Are the categories of sets, abelian groups, and commutative rings unique?

Question

Are the categories of sets, abelian groups, and commutative rings unique? Independence results like the independence of the generalized continuum hypothesis, the Whitehead problem, and the global dimension of $\prod_{n = 1}^\infty \mathbb{F}_2$ from ZFC seem to indicate no. And yet we don't say "Let $\mathbf{Set}$ be a category of sets" instead of "Let $\mathbf{Set}$ be the category of sets," etc. If these categories are not unique, then could they be if we wanted them to? And shouldn't they be unique, if the terms "set," "function," "abelian group," "commutative ring," etc., are to be well-defined?

EDIT: Questions have arisen as to what I mean by "unique". Unique as in one and only one? Unique as in unique up to equivalence of categories? Unique as in unique up to isomorphism of categories? Unique as in unique up to unique isomorphism of categories? Honestly, any of those four senses of "unique" is fine with me. Just pick one and answer that particular question. Does it make sense to consider $\mathbf{Set}$ as a category that is unique in some sense described above?

EDIT (SUMMARY): Answers to this question seem to have been of one of the following flavors.

1. Monism allows $\mathbf{Set}$ to be unique, but pluralism does not. Some mathematicians are monists, some are pluralists, while others think that both monism and pluralism are respectable philosophies of mathematics. Call these views "monism," "pluralism," and "monist-pluralist dualism."

2. Just as $\mathbb{Z}$ is unique up to iso in the category of rings, $\mathbf{Set}$ and $\mathbf{Ring}$ are unique up to category equivalence in the meta-category of categories, but there is no sense of uniqueness in an absolute sense. $\mathbb{Z}$ and $\mathbf{Set}$ and $\mathbf{Ring}$ are not unique in an absolute sense. If this is the case, then I'd argue that mathematical truth, too, is not absolute, but rather relative to a theory. Call this view "relativism." This view is new to me. I am not a relativist, at least as yet.

3. $\mathbb{Z}$ is unique up to iso, $\mathbf{Set}$ and $\mathbf{Ring}$ are unique up to category equivalence, etc. Mathematical truth is absolute and does not mean truth within a theory, and uniqueness is also absolute. Call this view "absolutism."

4. Conventionally, mathematicians today are operating under the assumptions of ZFC, and ZFC doesn't answer the question in the positive or in the negative, and so the question doesn't have a mathematical answer. Call this view "conventionism".

The answer one gives to the question apparently depends on what philosophical views one holds. But the same is true about questions like, "Is the axiom of choice true," "Does every nontrivial commutative ring have a maximal ideal," "Does every vector space have a basis?" A relativist would say, yes, if one is working under the assumptions of ZFC and the ordinary rules of mathematical proof. An absolutist would just say yes (if they believed that the axioms and theorems of ZFC were true). Others might just say yes, under the tacit assumption that most mathematicians today are operating under the assumptions of ZFC, while remaining agnostic about the status of mathematical truth.

Absolutists might see the question I posted as a valid mathematical question. A problem with asbolutism is that many important questions, like GCH and the Whitehead problem, have not been settled, at least as yet. A problem with relativism is, why are "if-then" statements true? What makes Boolean first-order logic absolute but not the rest of mathematics? Why are any mathematical proofs valid at all? Why not assume some non-Boolean constructive or intuitionistic logic? If my question is not a purely mathematical one and is partly "philosophical" and "open to interpretation", then isn't every mathematical question thus?

These aren't additional questions I'm asking for discussion here. I'm just trying to explain why I thought my question was a valid mathematical question and not a purely philosophical one. Thank you to those who tried to answer it, as your answers have much clarified my thinking about the problem. I've accepted Hamkins' answer because it was least biased, but I welcome other answers to the question and can always change what answer I accept. I felt pressured to accept an answer because of a vote to close the question as inappropriate for MO.

Answer 1

Introduction to pluralism

A version of this question lies at the heart of the ongoing dispute on pluralism in the philosophy of mathematics. Is there at bottom just one mathematical reality? Does every mathematical question, whether about arithmetic, about the real continuum, or about set theory, have a definite mathematical answer?

Most mathematicians (but not all) take the view that arithmetic assertions, for example, assertions such as the Riemann hypothesis or the question whether there are infinitely many prime pairs, have a definite answer. Either there are infinitely many prime pairs or there are not, full stop. According to this view, arithmetic questions have a determinate answer, whether we shall ever come to know it.

We know by Gödel's theorem, of course, that no effective axiomatic system (whether formalized in set theory, category theory, HoTT, or what have you) will be able to establish the truth of all the true arithmetic assertions. But this observation can be taken to be merely about the weakness of our formal theories, rather than necessarily about any kind of pluralism in the arithmetic facts of the matter. One can admit that any given theory is weak, even if one holds that true arithmetic is meaningful. Peano arithmetic PA is incomplete, but ZFC proves more arithmetic truths, and ZFC + large cardinals settles still more.

Characterizing structures in second-order logic

Support for this determinate view of arithmetic truth is often taken from the categoricity results. Dedekind proved, namely, that the natural number structure $\langle\mathbb{N},S,0\rangle$ is uniquely determined up to isomorphism by the three Dedekind axioms, that $0$ is not a successor, that the successor function is one-to-one, and that every number is generated from $0$ by successor, in the sense that every set of numbers containing $0$ and closed under successor contains all the numbers. Once one knows that the axioms determine the structure, then one knows that those axioms determine all arithmetic truth. I have argued that Dedekind's categoricity result is the beginning of structuralism in mathematics.

Other mathematicians observe that we have such categorical accounts of essentially all our familiar mathematical structures. The integer ring is uniquely determined up to isomorphism from the natural numbers, and the rational field. The real field is the unique complete ordered field. The complex numbers are the unique algebraic closure of the real field, or alternatively, they are the unique algebraically closed field of size continuum.

Thus, all our familiar mathematical structures are characterized uniquely in second-order logic. This can be taken as support for the view that mathematical truth generally is determinate in nature. The structures are determined by the (second-order) axioms, and thus the truths are determined.

Characterizing structures in first-order set theory

All these familiar categoricity arguments can be viewed as taking place in first-order set theory, provable as theorems of ZFC. In a sense, the first-order theory of sets provides a natural interpretation of the second-order logic of any particular structure, by providing the sets that will constitute the second-order part. So ZFC proves that there is a unique structure of arithmetic up to isomorphism, a unique real-closed field with a unique algebraic closure. Similarly, ZFC proves that the category of groups and the category of sets and so on is unique up to suitable isomorphism.

The philosophical difficulty here is that, meanwhile, ZFC is a first-order theory and thus subject to the incompleteness phenomenon. We know, for example, that if ZFC is consistent, then there can be models $M_0$ and $M_1$ of ZFC whose natural number structures $\mathbb{N}^{M_0}$ and $\mathbb{N}^{M_1}$ are not isomorphic to each other, even though each of them is thought to be the unique natural number structure in those respective set-theoretic universes. The philosophical difficulty is that, although ZFC proves that arithmetic truth is definite, nevertheless different models of ZFC can think that it is different arithmetic truths that come to be part of the definitely true arithmetic theory.

Characterizing structures in second-order set theory

In light of this, some logicians and philosophers of mathematics seek to apply the second-order categoricity results to set theory itself. Indeed, Zermelo proved that the models of second-order ZFC${}_2$ enjoy a quasi-categoricity result—they all agree on their common initial segments. Specifically, Zermelo proved that the models of second-order ZFC${}_2$ are exactly the models $V_\kappa$ for an inaccessible cardinal $\kappa$, or in other words, exactly the Zermelo-Grothendieck universes. These set-theoretic worlds are linearly ordered and all agree on the assertions expressible in their common parts.

Kreisel famously pointed out that indeed essentially all questions of classical mathematics are expressible as sentences that are absolute to low-level ranks of the cumulative hierarchy, which have the same truth value in all these Zermelo-Grothendieck universes. Thus, according to Kreisel, the continuum hypothesis has a determinate truth value in second-order set theory. It is either definitely true or definitely false, as a matter of (second-order) logic.

The universe view of sets

This is the beginning of the universe view, which holds that there is a unique set-theoretic reality underlying mathematics, and all statements have a definite truth value in this unique set-theoretic realm. This is the set-theoretic universe arising from the cumulative set-building process, where one iteratively computes the $V_\alpha$ hierarchy by adding all subsets at each stage and iterating through the ordinals. On this view, the answer to your question is Yes, there is a unique category of all groups, and it is the category of groups as defined in this final true set-theoretic universe $V$, and similarly with the category of rings and what have you.

Critics point out that second-order logic is simply a species of set theory. How can we establish the definiteness of our concept of finiteness by appealing to the comparatively murky concept of arbitrary set required in the second-order induction axiom? It seems hopeless to ground our concept of the finite this way. See my essay, A question for the mathematics oracle.

The multiverse view of sets

Set-theoretic pluralism offers an alternative perspective. According to the multiverse view, there are many concepts of set, each giving rise to a different set-theoretic realm. The continuum hypothesis might hold in some and not in others, and this situation is itself a kind of answer to the CH question—it holds and fails throughout the set-theoretic multiverse in a way that is quite deeply understood.

In regard to your question, these different set-theoretic universes each have their own (unique in that universe) categories of groups and categories of rings and so on, and these categories are not always isomorphic to each other across the universes. On the pluralist view, the answer to your question is negative.

Indeed, the question of whether they are isomorphic or not presumes a certain degree of set theory in the metatheory where those universes exist, and in this way one is led to the idea that there is a hierarchy of metatheoretic contexts. Indeed, every model of set theory provides a meta-theoretic context for the theories and models and categories which exist within that model. In this way, the traditional object-theory/meta-theory distinction is seen to break down as naive or crude, for we actually have a rich hierarchy of theories, each serving also as a metatheory.

More extreme and more moderate alternatives

Strong forms of the pluralist view extend to pluralism even in arithmetic as well as higher set theory. Many mathematicians prefer a kind of compromise position, taking arithmetic truth as definite, but allowing indeterminacy in higher set-theoretic truths. The universists hold that the set-theoretic universe is determinate all the way up, and the large cardinal hierarchy is pointing the way toward the one road upward.

So there are philosophical positions taken on all sides of this issue. I have written at length on these topics in various venues, but perhaps you might look at:

• My paper: Hamkins, Joel David, The set-theoretic multiverse, Rev. Symb. Log. 5, No. 3, 416-449 (2012). ZBL1260.03103.
• Chapter 8 of my book: Hamkins, Joel David, Lectures on the philosophy of mathematics, MIT Press, 2021.

Answer 2

Mathematics:

In mathematics as normally practiced in the most standard foundations, we take the notion of a "set" as given, and then we define everything else in math in terms of sets. We reason as though any statement we make has a definite truth value. We have some axioms (the ZFC axioms, let's say) which allow us to deduce which statements are true. We know that these axioms are not complete -- there will always be statements whose truth we can neither deduce nor refute from these axioms.

In terms of these most standard foundations, each of the categories you've mentioned is defined by a specific formula in the language of set theory [1], and so it is indeed unique. There's nothing special about categories among other first-order structures here [2]. There are statements about these categories whose truth we can neither prove nor refute. If you're not satisfied with this state of affairs, you're in good company (with Hilbert among others), but that's the way it is.

Metamathematics:

Of course, we can also do metamathematics and try to study things like the deductive system we're using to reason about sets, or try to change all the background assumptions discussed above. There are multiple areas of logic which study such questions. There are interesting philosophical positions which assert that the above "standard foundations" for mathematics are not the best way to look at things. But there's so much to talk about here that I'm afraid you'll have to ask a more specific question if you want a good answer. And at the end of the day, it's important to recognize that the above description of mathematics is the standard one -- even if you prefer to think about mathematics differently, you should familiarize yourself with how the foundations of mathematics work on the above view, and what the "rules of the game" are according to it.

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[1] Well, when it comes to these categories we do have to deal with size issues somehow. One option is to choose an arbitrary cutoff cardinal $\kappa$ and talk only about the sets / groups / whatever in $V_\kappa$ (maybe we also assume $\kappa$ is inaccessible and augment our axiom system with Grothendieck universes). Another option is to agree in our background theory that classes are also a thing, different from sets (and maybe use the NBG axioms to reason about things). But don't let these distractions obscure the main point: if you pick a sufficiently-expressive background language and metatheory and just agree on it, there's nothing going on here.

[2] See [1].

Answer 3

I believe a positive answer for ${\bf Set}$ lies in

Osius, G., Categorical set theory: A characterization of the category of sets, Journal of Pure and Applied Algebra Volume 4, Issue 1, February 1974, Pages 79-119, https://doi.org/10.1016/0022-4049(74)90032-2.

Osius considers augmentations of Lawvere's ETCS designed to capture various universes of sets ($Z$, $ZF$, $ZFC$, etc.), proceeding roughly by axiomatizing a sufficient chunk of the behavior of ${\bf Set}$ in each universe such that any category satisfying these axioms is equivalent to the category of sets in that universe, expressed initially by ${\bf Prop\ 7.20}$ for $Z$ and culminating in the construction of a 'functor' $\Phi$ which, when considered with sufficient additional axioms, is an equivalence from any model satisfying all axioms up to ${\bf 8.20}$ in his paper to the category of sets in the foundation $EWPT$.

None of the axioms of up to ${\bf 8.20}$ specifically reference the category of sets that I can see, so unless I'm mistaken Osius's characterization is a non-circular description of the category of sets up to equivalence of categories (the correct type of uniqueness for categories, IMO).

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Much has been made of the philosophical aspects of this question; here's my two cents. Mathematics is precise thought, and philosophy is necessarily 'blurry thought' to some extent -- when it becomes precise it becomes mathematics a-la Cantor, Gödel, Grothendieck, Lawvere, Hamkins, etc.

This question has philosophical aspects, but it has several interesting mathematical ones as well. Irrespective of our personal philosophical bents (I guess multiversal relativist for me?) I think we have an obligation to look towards the mathematical and away from the philosophical whenever possible on this site -- there will definitely be questions inextricably tied to philosophy in their answers, but hemming and hawing over 'how tied to philosophy' the answer is doesn't generally help us get at the mathematical content of the question.

Osius's paper provides the basic formula for 'characterizing the category of sets up to equivalence' in any set-theoretical universe -- find algebraic expressions for its behavior in terms of functions, then internalize these axioms in a category with sufficient structure to interpret the base language of that set theory until a canonically defined functor ($\Phi$ in his paper) becomes an equivalence between any category satisfying the internal versions of these axioms and the category of sets. Speaking as someone who believes in a multiverse of sets, this is a satisfying characterization.