Against Categoricity

Abstract

According to the universism position in set theory, there is only a Single Universe for set theory and all the models we build through forcing have instrumental value: they are useful only until we find the "true" one. The belief that this universe actually exists and the consequent dismissal of a pluralist conception is based on two main arguments: first, already in V we can simulate different universes, so there is no need to assume the existence of other universes: second, set theory is actually categorical, i.e. all the different models stemming from it are in fact isomorphic. I argue that both arguments are off target.

Full text

Against Categoricity
Subdisciplinary Area: Philosophy of Mathematics
January 31, 2020
Abstract
According to the universism position in set theory, there is only a
Single Universe for set theory and all the models we build through
forcing have instrumental value: they are useful only until we find
the “true” one. The belief that this universe actually exists and the
consequent dismissal of a pluralist conception is based on two main
arguments: first, already in Vwe can simulate different universes, so
there is no need to assume the existence of other universes: second, set
theory is actually categorical, i.e. all the different models stemming
from it are in fact isomorphic. I argue that both arguments are off
target.
1 Introduction
From its beginning, classic set theory has been quite the success story: it is
an axiomatic theory in which all the known mathematics can be expressed. It
was later discovered that its classical axiomatization, Z F C, was not enough
to settle key questions such as the Continuum Hypothesis (CH), that are
independent from the axioms of set theory. Since then, the quest for new
axioms to settle these independent questions began. This, in addition to the
development of forcing, gave rise to a plurality of set theoretic theories and
models: a multiverse.
The advocate of universism argues that all these theories and models
have only instrumental value, and that somewhere down the line we will
be able to decide which theory is the “right” set theory. This argument
takes strength from categoricity results and the possibility to simulate non
standard models of Z F C inside Vitself. I argue that these two arguments
are more problematic than it is usually thought, and that they don’t pose a
threat to the pluralistic conception of set theory.

2Against universism
A theory Tis categorical if and only if all of its model are isomorphic,
which is essentially to say that Thas only one model (up to isomorphism).
It is well known that no first-order axiomatisation of sets, such as Z F C, may
provide us with just one isomorphism type corresponding to the “structure
of all sets”. This is because first-order theories admit of a huge number
of non-isomorphic models, which makes it impossible for them to meet the
modelist’s desideratum. On the other hand, full second-order set theory
(such as, ZF C2, that is, Z F C with second-order versions of the Separation
and Replacement Axioms), may, in principle, fit the bill, thanks to Zermelo’s
quasi-categoricity theorem for ZF C2.
Zermelo proved that, given any two models Mand Nof ZF C2, either
M=N, or Mis a rank initial segment of N, that is, Mand Nmust have
the same width, but they might differ in height (and the height must, at
least, be that of Vκ, where κis a strongly inaccessible cardinal).12 So, the
universist might thrive on the quasi-categoricity result by asserting that any
sufficiently strong second-order theory of sets (such as ZF C2) will provide us
with a good approximation of the relevant isomorphism type of the ‘intended’
structure of all sets. Z F C2settles most of the independent propositions (in
particular the CH).
However, this second-order determinacy is seen by most set-theorists sim-
ply as a consequence of the in-built logical features of the theory, rather than
of the second-order axioms’ ability to express a more determinate concept of
set. Thus, set-theoretic indeterminacy has progressively come to be viewed
as an integral and stable feature of the current set-theoretic landscape, and
most of present set theoretic is carried out within (first-order) ZF C and,
indeed, systematically exploits its radical indeterminacy.
2 Against universism
As mentioned, universism is the thesis that there is only one set theoretic
universe, V(the canonical universe of set theory). This universe is the so
called “canonical” model of set theory, as opposed to all the others models,
the non-standard models. For example, the constructible universe Lis a
non-standard model of set theory.
Although it is true that set theorists make use all kinds of non standard
models of Z F C, universists typically insist that each of these models can
1In Zermelo (1930).
2A cardinal κis said to be inaccessible if: (1) it is regular and (2) limit. The least
inaccessible >ℵ0cannot be proved to exist in ZF C .κis strongly inaccessible if it is
inaccessible and, for all λ<κ,2λ< κ.

3Against categoricity
be simulated within V. An argument against pluralism is that in the single
universe Vwe can actually simulate any non standard model of set theory.
For instance, in Z F C we can simulate a model of Z F C +V=Lor a model
of ZF C +LCs (i.e. ZF C +Large Cardinals axioms), even though they are
incompatible.3However, we cannot simulate them at the same time. This
means that in Vwe can have a simulation of ZF C +V=Lin the canonical
model, but then, when trying to simulate Z F C +LCs we are forced to throw
away everything that was proved in the simulation of ZF C +V=L. The
main consequence of this fact is that we cannot compare two non standard
models at the same time. By contrast, all the different models are avail-
able in the set theoretic multiverse, at the same time, and we can prove
isomorphisms between their structures. Thus, compared with a pluralis-
tic conception of set theory, we are loosing the ability to investigate those
models synchronically. This means that, when comparing a single universe
prospective with a multiverse prospective using the principle MAXIMIZE4,
the latter will fare better than the former. Consequently, from a naturalistic
point of view, a multiverse conception of set theory is a more apt to include
and compare any mathematical object (this is the Generous Arena role from
Maddy, 2017) than the Single Universe.
3 Against categoricity
The multiversist might bite the bullet and insist that the study of isomor-
phisms between non standard models of set theory isn’t the proper subject of
set theory, and that when set theory is properly axiomatised, at the second-
order, it has only one model or, at least, it is quasi-categorical. More specifi-
cally, the argument assumes that set theory must be formulated at the second
order. That is, set theory must not only quantify on members of the domain,
but also on arbitrary subsets of the domain.
Now, the significance of Zermelo’s theorem is highly disputed.5As has
been frequently pointed out, the desired philosophical consequences of the
3V=Lis the Axiom of Constructability, that says that all the sets of the universe
can be build from simpler sets, and it is incompatible with the existence of most large
cardinals (LCs).
4According to this principle, when comparing two theories the one that can prove more
isomorphisms types is preferable, see Maddy, 1997.
5Zermelo himself does not encourage this interpretation of his theorem in Zermelo
(1930) (in Ewald (1996), pp. 1232-33), but the point is controversial. Other authors
have construed the quasi-categoricity of Z F C2as rather supporting: (1) the existence of a
single concept of set (Martin (2001)), or (2) the truth-value determinacy of all set-theoretic
statements (McGee (1997)).

4Against categoricity
theorem are available only if one adopts the full semantics of second-order
logic, while an alternative semantic approach, Henkin semantics, not con-
ducive to the quasi-categoricity result, is always possible. Moreover, Mead-
ows (2013)) makes the more general claim that, while categoricity proofs
(including Zermelo’s proof) may successfully prove that, under certain con-
ditions, given a realm of mathematical objects, there is certainly one theory
which picks up a unique structure of them, they cannot help establish that
there is one natural, intuitively given unique structure which incorporates all
the properties of that realm.6
Button and Walsh, elaborating upon McGee (1997), Parsons (1990) and,
partly, Putnam (1979), have come up with a reformulation of this argument
which relies, instead, upon an internal version of categoricity. Internal cate-
goricity is, very roughly, the idea that categoricity may be formulated, within
a suitable second-order logical framework, in a purely deductive way, with-
out having to deal with and/or endorse semantic facts whose philosophical
value is controversial. In particular, supporters of internal categoricity will
place great emphasis on what McGee calls the “second-order intolerance to
truth-value indeterminacy”, that is the fact that, for all ϕexpressed in the
language of second-order set theory, it is possible to prove that either ϕor
¬ϕis a theorem, and none on Zermelo’s quasi-categoricity theorem.
McGee and Martin have tried to revamp this argument, arguing that
determinacy it is not a feature of the logic used to formulate the theory, but
a consequence of the axioms itself. To this end, McGee (1997) proves that
second order set theory with urelements7(ZF C U2) is also categorical, while
Martin (2001) argues that the notions of set and membership are unique and
thus can be represented only by a categorical axiomatization.
The problem with both of these arguments is that they both rely on ex-
tremely strong assumptions: McGee’s theorem is based upon the Urelements
Axiom (that says that there exist urelements), while Martin arguments are
based on the Uniqueness Postulate (a form of extensionality that applies not
only in a structure, but across all structures). It can be argued that both
assumptions have an ad hoc character: while they help proving categoricity
results, they are not really used by mathematicians. More precisely, they are
not needed to prove standard mathematical results.8
More generally, the main problem with categoricity results is that the
concept of set is an algebraic concept. Briefly, this means that set theory
6As the latter belief already implies that either the concept of those objects is unique,
or that any further underpinning of the properties derivable from that concept is unique.
7An urelement is an object that it is not a set, but could be a member of a set.
8In particular, Martin’s Uniqueness Postulate is exceedingly strong: it allows one to
prove that even first-order set theory is categorical!

5REFERENCES
is no different from group theory or any other algebraic theory. The main
intention when formulating these theories is to admit non isomorphic mod-
els, since mathematicians need different models for different purposes9. All
these models are considered totally legitimate by the majority of working
mathematicians. The main consequence of this attitude is that second-order
determinacy is seen by most set-theorists simply as a consequence of the
in-built logical features of the theory, rather than as a consequence of the
second-order axioms’ ability to express a more determinate concept of set.
Thus, set-theoretic indeterminacy has progressively come to be viewed as an
integral part and stable feature of the current set-theoretic landscape.10
4 Conclusion
In conclusion, universism faces serious difficulties. The main argument usu-
ally considered in its defense, categoricity, is quite problematic, and doesn’t
give a satisfactory solution to the foundational problems in mathematics and
set theory.
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10As argued by Mostowski (1967) and, more recently, Hamkins (2012).

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