Categorical large cardinals and the tension between categoricity and set-theoretic reflection
aa r X i v : . [ m a t h . L O ] S e p CATEGORICAL LARGE CARDINALS AND THE TENSIONBETWEEN CATEGORICITY AND SET-THEORETICREFLECTION
JOEL DAVID HAMKINS AND HANS ROBIN SOLBERG
Abstract.
Inspired by Zermelo’s quasi-categoricity result characterizing themodels of second-order Zermelo-Fraenkel set theory ZFC , we investigate whenthose models are fully categorical, characterized by the addition to ZFC ei-ther of a first-order sentence, a first-order theory, a second-order sentence or asecond-order theory. The heights of these models, we define, are the categor-ical large cardinals. We subsequently consider various philosophical aspectsof categoricity for structuralism and realism, including the tension betweencategoricity and set-theoretic reflection, and we present (and criticize) a cat-egorical characterization of the set-theoretic universe h V, ∈i in second-orderlogic. Categorical accounts of various mathematical structures lie at the very core ofstructuralist mathematical practice, enabling mathematicians to refer to specificmathematical structures, not by having carefully to prepare and point at speciallyconstructed instances—preserved like the one-meter iron bar locked in a case inParis—but instead merely by mentioning features that uniquely characterize thestructure up to isomorphism.The natural numbers h N , , S i , for example, are uniquely characterized by theDedekind axioms, which assert that 0 is not a successor, that the successor func-tion S is one-to-one, and that every set containing 0 and closed under successorcontains every number [Ded88, Ded01]. We know what we mean by the naturalnumbers—they have a definiteness—because we can describe features that com-pletely determine the natural number structure. The real numbers h R , + , · , , i similarly are characterized up to isomorphism as the unique complete ordered field[Hun03]. The complex numbers h C , + , ·i form the unique algebraically closed fieldof characteristic 0 and size continuum, or alternatively, the unique algebraic closureof the real numbers. In fact all our fundamental mathematical structures enjoy suchcategorical characterizations, where a theory is categorical if it identifies a uniquemathematical structure up to isomorphism—any two models of the theory are iso-morphic. In light of the L¨owenheim-Skolem theorem, which prevents categoricityfor infinite structures in first-order logic, these categorical theories are generallymade in second-order logic.In set theory, Zermelo characterized the models of second-order Zermelo-Fraenkelset theory ZFC in his famous quasi-categoricity result: Commentary can be made about this article on the first author’s blog athttp://jdh.hamkins.org/categorical-large-cardinals.
Theorem 1 (Zermelo [Zer30] ) . The models of
ZFC are precisely those isomorphicto a rank-initial segment h V κ , ∈i of the cumulative set-theoretic universe V cut offat an inaccessible cardinal κ . To prove this, Zermelo observed that if M is a model of the second-order axiom-atization of set theory ZFC , with the full second-order replacement axiom, then M will be well-founded, since it will contain all its ω -sequences; so it will be (iso-morphic to) a transitive set; it will be correct about power sets; consequently, M ’sinternal cumulative V α hierarchy will agree with the actual V α hierarchy, and theheight κ = Ord M will have to be both regular and a strong limit. So M will be V κ for some inaccessible cardinal; and conversely, all such V κ are models of ZFC .It follows that for any two models of ZFC , one of them is isomorphic to aninitial segment of the other. These set-theoretic models h V κ , ∈i have now cometo be known as Zermelo-Grothendieck universes , in light of Grothendieck’s use ofthem in category theory (a rediscovery several decades after Zermelo); they featurein the universe axiom , which asserts that every set is an element of some such V κ ,or equivalently, that there are unboundedly many inaccessible cardinals.In this article, we seek to investigate the extent to which Zermelo’s quasi-categoricity analysis can rise fully to the level of categoricity, in light of the ob-servation that many of the V κ universes are categorically characterized by theirsentences or theories. Main Question 2.
Which models of
ZFC satisfy fully categorical theories? If κ is the smallest inaccessible cardinal, for example, then up to isomorphism h V κ , ∈i is the unique model of ZFC satisfying the first-order sentence “there areno inaccessible cardinals.” The least inaccessible cardinal is therefore an instanceof what we call a first-order sententially categorical cardinal. Similar ideas apply tothe next inaccessible cardinal, and the next, and so on for quite a long way. Manyof the inaccessible universes thus satisfy categorical theories extending ZFC by asentence or theory, either in first or second order, and we should like to investigatethese categorical extensions of ZFC .In addition, we shall discuss the philosophical relevance of categoricity and pointparticularly to the philosophical problem posed by the tension between the wide-spread support for categoricity in our fundamental mathematical structures withset-theoretic ideas on reflection principles, which are at heart anti-categorical.1. Main definition and preliminary results
Our main theme concerns these notions of categoricity:
Main Definition 3. (1) A cardinal κ is first-order sententially categorical , if there is a first-ordersentence σ in the language of set theory, such that V κ is categorically char-acterized by ZFC + σ . Zermelo was more generally concerned with the urelement-based versions of set theory, andwhat he proved in [Zer30] is that the models of second-order ZF − infinity with urelements aredetermined by two cardinals—their inaccessible cardinal height (allowing ω when the infinityaxiom fails) and the number of urelements. In this article, we shall focus on the case of ZFC ,where there are no urelements and the axiom of infinity holds. ATEGORICAL LARGE CARDINALS 3 (2) A cardinal κ is first-order theory categorical , if there is a first-order theory T in the language of set theory, such that V κ is categorically characterizedby ZFC + T .(3) A cardinal κ is second-order sententially categorical , if there is a second-order sentence σ in the language of set theory, such that V κ is categoricallycharacterized by ZFC + σ .(4) A cardinal κ is second-order theory categorical , if there is a second-ordertheory T in the language of set theory, such that V κ is categorically char-acterized by ZFC + T .One may easily refine and extend these definitions by stratifying on complexity.Thus, we will have natural notions of Σ mn categoricity in m th-order set theory,including Σ αn categoricity for transfinite order α , for either theories or sentences.And one may also consider categoricity in infinitary languages and so on. In thisarticle, however, let us focus on the categoricity notions in the main definition.Since Zermelo characterized the inaccessible cardinals κ as those for which V κ | =ZFC , all the cardinals κ in the main definition above are inaccessible. We couldequivalently have defined that κ is first-order sententially categorical if there is afirst-order sentence σ such that κ is the only inaccessible cardinal for which V κ | = σ .And similarly with the other kinds of categorical cardinals. In this sense, the topicis about categorical characterizations of V κ for inaccessible κ .We find it interesting to notice that theory categoricity is akin to Leibniziandiscernibility, since κ is theory categorical when V κ can be distinguished from othercandidates V λ by a sentence.If there are any inaccessible cardinals at all, then there will be easy examples ofcategorical cardinals. As we mentioned earlier, the least inaccessible cardinal κ ischaracterized over ZFC by “there are no inaccessible cardinals.” The next inac-cessible cardinal is characterized by the first-order sentence, “there is exactly oneinaccessible cardinal.” More generally, the α th inaccessible cardinal (if we indexfrom 0) is characterized by the assertion “there are exactly α many inaccessiblecardinals.” If α is sufficiently absolutely definable, then this assertion can be madewithout parameters and so the α th inaccessible cardinal will be sententially cate-gorical. So the categorical cardinals proceed from the beginning for quite a longway, up to the ω CK th inaccessible cardinal and beyond. Since we have observedthat the smallest large cardinals are generally categorical, there is a sense in whichcategoricity is a smallness notion, while non-categoricity is a largeness notion. Yet,the ω th inaccessible cardinal if it exists is sententially categorical, and the ω nd,and more. And thus we seem to open the door to the possibility of gaps in thecategorical cardinals, since there are only countably many sentences. Observation 4.
Categoricity is downward absolute from V to any V θ . That is, if κ is categorical in one of the four manners of the main definition and θ > κ , thenthe structure V θ knows that κ is categorical in that way.Proof. The point is that V θ can verify that V κ has whatever theory it has and thereare fewer challenges to categoricity in V θ than in V . Every inaccessible cardinal δ in V θ is also inaccessible in V , and consequently differs in its theory from V κ , and V θ can see this. (cid:3) Upward absoluteness might seem at first to be too much to ask for, since perhapsa cardinal κ can be categorical inside V θ only because θ is not large enough to reveal JOEL DAVID HAMKINS AND HANS ROBIN SOLBERG the other models V λ that satisfy that same characterization. But for sententialcategoricity, it turns out that this situation never actually arises, and consequentlysentential categoricity is fully absolute between V and V θ . Theorem 5.
Sentential categoricity (first and second order) is absolute between V and any V θ , both upward and downward. That is, if κ < θ , then κ is sententiallycategorical in V if and only if it has the same sentential categoricity in V θ .Proof. If κ is sententially categorical in V , then it is sententially categorical in V θ by the same sentence, since there are fewer competitors inside V θ than in V .Conversely, suppose that κ is sententially categorical inside V θ with κ < θ .So there is a sentence σ , such that V κ | = σ and no other V λ in V θ satisfies σ . Inparticular, V κ is the first inaccessible level to satisfy σ , and so the sentence σ +“thereis no inaccessible δ with V δ | = σ ” is a categorical characterization of V κ in the fulluniverse V . No larger inaccessible level can satisfy this sentence, since V κ | = σ . So κ is sententially categorical in V . (cid:3) Another way to describe the upward absoluteness is that failures of sententialcategoricity are always witnessed by smaller as opposed to larger inaccessible car-dinals with the same sentence. Thus, every failure of sentential categoricity leadsto instances of inaccessible reflection. In the first-order case, one can view this as aweak form of Mahloness, since every Mahlo cardinal also has this property. In thesecond-order case, it is a weak form of indescribability.
Theorem 6.
If an inaccessible cardinal κ is not sententially categorical (either firstor second order), then every sentence σ of that order that is true in V κ is also truein V δ for some smaller inaccessible cardinal δ < κ .Proof. If σ is true in V κ , then it cannot be the first time this happens at an inacces-sible cardinal, since otherwise this very situation could be described in a sentence,providing a categorical characterization of κ . (cid:3) The corresponding fact is not true for theory categoricity, if one expects toreflect the whole theory, because failures of theory categoricity arise when thereis some κ < λ with V κ and V λ having the same theory. But in this case, therewill be some smallest κ with that theory, and this κ is not theory categorical,but this is not witnessed by any smaller δ < κ precisely because κ was alreadythe smallest with that theory. In particular, the analogue of theorem 5 also failsfor theory categoricity, because if κ is the least inaccessible cardinal that is nottheory categorical, then there will be some λ > κ with the same theory, and if λ is smallest with this property, then κ will be theory categorical inside V λ , but notin V . Nevertheless, there is an approximation version of downward reflection forfailures of theory categoricity, which can be seen as a somewhat stronger version ofweak Mahloness. Theorem 7.
If an inaccessible cardinal κ is not first-order theory categorical, thenfor every natural number n , there is a smaller inaccessible cardinal δ < κ forwhich V δ has the same Σ n theory. And similarly, if κ is not second-order theorycategorical, there for every n there is a smaller inaccessible V δ with the same Σ n theory.Proof. Suppose that κ is not first-order theory categorical. Then there is someother inaccessible cardinal λ for which V κ and V λ have the same theory. If λ < κ , ATEGORICAL LARGE CARDINALS 5 then we are done immediately. So assume λ > κ . Notice that V λ thinks that itsΣ n theory is shared by V κ . So it is part of the theory of V λ that there is a smallerinaccessible cardinal with the same Σ n theory. So this statement is also true in V κ ,as desired. The argument works the same with first-order or second-order. (cid:3) The general phenomenon is that failures of categoricity lead to inaccessible re-flection, which we view as weak forms of Mahloness and indescribability.Let us now establish an upward transmission effect of categoricity. For anycardinal κ , we use the boldface successor notation κ to denote the next inaccessiblecardinal above κ . Theorem 8. If κ is second-order sententially categorical, then κ is first-ordersententially categorical.Proof. If ψ is the second-order sentence characterizing V κ , then the next inaccessiblecardinal V κ can see that there is a largest inaccessible cardinal κ and that V κ satisfies ψ , and this property characterizes κ . (cid:3) Theorem 9. If κ is inaccessible and the sententially categorical cardinals are un-bounded in the inaccessible cardinals below κ , then κ is first-order theory categorical.Proof. Note that we are not assuming that κ is a limit of inaccessible cardinals.The first-order theory of V κ includes the assertions that those smaller inaccessiblecardinals satisfy the sentences that characterize them, whether those are first orsecond order. Therefore no smaller inaccessible cardinal δ < κ can have V δ withthe same theory as V κ . And no larger θ > κ can have the same theory either, sincein V θ either there are new sententially categorical cardinals, and the assertion thatthey exist would be part of the theory of V θ not true in V κ , or else the sententiallycategorical cardinals will not be unbounded in the inaccessible cardinals below θ ,which will itself be a statement true in V θ that is not true in V κ . (cid:3) Theorem 10.
No Mahlo cardinal is first-order theory categorical.Proof. If κ is Mahlo, then V δ ≺ V κ for a closed unbounded set of δ , which thereforeincludes many inaccessible cardinals. So V κ is not characterized by any first-ordersentence or theory. (cid:3) Theorem 11.
The least Mahlo cardinal is second-order sententially categorical,but not first-order theory categorical.Proof.
Being Mahlo is a Π property: every club C ⊆ κ has a regular cardinal.So the least one is second-order sententially categorical. But no Mahlo cardinal isfirst-order sententially or theory categorical by the above. (cid:3) Let us consider versions of theorem 10 for weaker large cardinal notions. Anordinal κ is otherworldly if V κ ≺ V β for some ordinal β > κ . (It is an interestingexercise to show that otherworldly cardinals are in fact also worldly , which means V κ | = ZFC, but note that otherworldly cardinals, like the worldly cardinals, neednot be regular and hence are not necessarily inaccessible, although they are stronglimit cardinals and i -fixed points.) Every inaccessible cardinal δ is the limit ofa closed unbounded set of otherworldly cardinals, each of which is otherworldlyup to δ , meaning that the targets β can be found cofinally in δ . A cardinal κ is totally otherworldly if V κ ≺ V β for arbitrarily large ordinals β . Recall from [HJ14]that a cardinal κ is uplifting if it is inaccessible and V κ ≺ V β for arbitrarily large JOEL DAVID HAMKINS AND HANS ROBIN SOLBERG inaccessible cardinals β ; the cardinal κ is pseudo-uplifting if V κ ≺ V β for arbitrarilylarge β , without insisting that β is inaccessible. So the pseudo-uplifting cardinalsare the same as the inaccessible totally otherworldly cardinals. All of these cardinalsare weaker in consistency strength than a Mahlo cardinal, since if δ is Mahlo, then V δ has a proper class of uplifting cardinals, which are therefore also pseudo-upliftingand inaccessibly totally otherworldly. None of these types of cardinals, it turns out,can be second-order theory categorical, in light of the following theorem. Theorem 12. If V κ ≺ M for some transitive set M with V κ +1 ⊆ M , then κ is notsecond-order theory categorical.Proof. Assume V κ ≺ M for a transitive set M with V κ +1 ⊆ M . Let T be thesecond-order theory of V κ , and observe that M thinks “there is an inaccessiblecardinal δ such that the second-order theory of V δ is T ,” since this is true in M of δ = κ . So this statement must also be true in V κ , and so there is δ < κ with V δ having the same theory as V κ . So κ is not second-order theory categorical. (cid:3) It follows that no measurable cardinal is second-order theory categorical, nor isany ( κ + 1)-strongly unfoldable cardinal κ , nor any Π -indescribable cardinal, norany uplifting cardinal, nor any otherworldly cardinal, since all these types of largecardinals satisfy the comparatively weak hypothesis of theorem 12.Let us prove the following bound on the categorical cardinals. A cardinal θ isΣ correct if V θ ≺ Σ V , that is, if every statement of complexity Σ (allowingparameters) is absolute between V θ and the full set-theoretic universe V . Everytotally otherworldly cardinal is Σ -correct, as is every strong cardinal and hencealso every supercompact cardinal and every extendible cardinal. Theorem 13.
Every second-order theory categorical cardinal is below the least Σ -correct cardinal.Proof. Suppose that θ is Σ -correct, which means that any Σ assertion aboutobjects in V θ that is true in V is true in V θ . Suppose that κ is second-order theorycategorical, and let T be the second-order theory of V κ . The assertion that there is acardinal κ for which V κ | = T is a Σ assertion about T in the L´evy hierarchy, becauseit can be verified inside any sufficiently large V η (see [Ham14a] for background onlocally verifiable properties). Therefore, κ must be inside V θ . So every second-ordertheory categorical cardinal must be below every Σ -correct cardinal. (cid:3) The proof works just as well with third-order, fourth-order, and so on, up to α thorder for any α that is itself Σ definable, and so these kinds of categoricity arealso bounded by the Σ -correct cardinals.Theorem 13 seems to justify the view of categoricity as a smallness notion forlarge cardinals, since they must all be below the Σ correct cardinals, and so theyare all therefore smaller than every strong cardinal, every supercompact cardinal,every extendible cardinal, every totally otherworldly cardinal, and more, since thesecardinals are Σ correct. This view in turn leads to some philosophical tensionswith the idea that mathematicians seek categorical accounts of all their fundamentalstructures. After all, for us to adopt one of the categorical axiomatizations thatcharacterize these categorical cardinals would be to adopt a theory that we knowmust be describing a smallish set-theoretic universe. This is the opposite of what weare trying to do in our set-theoretic foundations—we seek instead upward-reachingaxioms that will maximize our foundational realm and make it as large as possible, ATEGORICAL LARGE CARDINALS 7 so as more fully to accommodate arbitrary mathematical structure and ideas. Weshall discuss this philosophical tension more fully in section 7.Let us introduce the rank elementary forest on inaccessible cardinals, the relationby which κ (cid:22) λ if and only if V κ ≺ V λ , for inaccessible cardinals κ and λ . This is apartial order on inaccessible cardinals, and it is a forest, since the predecessors ofany node are linearly ordered. Theorem 14.
Every first-order theory categorical cardinal is a stump in the rankelementary forest, that is, a disconnected root node with nothing above it.Proof.
This is almost immediate from the definition. If κ is first-order theorycategorical, then we cannot have V κ ≺ V λ nor V δ ≺ V κ , and so κ must be astump. (cid:3) The converse is not true, since we can have V κ ≡ V λ without V κ ≺ V λ , and so itseems possible to violate first-order theory categoricity while κ is still a stump. Tosee this, let κ be the least inaccessible cardinal for which there is some ordinal λ for which V κ ≺ V λ , and let λ be least with that property. It follows that V λ thinksthat the theory T = Th( V κ ) is realized at an inaccessible cardinal, namely at κ , andso V κ should also think this about T . So there will be some inaccessible cardinal δ < κ with V δ ≡ V κ . So V λ will think that κ is a stump in the rank elementaryforest, but not first-order theory categorical.Ultimately theory categoricity is about the relation of elementary equivalence V κ ≡ V λ rather than the relation of elementary substructure V κ ≺ V λ , and so it issensible to consider the alternative forest, by which κ λ for inaccessible cardinals,just in case κ ≤ λ and V κ ≡ V λ . In this case, a cardinal is first-order theorycategorical just in case it is a stump in the forest. And there is a correspondingforest order for second-order elementary equivalence V κ ≡ V λ , by which thesecond-order theory categorical cardinals are exactly the stumps of the -forest.2. Gaps in the categorical cardinals
Let us now prove that there are various kinds of gaps in the categorical cardinals.If there are sufficiently many inaccessible cardinals, then the categorical cardinals(of any type) do not form an initial segment of the inaccessible cardinals.
Observation 15.
If there are uncountably many inaccessible cardinals, then thereare some inaccessible cardinals that are neither first-order nor second-order senten-tially categorical.Proof.
This is clear, because there are only countably many sentences, in eitherfirst or second order, and we may associate each sententially categorical cardinalwith the sentence that characterizes it. This association is one-to-one betweenthe sententially categorical cardinals and a countable set. So there must be someinaccessible cardinals that are not sententially categorical. (cid:3)
Theorem 16.
If there are any inaccessible cardinals that are not sententially cat-egorical, then there are gaps in the sententially categorical cardinals.(1) If κ is the least inaccessible cardinal that is not first-order sententially cat-egorical, then κ , if it exists, is first-order sententially categorical. This is not an instance of the Math Tea argument, as discussed in [HLR13], since we are notreferring here to truth-in-the-universe V , but only to truth in set structures. JOEL DAVID HAMKINS AND HANS ROBIN SOLBERG (2) If κ is the least inaccessible cardinal that is not second-order sententiallycategorical, then κ , if it exists, is first-order sententially categorical.Proof. Suppose that κ is the least inaccessible cardinal that is not first-order sen-tentially categorical. By theorems 5 and 6, this is observable inside any larger V θ .In particular, if κ exists, then V κ can see that κ is not first-order sententiallycategorical. Therefore, V κ is categorically characterized by the sentence asserting,“there is a largest inaccessible cardinal and it is the only inaccessible cardinal thatis not first-order sententially categorical.”Essentially the same argument works if κ is the least inaccessible cardinal thatis not second-order sententially categorical. In this case, we still find a first-order sentential characterization of V κ , since it will be the only inaccessible model whichthinks there is a largest inaccessible cardinal which also is the only such cardinalthat is not second-order sententially categorical, and the point is that this is a first-order assertion in V κ since this model can define the truth predicate for V κ , whichis a mere set in V κ . (cid:3) If there are sufficiently many inaccessible cardinals, then the gaps become ar-bitrarily complicated and self-reflecting, since if all the gaps in the sententiallycategorical cardinals had the same simple nature, we could recognize the end ofthe sententially categorical cardinals—the top gap in a sense—and thereby find aninaccessible cardinal above it that would be characterized by seeing that there wassuch a new large gap in the sententially categorical cardinals.A similar analysis works for theory categoricity, even though we lack the ana-logues of theorems 5 and 6 for theory categoricity.
Theorem 17.
If there are at least c + many inaccessible cardinals, then there aregaps in the theory categorical cardinals. Indeed, there is a first-order sententiallycategorical cardinal that is larger than some inaccessible cardinal that is not cate-gorical either by sentences or theories, either first or second order.Proof. Assume that there are at least c + many inaccessible cardinals. Since thereare only continuum many possible theories, there must be an inaccessible cardinalamong the first c + many that is not second-order theory categorical. If there isactually a c + th inaccessible cardinal κ , then V κ is characterized by the first-orderassertion that the order type of the inaccessible cardinals is exactly c + . So we mayassume that there is no c + th inaccessible, and so the order type of the inaccessiblecardinals in V is exactly c + . In this case, there will be some inaccessible cardinal θ that can see that there is some cardinal that is not second-order theory categorical,since above the least such cardinal, we need then also only to get above the othercardinal with the same second-order theory as it to see that it is not. Let θ bethe least inaccessible cardinal such that V θ can see that there is some inaccessiblecardinal that is not second-order theory categorical. We can express this propertyas a first-order assertion in V θ as follows: V θ | =“There is an inaccessible cardinal that is not second-ordertheory categorical, but every smaller V δ for δ inaccessible thinksevery inaccessible cardinal is second-order theory categorical.”This is a first-order sentential characterization of θ , since no other inaccessiblecardinal can think that it also is the least to see this situation. So we have founda first-order sententially categorical cardinal above an inaccessible cardinal that isnot second-order theory categorical. (cid:3) ATEGORICAL LARGE CARDINALS 9 On the number of categorical cardinals
There are, as we have mentioned, at most countably many sententially categoricalcardinals, either first or second order, simply because there are only countably manysentences. And it is clear that if there are infinitely many inaccessible cardinals,then there are infinitely many sententially categorical cardinals, because there is thefirst one, the second, the third and so on, and these are each sententially categorical,each characterized by the statement that there are exactly n inaccessible cardinals.The first ω many inaccessible cardinals are all first-order sententially categorical inthis way.Similarly, we have mentioned that because there are at most continuum many dif-ferent theories, there will be at most continuum many theory categorical cardinals.But how many different theory categorical cardinals must there be? If there areuncountably many inaccessible cardinals, must there be uncountably many theorycategorical cardinals? Are each of the first ω many inaccessible cardinals theorycategorical? If there are at least continuum many inaccessible cardinals, must therebe continuum many theory-categorical cardinals? Surprisingly, the answers to allthese questions can be negative. Theorem 18.
It is relatively consistent with
ZFC that the inaccessible cardinalsform a proper class but there are only countably many theory categorical cardinals.
This theorem is an immediate consequence of the following more specific theorem.
Theorem 19.
Every model of
ZFC has a forcing extension, preserving all inacces-sible cardinals and creating no new ones, in which there are only countably manysecond-order theory categorical cardinals.Proof.
Let G ⊆ Coll( ω, c ) be V -generic for the forcing to collapse the continuumto ω , and consider the forcing extension V [ G ]. This forcing is small relative to anyinaccessible cardinal, and so they are all preserved in the extension (and forcingcan never create new inaccessible cardinals). Furthermore, because this forcing ishomogenenous, it follows that the Boolean value [[ ϕ (ˇ a ) ]] of any statement ϕ usingonly check-name parameters ˇ a from the ground model is either 0 or 1. And sincethe forcing is definable in V , assertions about the Boolean value are expressiblein the language of set theory. Indeed, the theory of V [ G ] κ is the same as theset of sentences ϕ for which the statement ‘[[ ϕ ]] = 1’ is in the theory of V κ inthe ground model. Therefore, the forcing does not create any new instances ofsentential or theory categoricity. Since the number of theory categorical cardinalswas at most continuum in the ground model, and this cardinal has been collapsedto ω , it follows that there are only countably many theory categorical cardinals inthe forcing extension V [ G ]. (cid:3) Conversely, there can also be continuum many theory categorical cardinals.
Theorem 20.
If there are at least continuum many inaccessible cardinals, thenthere is a forcing extension, preserving the continuum and having exactly the sameinaccessible cardinals as the ground model, in which the first continuum many in-accessible cardinals are all first-order theory categorical.Proof.
Assume that there are at least continuum many inaccessible cardinals in V .Let κ α be the α th inaccessible cardinal. Similarly, fix an enumeration h A α | α < c i of the subsets A α ⊆ ω . Since κ α is regular and much larger than c , none of the κ α for α < c are limits of inaccessible cardinals. Therefore, in every such V κ α , theinaccessible cardinals will be strictly bounded below κ α . We can therefore force soas to code A α into the GCH pattern at the first ω many regular cardinals above thesupremum of the inaccessible cardinals below κ α . Using an Easton product, we canperform all this forcing at once, making a forcing extension V [ G ] in which all thecoding is done, while neither creating nor destroying any inaccessible cardinals. In V [ G ], the set A α is definable in V κ α , since this model can define the supremum ofthe inaccessible cardinals below κ α and can observe the GCH pattern at the next ω many successor cardinals. In particular, the particular sentences saying that the n th successor above that supremum does have the GCH or does not are part of thetheory of V κ α . Since the particular patterns are all different, this means that thesecardinals all have different theories. And since they also each think that there arefewer than continuum many inaccessible cardinals, these theories will also differfrom those of any V κ above every κ α . And so all these cardinals are first-ordertheory categorical in V [ G ]. (cid:3) Let us also mention that if the GCH holds in the ground model, then the forcingalso preserves all cardinals and cofinalities.We can also arrange that the number of theory categorical cardinals is strictlybetween ℵ and the continuum. Theorem 21.
It is relatively consistent that the number of first-order theory cate-gorical cardinals is ω , even when the continuum is larger than this, and even whenthe inaccessible cardinals form a proper class.Proof. By theorem 20, we may begin with a model of set theory V in which thereare continuum many inaccessible cardinals that are first-order theory categorical.And we may suppose as well that V has abundant inaccessible cardinals, if welike. By forcing if necessary, we may also assume that the continuum hypothesisholds, since this forcing is small, it neither creates nor destroys any inaccessiblecardinals, and if the first-order theory categorical cardinals are categorical in theway described in theorem 20, by coding reals into the GCH pattern in the blocksbelow the first continuum (now ω ) many inaccessible cardinals, then these cardinalswill remain first-order theory categorical after forcing the CH. So we have a modelwith exactly ω many first-order theory categorical cardinals. We may now forceto V [ G ] by adding any number of Cohen reals via Add( ω, θ ), for some definablecardinal θ , such as θ = ℵ . This forcing is small, definable and homogeneous, andso it preserves all the inaccessible cardinals, preserves the coding making the first ω many inaccessible cardinals first-order theory categorical, and it creates no newinstances of categoricity. So V [ G ] has the desired features. (cid:3) The argument is extremely flexible, and we could have arranged to have exactly ℵ many first-order theory categorical cardinals, while the continuum is ℵ ω +5 , orwhatever, in diverse other possible combinations.4. Complete implication diagram
Every first-order sententially categorical cardinal is of course also first-order the-ory categorical, since the sentence is part of the theory, and similarly with second-order; and first-order categoricity immediately implies second-order categoricityfor sentences or theories, since first-order assertions count as (trivial) instances
ATEGORICAL LARGE CARDINALS 11 of second-order assertions. What we aim to do now is prove that beyond theseimmediate implications, there are no other provable implications.
Theorem 22.
Assuming the consistency of sufficiently many inaccessible cardinals,the complete provable implication diagram for the categoricity notions is as follows: κ is first-ordersententially categorical κ is first-ordertheory categorical κ is second-ordersententially categorical κ is second-ordertheory categorical None of these implications are reversible and no other implications are provable.
In fact, we shall prove the following more refined result, which shows that wecannot even get new implications by combining components of the diagram.
Theorem 23.
Implications between the various categoricity notions are those shownin the following Venn diagram, and if there are at least c + many inaccessible car-dinals, then every cell of the diagram is inhabited. second-ordertheorycategorical second-ordersententiallycategoricalfirst-ordertheorycategorical first-ordersententiallycategorical The positive implications of the diagram are in each case easy to prove, andthese correspond to the inclusions indicated in the Venn diagram. What remainsis to prove that all the various cells of the diagram are inhabited. To begin withthat, we have noted that if there are any inaccessible cardinals at all, then the leastinaccessible cardinal is first-order sententially categorical, and so the yellow regionat bottom is inhabited. Next, theorem 11 shows that the least Mahlo cardinal issecond-order sententially categorical but not first-order theory categorical, whichshows that the blue region at the right is inhabited. But in fact we can weaken thehypothesis necessary for this as follows.
Theorem 24.
If there is an inaccessible cardinal that is not first-order theorycategorical (for example, if there are at least c + many inaccessible cardinals), thenthere is an inaccessible cardinal that is second-order sententially categorical, but notfirst-order theory categorical. Proof.
If there are at least c + many inaccessible cardinals, then there must bean inaccessible cardinal that is not first-order theory categorical (nor even second-order theory categorical), since there are at most continuum many possible theories.If there is an inaccessible cardinal that is not first-order theory categorical, thenthere are inaccessible cardinals δ < κ , such that V δ and V κ have the same first-ordertheory. Let κ be least such that this situation arises. So certainly κ is not first-order theory categorical. Nevertheless, the cardinal κ is characterized by a certainproperty of the first-order truth predicate of V κ , which is second-order definable.With a single second-order sentence, we can assert that there is some inaccessiblecardinal δ < κ for which V δ has the same first-order theory as V κ , and that κ is leastfor which this situation occurs. So κ is second-order sententially categorical. (cid:3) Next, to show that the red region at the left is inhabited, consider the followingtheorem.
Theorem 25.
If there are uncountably many inaccessible cardinals, then there is afirst-order theory categorical cardinal that is not second-order sententially categor-ical.Proof.
This is an instance of theorem 9, but let us give the argument. Suppose thatthere are uncountably many inaccessible cardinals. Since there are only countablymany second-order sentences, there are also only countably many second-ordersententially categorical cardinals. In particular, there are only countably manysecond-order sententially categorical cardinals amongst the first ω many inacces-sible cardinals. And so there will be a first inaccessible cardinal κ that is largerthan all of those. It is part of the first-order theory of V κ that those other smallersententially categorical cardinals exist, that there are only countably many inacces-sible cardinals, and there are no inaccessible cardinals above all of the sententiallycategorical cardinals. So the theory of V κ characterizes κ , since no other cardinalcan have exactly the same collection of second-order sententially categorical cardi-nals and view itself as the next inaccessible cardinal after them. So κ is first-ordertheory categorical. Since also it is amongst the first ω many inaccessible cardinalsand strictly larger than all sententially categorical cardinals in that interval, it isnot itself sententially categorical. (cid:3) Corollary 26.
If the class of inaccessible cardinals has uncountable cofinality, thenthe supremum of the first-order theory categorical cardinals is strictly larger thanthe supremum of the second-order sententially categorical cardinals.Proof.
We can simply run the argument of theorem 25 by taking the first inaccessi-ble cardinal κ larger than all second-order sententially categorical cardinals (insteadof just the first ω many). Since there are only countably many such sententiallycategorical cardinals and the cofinality of the inaccessible cardinals is uncountable,there will be such a cardinal κ . And now the argument of theorem 25 shows that κ is first-order theory categorical, but strictly larger than all second-order sententiallycategorical cardinals. (cid:3) Let us now prove that if there are sufficiently many large cardinals, then thecentral dark purple region of the Venn diagram is inhabited. For a first argument,we may modify the Mahlo cardinal argument of theorem 11 by defining that acardinal κ is (first-order) definably Mahlo , if every closed unbounded set C ⊆ κ that is definable in V κ from parameters in V κ contains a regular cardinal. Another ATEGORICAL LARGE CARDINALS 13 way to say this is that κ exhibits the Mahloness property for club sets definablein V κ . Theorem 27.
The least inaccessible definably Mahlo cardinal κ is second-ordersententially categorical and first-order theory categorical but not first-order senten-tially categorical.Proof. Let κ be the least inaccessible first-order definably Mahlo cardinal. For eachnatural number n , there is by the reflection theorem a definable club of cardinals δ < κ with V δ ≺ Σ n V κ . Since κ is definably Mahlo, this implies that there is someinaccessible cardinal δ < κ with the same Σ n theory as V κ . So κ is not first-ordersententially categorical.Meanwhile, the fact that κ is definably Mahlo is a property of its first-ordertheory, because every instance of the definably Mahlo scheme is a first-order as-sertion. And it is also part of the theory of V κ that no smaller δ < κ is definablyMahlo. So κ is first-order theory categorical, since no other cardinal can have thiscombination.Finally, κ is second-order sententially categorical, since the assertion that thetheory of V κ contains that combination of statements is a single second-order as-sertion about V κ , namely, the assertion that, “in the unique truth predicate forfirst-order truth, every instance of the definably Mahlo scheme comes out true, aswell as the assertion that no smaller inaccessible cardinal is definably Mahlo.”So the least inaccessible definably Mahlo cardinal is second-order sententiallycategorical, first-order theory categorical, but not first-order sententially categori-cal. (cid:3) Meanwhile, as with theorem 24, we can also provide an example from a muchweaker large cardinal hypothesis.
Theorem 28.
If there is an inaccessible cardinal that is not first-order sententiallycategorical (for example, if there are uncountably many inaccessible cardinals), thenthe least such cardinal is first-order theory categorical and second-order sententiallycategorical, but not first-order sententially categorical.Proof.
Suppose that κ is the smallest inaccessible cardinal that is not first-ordersententially categorical. By theorem 6, this means κ is smallest with the propertythat every first-order sentence σ true in V κ is also true in some smaller inaccessible V δ . We claim that κ is first-order theory categorical, since it thinks that no smallerinaccessible cardinal has the property we just described, and yet every instanceof the defining property of κ is part of the first-order theory of V κ . So amonginaccessible cardinals, only V κ will have that combination in its theory. Finally, weclaim also that κ is second-order sententially categorical, because first-order truthin V κ is second-order definable, and so the property that every first-order sentence σ true in V κ reflects to some inaccessible V δ below is a single second-order assertionabout V κ . So we can characterize V κ by that property plus the assertion that nosmaller inaccessible cardinal has that property. (cid:3) Thus, if there are sufficiently many inaccessible cardinals, then the central darkpurple region of the Venn diagram is inhabited.Finally, let us show that the light purple region at the top of the Venn diagramalso is inhabited.
Theorem 29.
If there is an inaccessible cardinal that is not second-order theorycategorical (for example, if there are at least c + many inaccessible cardinals), thenthere is an inaccessible cardinal that is second-order theory categorical, but neithersecond-order sententially categorical nor first-order theory categorical.Proof. If there is an inaccessible cardinal that is not second-order theory categorical,then there are inaccessible cardinals δ < λ for which V δ and V λ have the samesecond-order theory. In particular, λ has the property (as in theorem 7) that forevery natural number n , there is a smaller inaccessible cardinal δ < λ for which V δ has the same Σ n theory as V λ .Let κ be the smallest inaccessible cardinal with that property, so that for anynatural number n , there is a smaller inaccessible cardinal δ for which V δ has thesame Σ n theory as V κ . It follows that any second-order sentence true in V κ is alsotrue in such a V δ , and so κ is not second-order sententially categorical. But also,we claim, κ is not first-order theory categorical, because the first-order theory of V κ is part of the Σ theory of V κ , as the truth predicate is definable at this level(indeed, first-order truth has complexity ∆ ). Namely, a first-order sentence ψ istrue in V κ if and only if there is a class T obeying the Tarskian truth recursion—soit is a truth predicate—according to which ψ is declared true. If V δ has the sameΣ theory as V κ , then they agree on the entire first-order theory, and so κ is notfirst-order theory categorical. So this cardinal κ is as desired. (cid:3) Thus, we have established theorem 23 and therefore also theorem 22.5.
Categoricity and forcing
Let us make a few observations about the non-absoluteness of categoricity be-tween a model of set theory and its forcing extensions. Any inaccessible cardinalcan be made into the least inaccessible cardinal of a forcing extension (see [Car17]),and this will be first-order sententially categorical. Can we do it, however, whilepreserving all inaccessible cardinals?
Question 30.
Can every inaccessible cardinal become first-order sententially cate-gorical in a forcing extension with the same inaccessible cardinals?
Yes, indeed, this is possible.
Theorem 31. If κ is an inaccessible cardinal, then there is a forcing extensionwith exactly the same inaccessible cardinals in which κ is first-order sententiallycategorical.Proof. We claim that every inaccessible cardinal can be characterized in a forcingextension as the least inaccessible cardinal that is a limit of failures of the GCH. Tosee this, suppose that κ an inaccessible cardinal. Let V [ C ] be the forcing extensionarising from the forcing that shoots a club set C ⊆ κ avoiding the inaccessiblecardinals. Conditions are closed bounded sets in κ with no inaccessible cardinals,ordered by end-extension. This forcing has δ -closed dense sets for every δ < κ , andtherefore it adds no new <κ -sequences over the ground model. It therefore preserves V κ and hence all inaccessible cardinals below κ ; and it preserves the inaccessibilityof κ itself; and being size κ , it also preserves all larger inaccessible cardinals. In V [ C ], therefore, we have ensured that κ is inaccessible, but not Mahlo, because wenow have a club in κ avoiding the inaccessible cardinals. In particular, κ is a limit ATEGORICAL LARGE CARDINALS 15 of elements of C , but no other inaccessible cardinal is a limit of elements of C . Let V [ C ][ G ] be the subsequent forcing extension that forces the GCH up to κ , exceptat the successors of cardinals in C , where the continuum is the double successor.This forcing preserves all inaccessible cardinals.In V [ C ][ G ], the cardinal κ is an inaccessible limit of cardinals at which the GCHfails, namely, the successors of the elements of C . But no smaller inaccessiblecardinal has that property, because C is bounded below every smaller inaccessiblecardinal. So κ is the least inaccessible cardinal that is a limit of failures of the GCH,and this property provides a first-order sentential categorical characterization of κ in the forcing extension V [ C ][ G ]. (cid:3) The theorem is an instance of the large cardinal killing-them-softly phenomenonpromulgated by Erin Carmody [Car17], who proved in a variety of cases that onecan often slightly reduce (as little reduced as possible) the large cardinal strengthof a large cardinal in a forcing extension. Theorem 31 achieves this, if we view non-categoricity as a largeness notion—we have killed the non-categoricity of κ whilepreserving its inaccessibility. But one might aspire to sharper (or we should say softer ) categoricity killing-them-softly results. For example, can we force any inac-cessible cardinal that is not second-order theory categorical to become second-ordertheory categorical, but neither first-order theory categorical nor second-order sen-tentially categorical? Can we force any inaccessible cardinal that is not first-ordertheory categorical to become first-order theory categorical, but not second-ordersententially categorical? Can we force any inaccessible cardinal that is not second-order sententially categorical to become second-order sententially categorical, butnot first-order theory categorical? These would be softer killings of non-categoricitythan what we achieved in theorem 31.6. Generalization to other cardinal notions
Let us prove a version of Zermelo’s quasi-categoricity theorem for the class ofworldly cardinals, where a cardinal κ is worldly if V κ | = ZFC, meaning here justthe first-order theory ZFC. Theorem 32.
The models of
ZFC+
Zermelo , that is, first-order ZFC with second-order Zermelo set theory, are precisely the models V κ for a worldly cardinal κ .Proof. If κ is worldly, then V κ is a model of ZFC which is correct about power sets,and so it satisfies the second-order separation axiom and hence the second-orderZermelo theory. Conversely, if M is a model of ZFC plus second-order Zermeloset theory, then because of the second-order separation axiom, it will be correctabout power sets, and so the model will be well-founded—we may assume withoutloss that it is transitive—and so the internal computation of the cumulative V α hierarchy will be correct. So M = V κ for some ordinal κ for which V κ | = ZFC,meaning that κ is worldly. (cid:3) The theory ZFC + Zermelo can be equivalently described as ZFC + Separation ,that is, with the second-order separation axiom, since the only part of the second-order Zermelo theory that adds something over first-order ZFC is the second-orderseparation axiom.Most of the arguments and analysis of this article will simply carry over tothe worldly cardinals. More generally, even without an explicit quasi-categoricity result, we may consider the notions of categoricity relative to any fixed class ofcardinals A . For example, we can define that a cardinal κ is sententially categoricalrelative to A , if there is a sentence such that V κ | = σ and κ is only element of A with that feature; and similarly with theory categoricity and so on. In this way,versions of our analysis will apply to the class of weakly compact cardinals, say, orthe measurable cardinals or the supercompact cardinals or what have you.7. Three philosophical issues
For the rest of the article, we should like to engage with several matters of amore philosophical nature.7.1.
Internal vs. meta-theoretic categoricity.
The first matter concerns thedifference between using an internal account of categoricity as opposed to an ex-ternal or metatheoretic account. To explain, notice that in the main definition ofthis article we defined what it means for a cardinal to be first-order sententiallycategorical in the same way that we would make any definition in mathematics,using a background set theory of ZFC, for example. This is an internal account, adefinition made inside the theory, and it can be sensibly applied inside any givenmodel of ZFC. Every model of ZFC comes after all with its own notions of what itmeans to be an ordinal or an inaccessible cardinal, and with its own notions of whatit means to be a sentence or for a sentence to be true in a structure such as V κ .The relevance of this is that some models of ZFC are nonstandard, perhaps even ω -nonstandard, which means they have nonstandard natural numbers and there-fore also nonstandard sentences and formulas in the language of set theory. Is itconceivable that such a model might think that a cardinal is sententially categori-cal, but the sentences witnessing this are all nonstandard? In this case, the modelwould think the cardinal is sententially categorical, but outside the model we wouldnot be able to write down any characterizing sentence.Indeed, let us prove that this can definitely happen. Suppose that our set-theoretic universe V has infinitely many inaccessible cardinals, and let us performthe ultrapower construction by a nonprincipal ultrafilter on ω . In the resultingmodel V ∗ , let n be a nonstandard natural number, and let κ be the n th inaccessiblecardinal in V ∗ (indexing from 0). This will be a first-order sententially categoricalcardinal in V ∗ , according to our main definition, because V ∗ κ will be characterizedby the assertion “there are exactly n inaccessible cardinals.” But because n isnonstandard, however, this is not a standard-length sentence—it is nonstandard.And furthermore, there is no way to make this assertion with a standard sentence,because then n would be definable in V ∗ , which it cannot be because it is notin the range of the ultrapower map. Similarly, there can be no standard finitesentence that characterizes V ∗ κ , since any such sentence would make κ definable in V ∗ , which would in turn make n definable, which as we have said is impossible. Sothis cardinal κ is categorical in V ∗ according to the internal notion of categoricityprovided by interpreting the main definition inside V ∗ , but it is not categoricalin the external model-theoretic sense of being characterized by an actual sentencein the language of set theory, even when we restrict the comparison only to otherinaccessible cardinals in V ∗ . More generally, any statement σ true in some n th V ∗ κ for a nonstandard n will also be true at infinitely many standard-index inaccessiblecardinals, by Lo´s’s theorem on ultrapowers. ATEGORICAL LARGE CARDINALS 17
Another easy way to see that internal and external notions of categoricity mustdiffer is by a simple cardinality argument. Namely, if V ∗ is an ultrapower of V by anonprincipal ultrafilter on ω , then standard arguments show that V ∗ will have con-tinuum many natural numbers (counted in V , externally to V ∗ ). If V has infinitelymany inaccessible cardinals, then V ∗ will have what it thinks is infinitely manyinaccessible cardinals, and this will include continuum many inaccessible cardinals(counted again in V ) amongst what it thinks are the first ω ∗ many. Although V ∗ thinks all these inaccessible cardinals are sententially categorical (since ZFC provesthat the first ω many inaccessible cardinals are sententially categorical), this is toomany for them all to be actually categorical there, since there are only countablymany sentences. So there is a definite difference between the internal notion andthe external meta-theoretic notion.Which is the right approach? In this article, we had intended to introduce andanalyze categorical cardinals as a large cardinal notion, and so it seemed correctto do so as an internal conception of ZFC, as one does with all the other largecardinal notions. This makes categoricity a set-theoretic notion that can be consid-ered sensibly inside any model of ZFC, and subject to questions about consistencyand forcing and so on. But if one were interested in categoricity exclusively asa foundational or philosophical matter, then it might seem reasonable that onewould want to insist on actually finite standard sentences to serve as the categor-ical characterizations. Ultimately, the choice between the two accounts hinges onhow one answers the question: Is a categoricity claim a metatheoretic claim or anobject-theoretic claim?We said that the two approaches are different, but in fact the external notion isstronger. If κ is inaccessible inside a model M and ( V κ ) M is characterized amongstits competitors in M by an actual sentence σ , that is, by a standard-finite sentence σ that we can write down in the metatheory, then the copy of this sentence inside M will serve as a categorical characterization of κ in M . So external categoricityimplies internal categoricity, once one has fixed the model.And yet, there is another sense in which the two approaches can be seen asessentially the same. Namely, one understanding of what it means to have aninterpretation of second-order logic is that one has essentially fixed a metatheoreticset-theoretic background. And if one were to treat this set-theoretic backgroundas a object-theoretic model of set theory, which is essentially to say, if one were totake that set-theoretic background seriously as a model of set theory, then whatwas previously the external conception of categoricity becomes the internal notionrelative to this newly derived model of set theory.The first author has described how this kind of move—transforming object theoryto metatheory or vice versa—is a natural outcome of the pluralist position in settheory:
The multiverse perspective ultimately provides what I view as an enlarge-ment of the theory/metatheory distinction. There are not merely twosides of this distinction, the object theory and the metatheory; rather,there is a vast hierarchy of metatheories. Every set-theoretic context, af-ter all, provides in effect a metatheoretic background for the models andtheories that exist in that context—a model theory for the models and The second author emphasizes that one might alternatively view second-order logic throughthe lens of plural quantification, Fregean concepts, or some other method conceptually distinctfrom set theory, whereas the first author views these still as nevertheless essentially set-theoretic. theories one finds there. Every model of set theory provides an interpre-tation of second-order logic, for example, using the sets and predicatesexisting there. Yet a given model of set theory M may itself be a modelinside a larger model of set theory N , and so what previously had beenthe absolute set-theoretic background, for the people living inside M ,becomes just one of the possible models of set theory, from the perspec-tive of the larger model N . Each metatheoretic context becomes justanother model at the higher level. In this way, we have theory, metathe-ory, metametatheory, and so on, a vast hierarchy of possible set-theoreticbackgrounds. [Ham21, p. 298] The deviation between the external, metatheoretic approach and the internal,object-theoretic approach to categoricity reveals a measure of nonabsoluteness forcategoricity—whether a structure is categorical or not depends on the set-theoreticbackground in which the second-order logic is interpreted. In this way, discussionsof categoricity become wrapped up with the debate on set-theoretic pluralism.7.2.
Categoricity as semantic completeness.
Next, we should like to discusscategoricity as a strong form of semantic completeness. A categorical theory com-pletely determines the structure in which it holds, and in this sense, the theoryalso completely determines the truths of that structure. If T is categorical, afterall, then for any assertion ϕ in that language, either T | = ϕ or T | = ¬ ϕ , preciselybecause either ϕ is true in the unique model of T or it isn’t. Dedekind arithmetic,for example, is complete in this sense for arithmetic assertions, and the axioms ofa complete ordered field are complete for assertions about the real numbers.Kreisel [Kre67] argued similarly with second-order set theory, using Zermelo’squasi-categoricity result to argue that ZFC settles the continuum hypothesis. Sincethe truth or falsity of the continuum hypothesis is revealed very low in the set-theoretic hierarchy, at the level of V ω +2 , and since all the models of ZFC have theform V κ for an inaccessible cardinal κ and these agree on V ω +2 , it follows that allthe models of ZFC give the same answer for the continuum hypothesis. In thissense, ZFC settles the continuum hypothesis. Daniel Isaacson [Isa11] also defendsthis view.Similar reasoning shows that ZFC is complete with respect to nearly the en-tirety of classical mathematics, which takes place at comparatively low levels of theset-theoretic hierarchy—mathematicians have argued that V ω +5 is sufficient, butindeed even V ω + ω or V ω would be good enough—the argument shows that ZFC is semantically complete with respect to any mathematical question that can beresolved inside any V α up to the first inaccessible cardinal.So it would seem to be great news—our fundamental theory ZFC determinesthe answer to essentially every mathematical question! Fantastic! Let’s get straightto work with this theory.But wait, you say that it isn’t working? We are told that the theory ZFC determines the answer to CH and all other classical mathematical questions, butthe disappointment comes when we seem unable to use this theory in any wayto figure out what the answers actually are. The reason is that we lack a sound,complete, and verifiable proof system for second-order logic, and so we cannotactually use the completeness of the theory ZFC in any mechanistic manner ofreasoning to determine the answer. When working with a second-order theorywhat often happens in practice is that one adopts as much of the second-order ATEGORICAL LARGE CARDINALS 19 theory as one can, by gathering together the set-existence principles one views assound. But to do so is to adopt in the metatheory what amounts to a first-order settheory such as ZFC. And since this theory does not settle the continuum hypothesisor even every arithmetic question—it must be incomplete—we are forced to giveup the semantic completeness claim.The completeness of a first-order theory T (specified by a computable list ofaxioms) leads necessarily to a computable decision procedure for the entire contentof the theory. Namely, given any question ϕ , we can search systematically fora proof T ⊢ ϕ or a refutation T ⊢ ¬ ϕ ; if the theory is complete, then we willeventually find one of these and thereby come to the answer of whether ϕ holdsin the theory or not. In second-order logic, however, we have no such completeproof system and we must remain basically at a loss. For this reason, the semanticcompleteness of our second-order set theories is not as useful as it might seem.The objection we have made so far to the completeness claim for the second-ordertheory is about our inability to use it, rather than an ontological point about whatthere is and what is true. So let us mount another more serious objection. Namely,we claim that one cannot deduce the definiteness of our mathematical structures onthe basis of categorical characterizations in second-order logic. What we claim isthat any legitimate metatheoretic aparatus, whether it is second-order logic, pluralquantifiers, Fregean concepts or what have you, faces a certain dichotomy—eitherit will be incomplete in the metatheoretic account it provides, like using a first-order commitment such as ZFC, or else it will be complete, but in a way that begsthe question concerning the definite nature of the metatheoretic ontology. This isquestion-begging because we cannot establish the definiteness of the object-theoryset concept by appealing to a presumed definiteness of the meta-theory set concept.To do so is merely to put off the definiteness objection from the object theory tothe metatheory, but without in any way answering that objection. If someone saysthat our concept of set is definite and complete because they have a categoricalaccount of it in second-order logic, then we would simply want to know why theirmetatheoretic concept of set (or of pluralities or Fregean concepts or what haveyou) is definite and complete.The first author argues similarly as follows that the semantic completeness ofthe second-order theory ZFC is illusory, for all that has happened is that we havepushed off the incompleteness into the metatheory. Critics view this [Kreisel’s argument on the determinateness of CH] assleight of hand, since second-order logic amounts to set theory itself, inthe metatheory. That is, if the interpretation of second-order logic is seenas inherently set-theoretic, with no higher claim to absolute meaning orinterpretation than set theory, then to fix an interpretation of second-order logic is precisely to fix a particular set-theoretic background inwhich to interpret second-order claims. And to say that the continuumhypothesis is determined by [the] second-order set theory is to say that,no matter which set-theoretic background we have, it either asserts thecontinuum hypothesis or it asserts the negation. I find this to be like say-ing that the exact time and location of one’s death is fully determinatebecause whatever the future brings, the actual death will occur at a par-ticular time and location. But is this a satisfactory proof that the futureis “determinate”? No, for we might regard the future as indeterminate,even while granting that ultimately something particular must happen.
Similarly, the proper set-theoretic commitments of our metatheory areopen for discussion, even if any complete specification will ultimatelyinclude either the continuum hypothesis or its negation. Since differentset-theoretic choices for our interpretation of second-order logic will causedifferent outcomes for the continuum hypothesis, the principle remainsin this sense indeterminate. [Ham21, p. 288]
We should like to emphasize that this objection applies just as much to the moreordinary categorical characterizations of mathematical structure. The fact that theDedekind axioms for the natural numbers determine a definite structure h N , S, i and therefore determine all the arithmetic truths is not actually helpful for us todiscover those truths. Ultimately, number theorists will find themselves adoptingwhat amounts to a first-order theory such as PA or ZFC in the metatheory, andthese remain incomplete for arithmetic truth. For analogous reasons, the categor-ical accounts of the structures V κ on which we have focussed in this article mayultimately be less fulfilling than one hoped.7.3. Categoricity, reflection and realism.
Finally, we should like to call atten-tion to a certain perplexing tension we observe between two fundamental values inmathematics—the contradictory natures of categoricity and set-theoretic reflection;the matter deserves philosophical attention.On the one hand, mathematicians almost universally seek categorical accountsof their fundamental mathematical structures, from Dedekind’s axiomatization ofarithmetic to the characterization of the real numbers as a complete ordered field.Categoricity is taken as a positive value and a key general goal in mathematicalpractice. At least part of the explanation for this is that the categorical charac-terizations of our structures seem to give us reason to regard these structures asdefinite. We know what we mean by the natural numbers, on this view, precisely be-cause we can categorically describe the natural number structure. Indeed, becauseall our fundamental mathematical structures admit of such categorical characteri-zations, we thereby have reason to think of them as definite and real, and in thisway categoricity seems to lead to mathematical realism. At the same time, cat-egoricity seems also to implement structuralism, because the categorical accountsof our fundamental structures invariably do so only up to isomorphism, and so toregard every structure that fulfills the characterization as perfectly satisfactory isprecisely to adopt the structuralist stance.On the other hand, set theorists vigorously defend principles of set-theoretic re-flection, asserting in various ways that every truth of the full set-theoretic universereflects down to the same truth made in a set-sized structure. Reflection is oftendescribed as expressing a core feature of the set-theoretic universe, and indeed theL´evy-Montague reflection theorem is equivalent over a weak theory to the replace-ment axiom of ZFC. Reflection ideas are used not only to justify the ZFC axiomsof set theory, but also the existence of large cardinals [Rei74, Mad88].The puzzling conflict we aim to highlight between categoricity and reflection isthat reflection is at heart an anti-categorical principle—it asserts explicitly that nostatement characterizes the set-theoretic universe V , that every statement true in V is also true in a much smaller structure. The philosophical question to sort outhere is how we can regard categoricity as vitally important in all our fundamentalmathematical structures and yet simultaneously assert as a core principle thatthe set-theoretic universe itself is not categorical. Ultimately, there must be a ATEGORICAL LARGE CARDINALS 21 fundamental mis-match between the extent of the reflection phenomenon and thecomplexity of any categorical characterization of the set-theoretic universe, sincethe kinds of statements and theories that reflect clearly cannot encompass thecategorical characterization itself, which by the fact of categoricity does not reflectto any smaller structure.The tension manifests also in attitudes toward large cardinals. Set theorists com-monly defend a larger-is-better approach to large cardinals, pointing to the highlystructured tower of consistency strength that they provide and the explanatoryconsequences down low of even the strongest large cardinal notions. Yet, as wehave mentioned, categoricity for large cardinals is a smallness notion rather thana largeness notion. Theorem 13 shows that the categorical cardinals are all belowthe least Σ -correct cardinal, and consequently below every strong cardinal, everysupercompact cardinal, every extendible cardinal, every totally otherworldly cardi-nal and more. If we look upon categoricity as desireable in a foundational theory,therefore, we would seem to be pushed toward the low end of the large cardinalhierarchy, to the smallest large cardinals, to the set-theoretic universes satisfyingcategorical theories. For example, the theory ZFC +“there are no inaccessible car-dinals” is categorical, as is ZFC +“there are exactly ω + 5 inaccessible cardinals.”But in the philosophy of set theory, one finds instead general arguments againstsuch theories in the foundations of set theory—they are viewed as restrictive andlimiting. Penelope Maddy [Mad98], for example, formulates the maximize princi-ple and uses it to explain set theorist’s resistance to the axiom of constructibilityand to large cardinal nonexistence axioms in general on the grounds that they arerestrictive. According to maximize , we should rather seek open-ended, nonlimitingaxiomatizations of set theory. Even critics of Maddy’s position, such as the firstauthor in [Ham14b], retain an open-ended conception of set theory and do not pushfor categoricity in the set-theoretic universe.Theorem 13 and the idea to which it leads, that categoricity is for small universesonly, seem to suggest that we might not want or expect a categorical account of thefull set-theoretic universe V . According to the toy model perspective (describedin [Ham12], [Ham14b]), one studies the various set models of set theory and howthey relate to one another partly in order to gain insight into the nature of thelarger actual set-theoretic universe in the context of its multiverse including all itsvarious forcing extensions. The toy models serve as a proxy for the real thing, whichremains inaccessible to us—we look into the toy models to learn what might be truein V or what we would desire to see in V . When we consider the toy models of V κ for inaccessible cardinals κ , we see that it is only the small large cardinals thatrealize a categorical theory, while the larger large cardinals do not, and so the toymodel perspective together with maximize seems to incline us to think that the fulluniverse V should not fulfill a categorical theory. On the toy model perspective,we might adopt non-categoricity as a goal.The philosophical problem here is to explain this transition in attitude towardcategoricity. Why do we take categoricity as a fundamental value for smallishmathematical structures such as the natural numbers, the real numbers and so on,but not for the set-theoretic universe as a whole?Let us try to offer a solution by describing a way out of the impasse, even thoughultimately we shall take a different lesson from this analysis. What we claim is thatif one takes second-order logic to have a fixed meaning, one where the metatheoretic concept of set obeys something at least like ZFC, then the set-theoretic universe V does in fact have a categorical characterization in second-order logic. The existenceof this characterization ultimately places limits on the extent of reflection that ispossible for the set-theoretic universe to exhibit.Specifically, we claim, the class structure of the set-theoretic universe h V, ∈i ischaracterized up to isomorphism in second-order logic as the unique well-foundedextensional set-like relation realizing every subset of its domain. In more detail:(1) The membership relation ∈ is extensional. x = y ↔ ∀ z ( z ∈ x ↔ z ∈ y )(2) The membership relation ∈ is well founded. ∀ A (cid:2) ∃ a A ( a ) → ∃ a (cid:0) A ( a ) ∧ ∀ x ( x ∈ a → ¬ A ( x )) (cid:1)(cid:3) (3) The membership relation ∈ is set-like. ∀ a ∃ A ∀ x (cid:0) x ∈ a ↔ A ( x ) (cid:1) (4) Every subset of V is realized. ∀ A ∃ a ∀ x (cid:0) x ∈ a ↔ A ( x ) (cid:1) The lower-case quantifers ∀ x quantify over the objects x of the domain V , whilethe upper-case quantifiers ∀ A are interpreted in second-order logic, ranging overall subsets of V . Note that V itself will be a proper class in the metatheory, nota set, and so we emphasize that ∀ A means for all sub sets A ⊆ V , which will notinclude the proper class subclasses of V , such as V itself or the class of ordinals of V . So this quantifier differs from the class quantifier used in G¨odel-Bernays andKelley-Morse set theory. Axiom (2) is the assertion that every nonempty subset A ⊆ V has an ∈ -minimal element. Axiom (3) asserts that the ∈ -members of any setin V form a set—that is, a set in the metatheory, using the second-order concept ofset. Axiom (4) asserts conversely that for every subset A ⊆ V there is an object a in V whose ∈ -elements are the same the elements of A . In this way, axioms (3) and(4) assert a kind of correspondence between the object theory and the metatheoryas to what the sets are. Note also that axiom (4) makes a stronger claim than thesecond-order separation axiom Separation , which asserts merely that every subsetof a set already in V is in V , whereas our axiom asserts that every subset of V isin V .The characterization is entirely about the interplay of the metatheoretic andobject-theoretic concepts of set, and it therefore relies critically on our having al-ready fixed an interpretation of second-order logic. At bottom what the axiomati-zation asserts—perhaps disappointingly—is that the set-theoretic structure h V, ∈i implements up to isomorphism exactly the wellfounded cumulative set hierarchyfrom the metatheory into the object theory. It copies the metatheory to the objecttheory.To begin the proof of categoricity, let us observe how the process of closing underall subsets—specified by axiom (4)—proceeds via the cumulative hierarchy. Theaxioms successively force certain kinds of objects into V in a transfinite recursiveprocess. Namely, we begin with nothing V = ∅ , and at successor stages V α +1 hasobjects realizing every possible subset of V α ; so it is in effect (isomorphic to) the full,actual power set of V α as provided by the metatheoretic set concept interpretingsecond-order logic. At limit stages λ , we gather together everything we’ve added ATEGORICAL LARGE CARDINALS 23 so far V λ = S α<λ V α . The point is that if this is a set, then the closure processof axiom (4) asserts that V λ itself must be realized by an element of V , and sothe closure process will continue to V λ +1 and so on. Ultimately, we build the V α hierarchy through all the ordinals of the metatheory and the resulting universeis V = S α V α . This structure fulfills axioms (1) and (2) by construction, sincewe assume the metatheoretic set concept is wellfounded and extensional; it fulfillsaxiom (3) because we only ever added objects were sets in the metatheory; and itfulfills axiom (4) since every subset will be bounded in rank, since we presume thatthe metatheoretic set concept fulfills the replacement axiom.To complete the proof of categoricity is now a simple induction on rank. Namely,if we have two structures (cid:10) V, ∈ V (cid:11) and (cid:10) W, ∈ W (cid:11) satisfying axioms (1), (2), (3) and(4), then we can construct an isomorphism of the corresponding levels V α ∼ = W α ofthe cumulative hierarchy. Since the membership relations are both well founded,the two conceptions of ordinals and their stages in the cumulative hierarchy willboth conform with the metatheoretic conception of ordinal. Both hierarchies beginwith nothing, and if we have an isomorphism at level α , then it extends uniquelyto an isomorphism at level α + 1, since any subset of V α can be transferred by ourpartial isomorphism to a subset of W α and vice versa. And the partial isomorphismsunion to an isomorphism at the limit stages. It cannot be that one structure stopsgrowing at a stage while the other continues, since the ordinal heights of the twostages are isomorphic and so one of them is a set in the metatheory if and only if theother also is. Finally, this cumulative recursive process must ultimately exhaust theobjects of V and W , since if there is an object a in V outside the V α hierarchy, thenits members form a set by axiom (3), and its members-of-members and so on, butthis iterated collection (which forms a set in the metatheory since we assume ZFCin the metatheory) cannot have an ∈ V -minimal element b outside the V α hierarchy,contrary to axiom (2), since any such object b would have all its ∈ V -members insidethe V α hierarchy, which would place b into the hierarchy at the supremum of thosestages. Thus, the cumulative V α and W α hierarchies exhaust all of V and W andwe therefore have an isomorphism of (cid:10) V, ∈ V (cid:11) with (cid:10) W, ∈ W (cid:11) . So this is a categoricalcharacterization. (This categoricity argument shares an essential similarity to thecentral argument of [Mar01], which can be read essentially as a categoricity thesis.)The key difference between our categorical account and the second-order set the-ory ZFC is that ZFC has only the second-order separation axiom rather than ouraxiom (4). The second-order separation axiom is insufficient to force the cumulativehierarchy to keep growing past inaccessible cardinals, since V κ for κ inaccessible sat-isfies the second-order separation axiom. But such a structure V κ does not satisfyour axiom (4), since κ and V κ are sets in the metatheory, and so must be added aselements, causing the hierarchy to keep growing. This key difference in the axiomsis why Zermelo achieves only a quasi-categoricity result, making stops at everyinaccessible cardinal, whereas we are able to achieve full categoricity, proceedingonward and upward to the full set-theoretic universe.Note also the difference in kind between the categorical account we have pro-vided above for the set-theoretic universe V and the notion of categoricity used forcategorical cardinals in the main definition. When defining the categorical cardi-nals, although we had used a second-order theory for the structure V κ in question,ultimately this amounts to a first-order notion in the set-theoretic universe V inwhich the definition is considered, since we are in effect quantifying over V κ +1 , which is a set in V . The categorical account of V , in contrast, is not first-order in V , but second-order over V . For this reason, the notion of categoricity provided inthe categorical account of V is not subject to the conclusion of theorem 13.Does this categoricity proposal resolve the tension between categoricity and re-flection? By providing a categorical account of the set-theoretic universe in second-order logic, it seems both to place categoricity as the more primary notion, and alsoto identify limitations on the possible extent of reflection. The reflection principlecannot rise fully to second-order logic, since one of the truths of the set-theoreticuniverse is that it is not a set, and this is a truth that cannot reflect to any actualset.But is this categorical account of the set-theoretic universe satisfactory? Does itenable us to secure a definite meaning for the universe of sets and definite accountfor which sets there are? No, not really. Let us criticise it. As we have mentioned,the characterization proceeds essentially by copying the metatheoretic concept of setinto the object-theoretic account. The categorical account is therefore only sensiblewhen we have already fixed a meaning of second-order logic, when we have in effectfixed a set concept in the metatheory. And if we had used a different metatheoreticconcept of set, for example, if a form of set-theoretic pluralism were the case, thenthe categorical characterization undertaken with that other concept of set wouldgive rise to that concept of set as the unique set-theoretic realm. For this reason, thecategorical characterization by itself is without power to refute pluralism or to tell uswhich sets there really are. We cannot establish the definiteness of our set concepton the basis of the categoricity result without presuming the definiteness of the setconcept arising from the interpretation of second-order logic in the metatheory. Toattempt to do so would be circular reasoning that merely pushes off the problemfrom the object theory to the metatheory.The problem is fundamentally similar to the objection we had raised earlier toKreisel’s observation that the continuum hypothesis is settled in second-order settheory. We objected that the argument is circular, because Kreisel’s observationshows merely that the CH is settled, provided that one has already fixed a completeconcept of set to be used in the metatheoretic interpretation of second-order logic.But if there were a choice of such metatheoretic set concepts, then it wouldn’tnecessarily be settled the same way by all of them. This is the same circularitythat seems to prevent us from using the categorical account of the set-theoreticuniverse to come to an understanding of which sets there are.The main lesson we propose to take from the categorical characterization of theset-theoretic universe h V, ∈i is that the obvious circularity and unacceptability of ithelp to show how the other second-order characterizations that we have in mathe-matics are similarly inadequate for establishing definiteness. That is, the circularityobjection applies just as much to Dedekind’s axiomatization of arithmetic and tothe characterization of the real numbers as the unique complete ordered field. Thefirst author explains it like this: Some philosophers object that we cannot identify or secure the definite-ness of our fundamental mathematical structures by means of second-order categoricity characterizations. Rather, we only do so relative to aset-theoretic background, and these backgrounds are not absolute. Theproposal is that we know what we mean by the structure of the nat-ural numbers—it is a definite structure—because Dedekind arithmeticcharacterizes this structure uniquely up to isomorphism. The objection
ATEGORICAL LARGE CARDINALS 25 is that Dedekind arithmetic relies fundamentally on the concept of ar-bitrary collections of numbers, a concept that is itself less secure anddefinite than the natural-number concept with which we are concerned.If we had doubts about the definiteness of the natural numbers, howcan we assuaged by an argument relying on the comparatively indefi-nite concept of “arbitrary collection”? Which collections are there? Thecategoricity argument takes place in a set-theoretic realm, whose owndefinite nature would need to be established in order to use it to estab-lish definiteness for the natural numbers. [Ham21, p. 32]
Because the second-order accounts presume a set-theoretic realm of second-orderlogic, the definiteness of mathematical structure that they provide is only as definiteas the set-theoretic account of the second-order logic itself.
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Professor of Logic, University of Oxford & Sir Peter StrawsonFellow, University College, Oxford
E-mail address : [email protected] URL : http://jdh.hamkins.org (Hans Robin Solberg) Doctoral student, Philosophy, University of Oxford
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