theorem 10 would return Kurt to his own
perspective, but he would need to know in advance that Michael is talking
about the same things he is – though perhaps getting some of their properties
and relations wrong – and it’s hard to see what grounds he could have for this
belief.737373Obviously, the concern of the previous footnote arises here as
well. So far, at least, the mathematics of internal categoricity hasn’t
delivered a first-order result that Kurt can deploy, hasn’t shown that Parsons
can avoid an appeal to second-order logic. This obviously isn’t conclusive,
but the challenge hasn’t yet been met.

On the philosophical aspects of Parsons’s effort, consider again the question
he set out to answer: is our shared concept of natural number univocal, or can
‘two [different] structures answer equally well to our conception of the
sequence of natural numbers’ (Parsons, 2008, §48, p. 272)? As we’ve seen, he
then characterizes our shared concept in terms of a set of rules that include
the open-schematic version of mathematical induction and offers his Hilbertian
intuitive picture of ever-increasing strings of strokes as an example of a
structure that answers to it. As noted earlier, Parsons’s account of this
picture calls on ordinary human psychology – perception of figure/ground or
temporal succession, ‘insight into experienced space and time’ (ibid., §29, p.
178) – so the number concept it answers to would seem to be one shared by
adult humans beyond the developmental levels of understanding counting (about
age 3) and realizing there’s no largest number (about age 8). It’s the shared
understanding of 1, 2, 3, … that we use every day, in contexts both mudane and
theoretical.

Now consider some recent psychological findings of Josephine Relaford-Doyle
and Rafael Núñez at UCSD (Relaford-Doyle Núñez (2017, 2018, 2021)). The
experimental subjects were college students, one group drawn from the general
population without training in mathematics and another from mathematics majors
with at least a B- in a course on proof techniques that included mathematical
induction. Subjects were shown a visual argument that the sum of the first $n$
odd numbers is $n^{2}$: represent the first odd number, $1$, by one dot on the
page. ($1=1^{2}$). Represent the second odd number, $3$, by adding three dots
so as to form a $2\times 2$ square with the original dot in the lower, left-
hand corner. ($1+3=2^{2}$) Represent $5$ by adding five dots in the same way
so as to form a $3\times 3$ square, again with the first dot in the lower
left-hand corner. ($1+3+5=5^{2}$). And so on. The recursive character of the
construction is clear; an induction to all natural numbers is implicit.

Or so it seems. The experimenters focused on subjects who demonstrated that
they understood the statement (the sum of the first $n$ natural numbers is
$n^{2}$) and were willing to generalize to nearby numbers (having gone through
the argument for the first five odd numbers, they believed it would hold for
seven or eight). These subjects were then asked a striking question: might
there be a large number for which the statement fails? The answer seems
obvious – surely not! – but that was the reaction of only a third of the
mathematically trained respondents and fewer than 10% of the rest; nearly 40%
of the untrained subjects and 10% of the trained respondents saw the existence
of such a counterexample as plausible. The thought was that ‘larger numbers
could be … outliers’ or ‘when it gets to really high numbers … it’s possible
that … maybe it gets kind of fuzzy. Because at extremes things tend not to
work as they do normally’ or perhaps most tellingly, ‘it’s too hard to draw a
million dots’ (Relaford-Doyle Núñez, 2021, p. 14). One subject, a math and
computer science major in the mathematically trained group, made explicit
reference to computational limitations: ‘Large numbers have problems. In
computer science … even if it works on small, medium, most numbers, when you
get to the large number, there’s not enough space and it doesn’t work anymore.
So large numbers are weird cases’ ((Relaford-Doyle Núñez, 2018, p. 248)).

Relaford-Doyle and Núñez summarize:

> These responses all suggest that these participants believe that large
> numbers may have qualitatively different properties than small numbers, such
> that rules that apply to small numbers may no longer work at larger
> magnitudes. This is a reasonable conclusion to draw – in practice there are
> many differences between small and large numbers: small numbers … are
> encountered more frequently, have simple numerical notation and lexical
> structure, and are easier to use in computations. However, this finding is
> surprising in that it is in opposition to the widely-held assumption in
> developmental psychology that ‘mature’ conceptualizations of natural number
> are consistent with the Dedekind-Peano axioms, in which the entire set of
> natural number is governed by the same logic. (Relaford-Doyle Núñez, 2021,
> p. 14)

They conclude that ‘many undergraduates may possess non-normative
conceptualizations of natural number’ (Relaford-Doyle Núñez, 2021, p. 18).

The subjects’ remarks suggest that the ‘reasonable conclusion’ they’ve drawn
is something like: the diagrammatic reasoning demonstrates the truth of the
target claim only for numbers to which it could be applied, numbers for which
there could be an array of dots, numbers up to which one could count – what we
might informally call ‘feasible’ numbers. This idea generates a sorites
paradox – 1 is feasible, if $n$ is feasible, so is $n+1$, but $10^{70}$ is not
– and nevertheless plays a leading role in Yessenin-Volpin’s strict finitist
program in the foundations of mathematics (Yessenin-Volpin (1970)) and appears
in computer science as ‘feasible computation’ (see, e.g., Dean ()). These
interconnections have been investigated philosophically by M. Dummett (1975)
and more recently and formally by Walter Dean (2018), but our concern here is
with the informal notion, the intuitive picture that appears to lie behind the
reactions of Relaford-Doyle and Núñez’s subjects. Dean demonstrates the
consistency of this picture with a nonstandard model of arithmetic in which an
‘infinite integer’ plays the role of an unfeasible number. This isn’t to say
that these students – let’s call them ‘feasibilists’ – are referring to a non-
standard model. Theirs is an alternative intuitive picture on a psychological
level analogous to that of Parsons’s Hilbertian notion,747474Again, we’re
assuming that Parsons’s Hilbertian intuitive picture is straightforwardly
psychological, with no transcendental element. (Recall footnote 51.) That’s
why we take empirical psychology to be relevant. not a model in the sense of
model theory, and that alternative intuition prompts them to disagree with
those hewing to the orthodox concept.

If these preliminary results are replicated and extended, one surprise would
be that the developmental story Relaford-Doyle and Núñez allude needs an
update: believing there’s no largest number, appreciating the endlessness of
the sequence, 1, 2, 3 … , isn’t always enough to equip one with the notion of
an orthodox omega sequence.757575The first author made something like this
mistake, e.g., in (Maddy, 2018, §2), though she left open the possibility that
our shared notion isn’t fully determinate. Thanks to Jessica Gonzales,
Christopher Mitsch, and Stella Moon (attendees at a seminar offered by Núñez
at UCSD) for calling the work of Relaford-Doyle and Núñez to her
attention.,767676This would be particularly interesting developmentally,
because it would locate a third and quite different linguistic, or perhaps
better social, contribution to the mathematical conception of the natural
numbers. In counting, the initially merely sing-song sequence of number words
ends up providing a bridge between two primitive cognitive systems that we
share with many other animals (one exact system for small numbers and one
approximate system for larger numbers). After that, coming to realize there’s
no largest number depends on increasing dexterity with producing ever larger
numerical expressions. This new and final third step would typically involve
inculcation with mathematical induction. For Parsons’s project, the surprise
appears to be just what he feared: that our shared concept of natural number
is not univocal; that it admits both feasibilist and more orthodox readings;
that it doesn’t include full mathematical induction in the usual sense. (The
feasibilist apparently does think proof by induction works for feasible
portion of the number sequence.) Parsons’s emphasis on the centrality of
induction suggests that he might react by denying the feasibilist has a
concept of natural number; he might restrict his circle of concept-sharing to
those who’ve been trained on and internalized full mathematical induction (or
perhaps come by it intuitively).777777Relaford-Doyle and Núñez (Relaford-Doyle
Núñez, 2021, p. 1) begins with the observation that ‘Studies have repeatedly
documented students’ considerable difficulties in learning mathematical
induction’. Along the way, they note that even some students trained in
mathematical induction understand it only at the ‘procedural’ level – that is,
as a sort of algebraic recipe – and not at the ‘conceptual’ level that would
allow them to recognize induction in the diagrammatic argument.

This path seems uninviting. The subjects in these experiments are well-
educated young people who operate successfully with numbers in their everyday
and intellectual lives, so it doesn’t seem plausible to claim that they lack
the concept of natural number. Of course, Parsons is within his rights to
limit the scope of his inquiry and to use the term ‘concept of natural number’
as he chooses, but his Hilbertian intuitive picture is intended to answer to
that concept, and the psychological description of the intuitive picture is
clearly rests on our most fundamental thinking about numbers. The feasibilists
among Relaford-Doyle and Núñez’s subjects surely perceive figure/ground
relations and temporal succession, surely experience space and time, so they
must share the picture of the endless series of strings of strokes – $\|$,
$\||$, $\|||$, … – with their mathematical betters. Rather than painting the
feasibilist as embracing an alternative to the Hilbertian picture, perhaps a
better way to describe the situation is to say that we do all share that
picture, but that the picture itself is vague in Parsons’s sense, that it can
be taken in (at least) two different ways, that it does, in a sense, ‘flicker
in the distance’. The threat of this kind of psychological divergence isn’t
the sort of thing susceptible to the proving of theorems, about categoricity
or anything else. But the empirical situation remains unsettled.

To sum up, if this general psychological question about our shared concept of
number is what Parsons ultimately cares about, and if the experimental results
hold, then his question – is it univocal? – would be answered in the negative.
If the experimental results don’t hold or if his interest is actually in the
narrower question of the more learned concept of number, we’re left with the
mathematical question of how Kurt can define his isomorphism within the bounds
of a first-order internal categoricity theorem – and the closest known
mathematical results, Theorem 9 and Theorem 10, fall short of that goal. In
the end, there seems room for doubt that our shared concept, Parsons’s own
Hilbertian intuition of the endless sequence of strokes, is as clear and
determinate as we think it is. And if there is this room for doubt, formal
categoricity theorems don’t seem to be the kind of thing that might
conceivably help. Given these open questions, both mathematical and
philosophical, Parsons’s appeal to categoricity arguments to establish ‘the
uniqueness of the natural numbers’ can’t yet be judged a success.

##  6 Button and Walsh in ‘Categoricity’ (2018)

###  6.1 What do they set out to accomplish?

As Kreisel points out (Section 4), anyone confident that the quantifiers of
second-order logic are determinate should take the second-order version of
Zermelo’s categoricity theorem, Theorem 4, to show that CH has a determinate
truth value, and in fact, various observers make exactly this
inference.787878See, e.g., Shapiro (Shapiro, 2012, p. 105), Hellman (Hellman,
1989, pp. 70-71). Timothy Button and Sean Walsh in their 2018 book Philosophy
and Model theory (Button Walsh (2018)) are not among them. Part B of that
book, ‘Categoricity’, includes an extended exploration of the possible bearing
of categoricity on the status of CH, with analyses of both external and
internal theorems. For most of the book, they adopt a philosophical
perspective they call ‘modelism’, but in Part B, they introduce an
alternative, ‘internalism’. Their official position is to endorse neither –

> We should emphasise right now … that we are not advocating internalism, any
> more than we were advocating modelism. Rather, we are presenting internalism
> in a speculative spirit. It is a fascinating position, worthy of attention,
> and we want to develop it as best we can. (Button Walsh (2018), p. 223)

– so our goal here is to examine their claims about what would follow if one
were to adopt one perspective or the other.

Modelism, then, is a philosophical perspective on the nature of model theory,
a branch of mathematics that takes place within set theory.797979See, e.g.,
Button Walsh (2018), p. 145: ‘Model theory is officially implemented within
set theory’. The modelist understands model theory as a piece of applied
mathematics (cf. fluid dynamics) whose target phenomenon is the semantics of
ordinary mathematical discourse (cf. the behavior of real world substances
like water). More fully, model theory is to provide a workable account of the
relations between the natural language of mathematics and a pre-theoretic
metaphysics of structures that mathematics refer to (e.g., Shapiro (2012)) or
concepts that mathematicians express (e.g., Hellman (1989)).808080See (Button
Walsh, 2018, §§6.2 and 6.4, respectively). A remark on terminology: applied
mathematicians often describe themselves as providing ‘models’ for worldly
phenomena, but it would obviously invite confusion to describe model theory as
providing a model of the mathematical portion of natural language. We avoid
this in the text with awkward circumlocutions like the one just employed –
model theory provides a ‘workable account’ – or it provides a ‘treatment’ or
some aspect of mathematical discourse is ‘idealized to’ such and such. The
reader is invited to interpret these passages as intending nothing other than
‘model’ in sense of applied mathematics rather than that of model theory. In
this way, the natural language of actual mathematics is idealized to a formal
language $L$ and the structures mathematicians discuss to set-theoretic
$L$-structures.818181 Here, and in what follows, we elide the distinction
between objects-modelism and concepts-modelism which Button and Walsh treat
separately; reformulations of structure-talk into concept-talk should be
relatively straightforward. Also, the modelism in question is ‘moderate’, that
is, one that ‘rejects all appeals to faculties of mathematical intuition, or
anything similar’ and appeals only to ‘human faculties … which could plausibly
both have evolved within a species, and also could have developed within an
individual creature as it grew from a fetus into an adult’ ((Button Walsh,
2018, p. 42)). Finally, these quotations all come in Button and Walsh’s
treatment of arithmetic rather than set theory, where the issue is the
moderate modelist’s appeal to the second-order version of Dedekind’s
categoricity theorem. When it comes to set theory, where appeal to the second-
order version of Zermelo’s theorem is what’s at issue, they write that ‘we
would re-run all the arguments [from the arithmetic context], occasionally
replacing the phrase “arithmetic” with “set theory”. Flogging this dead horse
would, though, be exhausting and unilluminating. So we shall pass over the
corpse of moderate modelism without further ado’ (ibid., pp. 181-182). Faced
with the modelist’s appeal to Zermelo’s result in its second-order form
(Theorem 5), Button and Walsh reach the familiar conclusion:

> invoking [the second-order theorem] is simply question-begging, since the
> use of the full semantics simply assumes precisely what was at issue … in
> appealing to full second-order logic, the … modelist is simply out of the
> frying pan, and into another frying pan. (Button Walsh, 2018, pp. 159-160)

Internalism is then offered as an alternative perspective, from which internal
categoricity theorems might gain better traction.

At the outset, all we’re told about internalism is that it’s a rejection of
modelism and that it ‘relies heavily on deduction’ (Button Walsh, 2018, p.
223).828282The deduction they have in mind is a second-order, what we’ve been
writing as $\vdash_{2}$. Again (see footnote 81), the following attempt to
characterize internalism is translated from the arithmetic context where it’s
explicit (Button Walsh, 2018, chapter 10), to the set-theoretic context where
it’s presumably implicit (ibid., chapter 11). Later, the contrast between
modelism and internalism is glossed in terms of their respective
formalizations of a pre-theoretic claim like ‘there is a set-theoretic
structure’: the former offers ‘there is an $X$ and a binary relation $R$ on
$X$ such that $(X,R)\models\mbox{ZFC}^{\hskip 1.0pt2}$’, while the latter
formulates a second-order formula conjoining the axioms of $\mbox{ZFC}^{\hskip
1.0pt2}$ as direct claims about variables $X$ and $R$ and opts for the second-
order statement $\exists X\exists R\hskip 2.0pt\mbox{ZFC}^{\hskip
1.0pt2}(X,R)$. If we imagine each of these as the antecedent of a categoricity
theorem, this happily coincides with our characterization of the
external/weakly internal distinction as a property of theorems in Subsection
2.3 – the former involves semantic notions and only mentions the axioms, while
the latter involves no semantic notions and uses the axioms directly – so
there’s a tight connection between Button and Walsh’s internalism and weakly
internal categoricity theorems.

Beyond this, we’re told that internalism is not if-thenism – because the
antecedent is asserted unconditionally – not platonism – because no stand is
taken on the nature of the pre-theoretic metaphysics of structures – and not
logicism – because it has no epistemological ambitions.838383See (Button
Walsh, 2018, pp. 236-237). And finally, ‘internalists need not fear the use of
metalanguages … [they] only oppose semantic ascent’ (Button Walsh, 2018, p.
238) – so they’re perfectly open to proof-theoretic metatheory. Still, for all
this, internalism remains somewhat underspecified, a point to which we return
in Subsection 6.3.

Given that Button and Walsh’s project is exploratory, they can’t be said to
begin with a specific goal in mind, but they do end up with the conclusion
that the mathematics they present … 848484For the record, Button and Walsh
have a second, ‘transcendental argument’ for a limited conclusion that a
leading argument for the indeterminateness of CH ‘walks a dangerously thin
path between falsity and incoherence’ (Button Walsh (2018), pp. 256). Since
this line of thought doesn’t involve categoricity considerations, we leave it
aside here.

> … applies pressure to those who regard the continuum hypothesis as
> indeterminate. Indeed, it is not obvious how best to sustain that attitude.
> (Button Walsh (2018), p. 255)

This stops short of a straightforward affirmation that CH has a determinate
truth value, despite leaning in that direction. We take this hedged conclusion
as their goal, lay out the mathematics in Subsection 6.2, and evaluate its
success in achieving that goal in Subsection 6.3.

###  6.2 What do they actually do (or can be done)?

We’ve seen that Button and Walsh take the semantic notions in Zermelo’s
Theorem 5 to render it ineffective in a defense of the determinateness of CH,
so naturally what they now pursue is an internal categoricity theorem. We’ve
also seen that their internalist is perfectly comfortable with syntactic
second-order logic.858585This is why they think Parsons should have settled
for Theorem 6 (recall footnote 62). Here they obviously differ from Parsons, a
difference rooted in their respective diagnoses of what goes wrong in an
attempt to apply Zermelo’s theorem to CH in this way: is the problem second-
order logic itself or just the full semantics? We won’t attempt to adjudicate
this dispute, but the result is that Parsons sees internalness as an
instrument for achieving a first-order result, while Button and Walsh see
internalness as an end in itself. This is why Parsons needs a better version
of Theorem 9 or Theorem 10, while Button and Walsh would be content with an
internal theorem for ZF2 that runs parallel to Theorem 8 for $\mbox{PA}^{2}$ –
or rather, they want an internal theorem for $\mbox{ZF}^{\hskip 1.0pt2}$
running parallel to a version of Theorem 8 that’s rephrased as a theorem of
pure second-order logic.

At the risk of pedantry, let’s make this explicit. First, suppose we’ve
adapted the notational conventions laid out before Definition 3 for the case
of $\mbox{PA}^{2}$ in the vocabulary $\\{S,0\\}$ and new symbols $N_{1}$,
$N_{2}$, $S_{1}$, $S_{2}$, $0_{1}$, $0_{2}$. Then, spelling out that
definition, Theorem 8 becomes:

$\mbox{PA}_{1}^{2(N_{1})}\cup\mbox{PA}_{2}^{2(N_{2})}\vdash_{2}\exists F\hskip
2.0pt\mbox{\rm ISO}(F,N_{1},N_{2}).$ (8)

As was noted in passing in Subsection 5.2, Definition 3 takes the form it does
to allow for infinite theories, so the formulation for the current case can be
simplified by exploiting the finiteness of $\mbox{PA}^{2}$. Let $\mbox{\bf
PA}^{2(N_{1})}_{1}=\bigwedge\\{\phi^{(N_{1})}(0_{1},S_{1}):\phi(0,S)\in\mbox{PA}^{2}\\}\cup\\{N_{1}(0_{1}),\forall
x(N_{1}(x)\to N_{1}(S_{1}(x)))\\}$ and $\mbox{\bf
PA}^{2(N_{2})}_{2}=\bigwedge\\{\phi^{(N_{2})}(0_{2},S_{2}):\phi(0,S)\in\mbox{PA}^{2}\\}\cup\\{N_{2}(0_{2}),\forall
x(N_{2}(x)\to N_{2}(S_{2}(x)))\\}$. Then (8) becomes:868686We treat the
constant symbols $0,0_{1},0_{2}$ as individual variables when they’re
quantified.

######  Theorem 11 (Theorem 8 restated, Internal categoricity for
$\mbox{PA}^{2}$).

$\vdash_{2}\forall N_{1},0_{1},S_{1},N_{2},0_{2},S_{2}((\mbox{\bf
PA}^{2(N_{1})}_{1}\wedge\mbox{\bf PA}^{2(N_{2})}_{2})\to\exists F\hskip
2.0pt\mbox{\rm ISO}(F,N_{1},N_{2})).$

This is the arithmetic result that Button and Walsh hope to imitate for the
case of set theory.878787See (Button Walsh, 2018, p. 228).

Before leaving arithmetic, though, we should pause to ask precisely how
Theorem 11 bears on the determinateness of arithmetic claims. From a
modelist’s perspective, a categoricity theorem in model theory has
implications for the pre-theoretic realm of mathematical structures – namely,
that the theory in question is satisfied by precisely one such
structure888888If these pre-theoretic structures can be isomorphic without
being identical, then: precisely one such structure up to isomorphism. – and
it follows that each of its claims has a determinate truth value.898989On the
assumption in the previous footnote: true in all such structures or false in
all such structures – which is to say that each of its claims has a
determinate truth value. But the internalist posits no such pre-theoretic
metaphysics; the internal Theorem 11 involves no semantic notions. As Button
and Walsh put it, ‘no internal categoricity result can show that a theory pins
down a unique $\mathcal{L}$-structure in the model-theorist’s sense (even up-
to-isomorphism)’ (Button Walsh, 2018, p. 229, emphasis in the original). But
an internal theorem can show what they call ‘intolerance’.

To see how that goes, consider this immediate consequence of Theorem 11:

######  Corollary 2.

Suppose $\phi(0,S)$ is a second-order sentence in the vocabulary $\\{0,S\\}$.
Then

$\begin{array}[]{l}\vdash_{2}\forall
N_{1},0_{1},S_{1},N_{2},0_{2},S_{2}((\mbox{\bf
PA}^{2(N_{1})}_{1}\wedge\mbox{\bf PA}^{2(N_{2})}_{2})\to\\\ \hskip
28.45274pt(\phi^{(N_{1})}(0_{1},S_{1})\leftrightarrow\phi^{(N_{2})}(0_{2},S_{2}))).\end{array}$

Button and Walsh’s ‘intolerance’ follows:

######  Theorem 12 (Button Walsh (2018)).

_(‘Intolerance’ of $\mbox{PA}^{2}$)_ Suppose $\phi(0,S)$ is a second-order
sentence in the vocabulary $\\{0,S\\}$. Then

$\vdash_{2}\forall N,0,S(\mbox{\bf PA}^{2(N)}\to\phi^{(N)}(0,S))\vee\forall
N,0,S(\mbox{\bf PA}^{2(N)}\to\neg\phi^{(N)}(0,S))).$

###### Proof.

By Corollary 2,

$\begin{array}[]{l}\vdash_{2}\forall
N_{1},0_{1},S_{1},N_{2},0_{2},S_{2}((\mbox{\bf
PA}^{2(N_{1})}_{1}\wedge\mbox{\bf PA}^{2(N_{2})}_{2})\to\\\ \hskip
28.45274pt(\phi^{(N_{1})}(0_{1},S_{1})\vee\neg\phi^{(N_{2})}(0_{2},S_{2}))).\end{array}$

By rearranging quantifiers and connectives, we obtain

$\begin{array}[]{l}\vdash_{2}\forall N_{1}0_{1}S_{1}(\mbox{\bf
PA}^{2(N_{1})}_{1}\to\phi^{(N_{1})}(0_{1},S_{1}))\vee\\\ \hskip
11.38092pt\forall N_{2}0_{2}S_{2}(\mbox{\bf
PA}_{2}^{2(N_{2})}\to\neg\phi^{(N_{2})}(0_{2},S_{2})))\end{array}$

from which the claim follows by change of bound variables. ∎

Another way to put this is:

######  Corollary 3.

If $\phi$ is a second-order sentence in the vocabulary $\\{0,S\\}$, then

$\mbox{\bf PA}_{1}^{2(N_{1})}\cup\mbox{\bf
PA}_{2}^{2(N_{2})}\cup\\{\phi^{(N_{1})}(0_{1},S_{1}),\neg\phi^{(N_{2})}(0_{2},S_{2})\\}$

is deductively inconsistent.

Button and Walsh take this to place ‘pressure on the Algebraic Attitude toward
arithmetic’(Button Walsh (2018), p. 236), that is, on the idea that arithmetic
is not univocal.

This is the line of thought Button and Walsh want to see duplicated for the
case of set theory. Categoricity will have to be replaced by quasi-
categoricity – as in Theorem 5, the two models must be of the same height –
but with that proviso, just such an internal theorem has been proved by the
second author and Tong Wang (Väänänen Wang (2015)). Begin with
$\mbox{ZF}^{\hskip 1.0pt2}$ in the usual vocabulary $\\{\in\\}$.909090Button
and Walsh actually use Scott-Potter set theory, but this makes little
difference for our purposes, so we stick with $\mbox{ZF}^{\hskip 1.0pt2}$ to
maintain consistency. Urelements are similarly irrelevant, so for simplicity,
we assume there are none. Let $E_{1}$ and $E_{2}$ be binary relation symbols
and $X_{1}$ and $X_{2}$ be unary relation symbols. If $\phi$ is a second-order
sentence in the vocabulary $\\{\in\\}$, let $\phi(E_{1})$ and $\phi(E_{2})$ be
translations of $\phi$ into the vocabularies $\\{E_{1}\\}$ and $\\{E_{2}\\}$,
respectively, and let $\phi^{(X_{1})}(E_{1})$ and $\phi^{(X_{2})}(E_{2})$ be
$\phi(E_{1})$ and $\phi(E_{2})$ be $\phi$ with its first- and second-order
quantifiers relativized to $X_{1}$ and $X_{2}$, respectively. Once again
exploiting finiteness, this time of $\mbox{ZF}^{\hskip 1.0pt2}$, let
$\mbox{\bf
ZF}^{2(X_{1})}(E_{1})=\bigwedge\\{\phi^{(X_{1})}(E_{1}):\phi(E_{1})\in\mbox{ZF}^{\hskip
1.0pt2}(E_{1})\\}$ and $\mbox{\bf
ZF}^{2(X_{2})}(E_{2})=\bigwedge\\{\phi^{(X_{2})}(E_{2}):\phi(E_{2})\in\mbox{ZF}^{\hskip
1.0pt2}(E_{2})\\}$. Finally, let IA be the second-order sentence in the
vocabulary $\\{X_{1},E_{1},X_{2},E_{2}\\}$ which says – assuming the axioms of
$\mbox{ZF}^{\hskip 1.0pt2}$ for $(X_{1},E_{1})$ and $(X_{2},E_{2})$ – that the
classes of inaccessible cardinals in the sense of $(X_{1},E_{1})$ and
$(X_{2},E_{2})$, respectively, are isomorphic, and let
ISO$((X_{1},E_{1}),(X_{2},E_{2})))$ be the second-order sentence that says
$(X_{1},E_{1})$ and $(X_{2},E_{2})$ are isomorphic. Then we have:

######  Theorem 13 (Väänänen Wang (2015)).

_(Internal quasi-categoricity of ZF 2)_

$\vdash_{2}(\mbox{\bf ZF}^{\hskip 1.0pt2(X_{1})}(E_{1})\wedge\mbox{\bf
ZF}^{\hskip 1.0pt2(X_{2})}(E_{2})\wedge\mbox{IA})\to\mbox{\rm
ISO}((X_{1},E_{1}),(X_{2},E_{2})).$

(Cf. Button Walsh (2018), p. 255.) As in the case of arithmetic (see
Subsection 5.2), this implies Zermelo’s categoricity theorem.

The counterpart to Corollary 2 follows immediately:

######  Corollary 4.

Suppose $\phi$ is a second-order sentence in the vocabulary $\\{\in\\}$. Then

$\vdash_{2}(\mbox{\bf ZF}^{\hskip 1.0pt2(X_{1})}(E_{1})\wedge\mbox{\bf
ZF}^{\hskip
1.0pt2(X_{2})}(E_{2})\wedge\mbox{IA})\to(\phi^{(X_{1})}(E_{1})\leftrightarrow\phi^{(X_{2})}(E_{2})).$

The counterpart to arithmetic intolerance – the idea that every sentence is
either true in all models or false in all models – requires fixing the number
of inaccessibles, their order type. Any order type will do, so for the
ultimate in simplicity, let $\mbox{IA}_{0}$ be the first-order sentence of set
theory saying that there aren’t any inaccessible cardinals, that every limit
cardinal $>\omega$ is singular. Finally, let

$\Gamma=\bigwedge(\mbox{ZF}^{\hskip 1.0pt2}\cup\\{\mbox{IA}_{0}\\}).$

Then, if $E$ is a new binary relation symbol and $X$ a new unary relation
symbol, we have

######  Theorem 14 (Button Walsh (2018)).

_(‘Intolerance’ of ZF 2)_ Suppose $\phi$ is a second-order sentence in the
vocabulary $\\{\in\\}$. Then

$\vdash_{2}\forall X,E(\Gamma^{(X)}(E)\to\phi^{(X)}(E))\vee\forall
X,E(\Gamma^{(X)}(E)\to\neg\phi^{(X)}(E)).$

###### Proof.

By Corollary 4,

$\vdash_{2}\forall
X_{1},E_{1},X_{2},E_{2}((\Gamma^{(X_{1})}(E_{1})\wedge\Gamma^{(X_{2})}(E_{2}))\to(\phi^{(X_{1})}(E_{1})\vee\neg\phi^{(X_{2})}(E_{2}))).$

By rearranging quantifiers and connectives, we obtain

$\vdash_{2}\forall
X_{1},E_{1}(\Gamma^{(X_{1})}(E_{1})\to\phi^{(X_{1})}(E_{1}))\vee\forall
X_{2},E_{2}(\Gamma^{(X_{2})}(E_{2})\to\neg\phi^{(X_{2})}(E_{2})))$

from which the claim follows by change of bound variables. ∎

Once again, ‘intolerance’ can be rephrased:

######  Corollary 5.

If $\phi$ is a second-order sentence in vocabulary $\\{\in\\}$, then the
theory

$\\{\Gamma^{(X_{1})}(E_{1}),\Gamma^{(X_{2})}(E_{2}),\phi^{(X_{1})}(E_{1}),\neg\phi^{(X_{2})}(E_{2})\\}$

is deductively inconsistent.

From this, Button and Walsh draw the expected conclusion:

> The present observation applies … pressure to those who regard the continuum
> hypothesis as indeterminate. Indeed, it is not obvious how best to sustain
> that attitude, in the face of this deductive inconsistency. (Button Walsh
> (2018), p. 255)

We examine this claim in Subsection 6.3.

Theorem 13 is both weakly internal and pure; of the desiderata present in
Parsons’s discussion, it fails only to be first-order. Of course, this doesn’t
trouble Button and Walsh, but it’s still worth observing that internal
categoricity and its consequences can be carried over to first-order ZF. The
source of categoricity is actually more transparent in the first-order than in
the second-order context because blanket appeal to the second-order
comprehension axiom is replaced by explicit, targeted features of the first-
order separation and replacement schemas.

To see this, begin with first-order ZF in the vocabulary $\\{\in\\}$ and use
the notation $\mbox{ZF}(E)$ for the result of replacing $\in$ with a binary
relation symbol $E$. Now suppose that $E_{1}$ and $E_{2}$ are new binary
relation symbols, $X_{1}$ and $X_{2}$ are new unary relation symbols, and
$\pi$ is a new unary function symbol. If $\phi(E)$ is a sentence in the
vocabulary $\\{E\\}$, let $\phi^{(X)}(E)$ be $\phi(E)$ with the first-order
quantifiers relativized to $X$. Now let $\mbox{ZF}^{(X_{1})}(E_{1})$ consist
of all $\phi^{(X_{1})}(E_{1})$, where $\phi\in\mbox{ZF}$, allowing in the
separation and replacement schemas formulas from the vocabulary
$\\{X_{1},E_{1},X_{2},E_{2},\pi,\in\\}$ with unrestricted (i.e. not
relativized to $X_{1}$) quantifiers – and similarly
$\mbox{ZF}^{(X_{2})}(E_{2})$. Let $\mbox{IO}_{\pi}$ be a first-order sentence
of the vocabulary $\\{X_{1},E_{1},X_{2},E_{2},\pi,\in\\}$ saying – assuming
the axioms ZF for $(X_{1},E_{1})$ and $(X_{2},E_{2})$ – that $\pi$ is an
isomorphism between the ordinals of $(X_{1},E_{1})$ and the ordinals of
$(X_{2},E_{2})$. Finally, given a formula $\phi(x,y)$, let $\mbox{\rm
ISO}_{\phi}((X_{1},E_{1}),(X_{2},E_{2}))$ be a first-order sentence that says