Abstract
Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a ‘structural’ perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure.
The authors wish to thank Ingo Blechschmidt, Andrew Brooke-Taylor, David Corfield, Patrik Eklund, Michael Ernst, Vera Flocke, Henning Heller, Deborah Kant, Cory Knapp, Maria Mannone, Jean-Pierre Marquis, Colin McLarty, Chris Scambler, Georg Schiemer, Stewart Sharpiro, Michael Shulman, Thomas Streicher, Oliver Tatton-Brown, Dimitris Tsementzis, Giorgio Venturi, Steve Vickers, Daniel Waxman, John Wigglesworth, and an anonymous reviewer for helpful discussion and comments, as well as an audience in Mussomeli, Italy at Filmat 2018. They would also like to thank the editors for putting together the volume. In addition, they are very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme).
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Notes
- 1.
This term is slightly coarse since it is ambiguous between material and categorial set theories (we will distinguish these later). For those that know the difference between the two types of set theory, we mean “material set theory” by “set theory” until we make the distinction, and lump categorial set theories in with category-theoretic foundations for now.
- 2.
See, for example, Awodey (1996).
- 3.
There is some dispute over the use of the term ‘categorial’ versus ‘structural’ when axiomatising sets in category theory. We use the term ‘categorial’ since we reserve structure-like terms for the philosophical notion of structure.
- 4.
- 5.
As with many notions in category theory, there are different arrow-theoretic ways of getting at the same idea. See, for example, Goldblatt (1984, Ch. 4) for some discussion.
- 6.
We are grateful to Michael Shulman and Dimitris Tsementzis for emphasising the importance of making this distinction.
- 7.
See Baldwin (2018, Ch. 1, esp. §1.2) for an argument that this is an often ignored distinction.
- 8.
We are grateful to Andrew Brooke-Taylor for bringing this example to our attention, and some further discussion of the issue. We would also like to thank Ingo Blechschmidt and Jean-Pierre Marquis for some further helpful conversations, in particular emphasising the pervasiveness of the non-concreteness phenomenon. For additional discussion see Marquis (2013), and for results showing how non-concreteness permeates see Di Liberti and Loregian (2018).
- 9.
A further simple (but somewhat silly) example is the following category which we define material-set-theoretically. The category has just one object {a, b}, and a single morphism defined by f(a) = f(b) = b. Here f = Id {a,b} (in the category), and so is trivially iso, but is nonetheless non-bijective.
- 10.
At least since Benacerraf (1965).
- 11.
A salient third option (especially given the topic of the current volume) is Homotopy Type Theory. Here type theory endowed with a homotopy-theoretic interpretation is employed, providing a foundation that meshes elegantly with category-theoretic methods. See the excellent (The Univalent Foundations Program 2013) for discussion.
- 12.
Good examples here are so called large cardinal axioms, as well as forcing axioms, and inner model hypotheses.
- 13.
Here we are playing slightly fast-and-loose with debates in the foundations of set theory; under a natural interpretation of Joel Hamkins’ multiverse perspective, set theory also should be understood as purely algebraic. See Hamkins (2012) for the original presentation of this view and Barton (2016) for an argument to the effect that this results in a purely algebraic interpretation.
- 14.
- 15.
- 16.
Moreover, one might think that category theory formalises these schematic types in a way that highlights privileged conceptual routes (such as when we know that a particular property is universal). Marquis, for example, writes:
“The point I want to make here is extremely simple: category theory, and not just its language, provides us with the proper code to represent the map of mathematical concepts.” (Marquis 2017a, p. 92)
- 17.
A second objection, one that we will not consider here, is the point raised by Mathias (2000, 2001) that category theory lacks the logical strength to discuss certain strong statements of analysis that relate to large cardinal axioms. While the objection merits a response, we set this aside for several reasons: (1) research is ongoing here, and it is unclear that category theory cannot do the job, (2) there are, in any case, logically strong category-theoretic statements (see below), and (3) the possible responses to the objection do not help us elucidate the philosophical role being played by category theory in terms of schematic types. See also Ernst (2017) for some discussion of these issues, as well as a general survey of the comparisons between categorial and set-theoretic foundations.
- 18.
We use the term ‘model’ in a loose and informal way here, and intend it to apply to possibly proper-class-sized structures. For example, we will at least allow (L, ∈) as a model, even though it is proper-class-sized.
- 19.
In the quotation above, Mac Lane is specifically interested in the first point we consider. However, the intuition expressed transfers naturally to other objections he makes, as outlined below.
- 20.
Similar remarks are made repeatedly in Mac Lane (1986), cf. pp. 359–360, 373.
- 21.
- 22.
The topos axiomatised by ETCS is that of a well-pointed topos with a natural number object and satisfying the categorial version of the Axiom of Choice.
- 23.
Of course, the material set-theorist might just accept the existence of proper-classes, hyper-classes, hyper-hyper-classes and so on. This is naturally interpretable in an ontology on which every universe can be extended in height, but there is also a question of whether the believer in one maximal unique universe of sets could also make use of nth-order hyper-classes. Normally it is assumed not, but this question remains philosophically open.
- 24.
Given the topic of the present volume, there is an interesting question as to the extent to which this difficulty is avoided in homotopy type theory. We thank Dimitris Tsementzis for the suggestion that this difficulty could possibly be overcome in this foundation.
- 25.
- 26.
- 27.
An example of such a structure would be the Skolemisation and Mostowski Collapse of any set-theoretic structure satisfying an appropriate amount of set theory.
- 28.
There may, nonetheless, be certain philosophical considerations here, as well as technical issues concerning how much higher-order reasoning we can capture using extensions. See Barton (2018) and Antos et al. for discussion.
- 29.
- 30.
This said, there are category-theoretic options here. See Bell (2011).
- 31.
Vladimir Voevodky himself was clear about this role for ZFC with respect to Homotopy Type Theory. See, for example, his abstract for the 2013 North American Meeting of the Association of Symbolic Logic, where he says:
Univalent foundations provide a new approach to the formal reasoning about mathematical objects. The languages which arise in this approach are much more convenient for doing serious mathematics than ZFC at the cost of being much more complex. In particular, the consistency issues for these languages are not intuitively clear. Thus ZFC retains its key role as the theory which is used to ensure that the more and more complex languages of the univalent approach are consistent. (Voevodsky 2014, p. 108)
We are grateful to Penelope Maddy for bringing this to our attention.
- 32.
The following analogy may be helpful. Viewing mathematics as describing a kind of quasi-computational enterprise, set theory is something like a theory of machine-code: It tells us what kinds of things can be built, and what we need to build them. Category theory on the other hand is like a high-level programming language, codifying what effects can be achieved by different structural relationships in different contexts. Both the set theorist as computational engineer and the category theorist as programmer have important roles to play in mathematics. This analogy (or, at least, something similar) was originally communicated to the first-author by David Corfield after a talk at the LSE in November 2013. He is also grateful to Dr. Corfield for subsequent discussion of categorial foundations.
- 33.
Strictly speaking, we have used a formulation of category theory here on which we have variables for objects as well as arrows. In a purely arrow-theoretic framework (where equality is only defined between parallel morphisms) one has the result that any two equivalent categories satisfy the same sentences. (We are grateful to Ingo Blechschmidt for helpfully forcing us to be precise about this issue.) However, if one wants to use parameters, for instance if one is developing the theory of modules in ETCS, or using more than one variable, then the result no longer holds. (Thanks here are due to Jean-Pierre Marquis for this useful comment.) There are some theories that aim to make it impossible to state non-isomorphism-invariant properties in their language. Two candidates here are Makkai’s FOLDS (see Makkai 1998; Marquis 2017b) and Univalent Foundations (see The Univalent Foundations Program 2013; Tsementzis 2017). We are grateful to Dimitris Tsementzis for stressing this use of Univalent Foundations and making us aware of the non-isomorphism invariance of category theory (in conversation and his Tsementzis 2017), as well as directing us to Theorem 8. We are also grateful (again) to Jean-Pierre Marquis for emphasising to us the interest of FOLDS.
- 34.
See here McLarty (1993, p. 495).
- 35.
See here, for example, Shapiro (1991).
- 36.
- 37.
An alternative would be to take the ‘wide sets’ approach of Menzel (2014) and modify Replacement.
- 38.
We make this assumption merely for technical hygiene since NBGS will do the job we want neatly. One could also drop this restriction, and use an impredicative comprehension scheme yielding a structural form of Morse-Kelley (call it MKS). This may well have interesting additional properties, such as the ability to define a satisfaction predicate for the universe and first-order formulas.
- 39.
In fuller formalism: If f is a set-theoretic isomorphism between s 0 and s 1, then there is an s such that s maps u α to \(u^{\prime }_\alpha \) iff f does.
- 40.
In fuller formalism: For any set X of urelements and set-theoretic functions \(f^X_{m,n}\), relations \(R^X_{m,n}\) on X, and constants \(c^X_n\) in X, there is an s such that U(s, u α) (for each u α ∈X), and structural relations \(f^s_{m,n}\), \(R^s_{m,n}\), and \(c^s_n\) equivalent to \(f^X_{m,n}\), \(R^X_{m,n}\), and \(c^X_n \) in the obvious way.
- 41.
This can be proved by the usual tedious induction on the complexity of ϕ.
- 42.
See, for example, Tent and Ziegler (2012) for a presentation of the usual proof.
- 43.
We are grateful to Steve Vickers for pressing this point.
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Barton, N., Friedman, SD. (2019). Set Theory and Structures. In: Centrone, S., Kant, D., Sarikaya, D. (eds) Reflections on the Foundations of Mathematics. Synthese Library, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-030-15655-8_10
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