Every worldly cardinal admits a Gödel-Bernays structure

My Oxford student Emma Palmer and I have been thinking about worldly cardinals and Gödel-Bernays GBC set theory, and we recently came to a new realization.

Namely, what I realized is that every worldly cardinal κ admits a Gödel-Bernays structure, including the axiom of global choice. That is, if κ is worldly, then there is a family X of sets so that Vκ,,X is a model of Gödel-Bernays set theory GBC including global choice.

For background, it may be helpful to recall Zermelo’s famous 1930 quasi-categoricity result, showing that the inaccessible cardinals are precisely the cardinals κ for which Vκ is a model of second-order set theory ZFC2.

If one seeks only the first-order ZFC set theory in Vκ, however, then this is what it means to say that κ is a worldly cardinal, a strictly weaker notion. That is, κ is worldly if and only if VκZFC. Every inaccessible cardinal is worldly, by Zermelo’s result. But more, every inaccessible is a limit of worldly cardinals, and so there are many worldly cardinals that are not inaccessible. The least worldly cardinal, for example, has cofinality ω. Indeed, the next worldly cardinal above any ordinal has cofinality ω.

Meanwhile, to improve slightly on Zermelo, we can observe that if κ is inaccessible, then Vκ is a model of Kelley-Morse set theory when equipped with the full second-order complement of classes. That is, Vκ,,Vκ+1 is a model of KM.

This is definitely not true when κ is merely worldly and not inaccessible, however, for in this case Vκ,,Vκ+1 is never a model of KM nor even GBC when κ is singular. The reason is that the singularity of κ would be revealed by a short cofinal sequence, which would be available in the full power set Vκ)+1=P(Vκ), and this would violate replacement.

So the question is:

Question. If κ is worldly, then can we equip Vκ with a suitable family X of classes so that Vκ,,X is a model of GBC?

The answer is Yes!

What I claim is that for every worldly cardinal κ, there is a definably generic well order of Vκ, so that the subsets definable in Vκ,, make a model of GBC.

To see this, consider the class forcing notion P for adding a global well order , as Vκ sees it. Conditions are well orders of some Vα for some α<κ, ordered by end-extension, so that lower rank sets always preceed higher rank sets in the resulting order.

I shall prove that there is a well-order that is generic with respect to dense sets definable in V,.

For this, let us consider first the case where the worldly cardinal κ has countable cofinality. In this case, we can find an increasing sequence κn cofinal in κ, such that each κn is Σn-correct in Vκ, meaning VκnΣnVκ.

In this case, we can build a definably generic filter G for P in a sequence of stages. At stage n, we can find a well order up to κn that meets all Σn definable dense classes using parameters less than Vκn. The reason is that for any such definable dense set, we can meet it below κn using the Σn-correctness of κn, and so by considering various parameters in turn, we can altogether handle all parameters below Vκn using Σn definitions. That is, the nth stage is itself an iteration of length κn, but it will meet all Σn definable dense sets using parameters in Vκn.

Next, we observe that the ultimate well-order of Vκ that arises from this construction after all stages is fully definably generic, since any definition with arbitrary parameters in Vκ is a Σn definition with parameters in Vκn for some large enough n, and so we get a definably generic well order . Therefore, the usual forcing argument shows that we get GBC in the resulting model Vκ,,Def(Vκ), as desired.

The remaining case occurs when kappa has uncountable cofinality. In this case, there is a club set Cκ of ordinals γC with VγVκ. (We can just intersect the clubs Cn of the Σn-correct cardinals.) Now, we build a well-order of Vκ that is definably generic for every Vγ for γC. At limits, this is free, since every definable dense set in V_lambda with parameters below is also definable in some earlier Vγ. So it just reduces to the successor case, which we can get by the arguments above (or by induction). The next correct cardinal γ above any ordinal has countable cofinality, since if one considers the next Σ1-correct cardinal, the next Σ2-correct cardinal, and so on, the limit will be fully correct and cofinality ω.

The conclusion is that every worldly cardinal κ admits a definably generic global well-order on Vκ and therefore also admits a Gödel-Bernays GBC set theory structure Vκ,,X, including the axiom of global choice.

The argument relativizes to any particular amenable class AVκ. Namely, if Vκ,,A is a model of ZFC(A), then there is a definably generic well order of Vκ such that Vκ,,A, is a model of ZFC(A,), and so by taking the classes definable from A and , we get a GBC structure X including both A and .

This latter observation will be put to good use in connection with Emma’s work on the Tarski’s revenge axiom, in regard to finding the optimal consistency strength for one of the principles.

The otherwordly cardinals

I’d like to introduce and discuss the otherworldly cardinals, a large cardinal notion that frequently arises in set-theoretic analysis, but which until now doesn’t seem yet to have been given its own special name. So let us do so here.

I was put on to the topic by Jason Chen, a PhD student at UC Irvine working with Toby Meadows, who brought up the topic recently on Twitter:

In response, I had suggested the otherworldly terminology, a play on the fact that the two cardinals will both be worldly, and so we have in essence two closely related worlds, looking alike. We discussed the best way to implement the terminology and its extensions. The main idea is the following:

Main Definition. An ordinal κ is otherworldly if VκVλ for some ordinal λ>κ. In this case, we say that κ is otherworldly to λ.

It is an interesting exercise to see that every otherworldly cardinal κ is in fact also worldly, which means VκZFC, and from this it follows that κ is a strong limit cardinal and indeed a -fixed point and even a -hyperfixed point and more.

Theorem. Every otherworldly cardinal is also worldly.

Proof. Suppose that κ is otherworldly, so that VκVλ for some ordinal λ>κ. It follows that κ must in fact be a cardinal, since otherwise it would be the order type of a relation on a set in Vκ, which would be isomorphic to an ordinal in Vλ but not in Vκ. And since ω is not otherworldly, we see that κ must be an uncountable cardinal. Since Vκ is transitive, we get now easily that Vκ satisfies extensionality, regularity, union, pairing, power set, separation and infinity. The only axiom remaining is replacement. If φ(a,b) obeys a functional relation in Vκ for all aA, where AVκ, then Vλ agrees with that, and also sees that the range is contained in Vκ, which is a set in Vλ. So Vκ agrees that the range is a set. So Vκ fulfills the replacement axiom. ◻

Corollary. A cardinal is otherworldly if and only if it is fully correct in a worldly cardinal.

Proof. Once you know that otherworldly cardinals are worldly, this amounts to a restatement of the definition. If VκVλ, then λ is worldly, and Vκ is correct in Vλ. ◻

Let me prove next that whenever you have an otherworldly cardinal, then you will also have a lot of worldly cardinals, not just these two.

Theorem. Every otherworldly cardinal κ is a limit of worldly cardinals. What is more, every otherworldly cardinal is a limit of worldly cardinals having exactly the same first-order theory as Vκ, and indeed, the same α-order theory for any particular α<κ.

Proof. If VκVλ, then Vλ can see that κ is worldly and has the theory T that it does. So Vλ thinks, about T, that there is a cardinal whose rank initial segment has theory T. Thus, Vκ also thinks this. And we can find arbitrarily large δ up to κ such that Vδ has this same theory. This argument works whether one uses the first-order theory, or the second-order theory or indeed the α-order theory for any α<κ. ◻

Theorem. If κ is otherworldly, then for every ordinal α<κ and natural number n, there is a cardinal δ<κ with VδΣnVκ and the α-order theory of Vδ is the same as Vκ.

Proof. One can do the same as above, since Vλ can see that Vκ has the α-order theory that it does, while also agreeing on Σn truth with Vλ, so Vκ will agree that there should be such a cardinal δ<κ. ◻

Definition. We say that a cardinal is totally otherworldly, if it is otherworldly to arbitrarily large ordinals. It is otherworldly beyond θ, if it is otherworldly to some ordinal larger than θ. It is otherworldly up to δ, if it is otherworldly to ordinals cofinal in δ.

Theorem. Every inaccessible cardinal δ is a limit of otherworldly cardinals that are each otherworldly up to and to δ.

Proof. If δ is inaccessible, then a simple Löwenheim-Skolem construction shows that Vκ is the union of a continuous elementary chain Vκ0Vκ1VκαVκ Each of the cardinals κα arising on this chain is otherworldly up to and to δ. ◻

Theorem. Every totally otherworldly cardinal is Σ2 correct, meaning VκΣ2V. Consequently, every totally otherworldly cardinal is larger than the least measurable cardinal, if it exists, and larger than the least superstrong cardinal, if it exists, and larger than the least huge cardinal, if it exists.

Proof. Every Σ2 assertion is locally verifiable in the Vα hierarchy, in that it is equivalent to an assertion of the form ηVηψ (for more information, see my post about Local properties in set theory). Thus, every true Σ2 assertion is revealed inside any sufficiently large Vλ, and so if VκVλ for arbitrarily large λ, then Vκ will agree on those truths. ◻

I was a little confused at first about how two totally otherwordly cardinals interact, but now everything is clear with this next result. (Thanks to Hanul Jeon for his helpful comment below.)

Theorem. If κ<δ are both totally otherworldly, then κ is otherworldly up to δ, and hence totally otherworldly in Vδ.

Proof. Since δ is totally otherworldly, it is Σ2 correct. Since for every α<δ the cardinal κ is otherworldly beyond α, meaning VκVλ for some λ>α, then since this is a Σ2 feature of κ, it must already be true inside Vδ. So such a λ can be found below δ, and so κ is otherworldly up to δ. ◻

Theorem. If κ is totally otherworldly, then κ is a limit of otherworldly cardinals, and indeed, a limit of otherworldly cardinals having the same theory as Vκ.

Proof. Assume κ is totally otherworldly, let T be the theory of Vκ, and consider any α<κ. Since there is an otherworldly cardinal above α with theory T, namely κ, and because this is a Σ2 fact about α and T, it follows that there must be such a cardinal above α inside Vκ. So κ is a limit of otherworldly cardinals with the same theory as Vκ. ◻

The results above show that the consistency strength of the hypotheses are ordered as follows, with strict increases in consistency strength as you go up (assuming consistency):

  • ZFC + there is an inaccessible cardinal
  • ZFC + there is a proper class of totally otherworldly cardinals
  • ZFC + there is a totally otherworldly cardinal
  • ZFC + there is a proper class of otherworldly cardinals
  • ZFC + there is an otherworldly cardinal
  • ZFC + there is a proper class of worldly cardinals
  • ZFC + there is a worldly cardinal
  • ZFC + there is a transitive model of ZFC
  • ZFC + Con(ZFC)
  • ZFC

We might consider the natural strengthenings of otherworldliness, where one wants VκVλ where λ is itself otherworldly. That is, κ is the beginning of an elementary chain of three models, not just two. This is different from having merely that VκVλ and VκVη for some η>λ, because perhaps Vλ is not elementary in Vη, even though Vκ is. Extending successively is a more demanding requirement.

One then naturally wants longer and longer chains, and ultimately we find ourselves considering various notions of rank in the rank elementary forest, which is the relation κλVκVλ. The otherworldly cardinals are simply the non-maximal nodes in this order, while it will be interesting to consider the nodes that can be extended to longer elementary chains.

Worldly cardinals are not always downwards absolute

 

UniversumI recently came to realize that worldly cardinals are not necessarily downward absolute to transitive inner models. That is, it can happen that a cardinal κ is worldly in the full set-theoretic universe V, but not in some transitive inner model W, even when W is itself a model of ZFC. The observation came out of some conversations I had with Alexander Block from Hamburg during his recent research visit to New York. Let me explain the argument.

A cardinal κ is inaccessible, if it is an uncountable regular strong limit cardinal. The structure Vκ, consisting of the rank-initial segment of the set-theoretic universe up to κ, which can be generated from the empty set by applying the power set operation κ many times, has many nice features. In particular, it is transitive model of ZFC. The models Vκ for κ inaccessible are precisely the uncountable Grothendieck universes used in category theory.

Although the inaccessible cardinals are often viewed as the entryway to the large cardinal hierarchy, there is a useful large cardinal concept weaker than inaccessibility. Namely, a cardinal κ is worldly, if Vκ is a model of ZFC. Every inaccessible cardinal is worldly, and in fact a limit of worldly cardinals, because if κ is inaccessible, then there is an elementary chain of cardinals λ<κ with VλVκ, and all such λ are worldly. The regular worldly cardinals are precisely the inaccessible cardinals, but the least worldly cardinal is always singular of cofinality ω.

The worldly cardinals can be seen as a kind of poor-man’s inaccessible cardinal, in that worldliness often suffices in place of inaccessibility in many arguments, and this sometimes allows one to weaken a large cardinal hypothesis. But meanwhile, they do have some significant strengths. For example, if κ is worldly, then Vκ satisfies the principle that every set is an element of a transitive model of ZFC.

It is easy to see that inaccessibility is downward absolute, in the sense that if κ is inaccessible in the full set-theoretic universe V and WV is a transitive inner model of ZFC, then κ is also inaccessible in W. The reason is that κ cannot be singular in W, since any short cofinal sequence in W would still exist in V; and it cannot fail to be a strong limit there, since if some δ<κ had κ-many distinct subsets in W, then this injection would still exist in V. So inaccessibility is downward absolute.

The various degrees of hyper-inaccessibility are also downwards absolute to inner models, so that if κ is an inaccessible limit of inaccessible limits of inaccessible cardinals, for example, then this is also true in any inner model. This downward absoluteness extends all the way through the hyperinaccessibility hierarchy and up to the Mahlo cardinals and beyond. A cardinal κ is Mahlo, if it is a strong limit and the regular cardinals below κ form a stationary set. We have observed that being regular is downward absolute, and it is easy to see that every stationary set S is stationary in every inner model, since otherwise there would be a club set C disjoint from S in the inner model, and this club would still be a club in V. Similarly, the various levels of hyper-Mahloness are also downward absolute.

So these smallish large cardinals are generally downward absolute. How about the worldly cardinals? Well, we can prove first off that worldliness is downward absolute to the constructible universe L.

Observation. If κ is worldly, then it is worldly in L.

Proof. If κ is worldly, then VκZFC. This implies that κ is a beth-fixed point. The L of Vκ, which is a model of ZFC, is precisely Lκ, which is also the Vκ of L, since κ must also be a beth-fixed point in L. So κ is worldly in L. QED

But meanwhile, in the general case, worldliness is not downward absolute.

Theorem. Worldliness is not necessarily downward absolute to all inner models. It is relatively consistent with ZFC that there is a worldly cardinal κ and an inner model WV, such that κ is not worldly in W.

Proof. Suppose that κ is a singular worldly cardinal in V. And by forcing if necessary, let us assume the GCH holds in V. Let V[G] be the forcing extension where we perform the Easton product forcing P, so as to force a violation of the GCH at every regular cardinal γ. So the stage γ forcing is Qγ=Add(γ,γ++).

First, I shall prove that κ is worldly in the forcing extension V[G]. Since every set of rank less than κ is added by some stage less than κ, it follows that VκV[G] is precisely γ<κVκ[Gγ]. Most of the ZFC axioms hold easily in VκV[G]; the only difficult case is the collection axiom. And for this, by considering the ranks of witnesses, it suffices to show for every γ<κ that every function f:γκ that is definable from parameters in VκV[G] is bounded. Suppose we have such a function, defined by f(α)=β just in case φ(α,β,p) holds in VκV[G]. Let δ<κ be larger than the rank of p. Now consider Vκ[Gδ], which is a set-forcing extension of Vκ and therefore a model of ZFC. The fail forcing, from stage δ up to κ, is homogeneous in this model. And therefore we know that f(α)=β just in case 1 forces φ(αˇ,βˇ,pˇ), since these arguments are all in the ground model Vκ[Gδ]. So the function is already definable in Vκ[Gδ]. Because this is a model of ZFC, the function f is bounded below κ. So we get the collection axiom in VκV[G] and hence all of ZFC there, and so κ is worldly in V[G].

For any Aκ, let PA be the restriction of the Easton product forcing to include only the stages in A, and let GA be the corresponding generic filter. The full forcing P factors as PA×PκA, and so V[GA]V[G] is a transitive inner model of ZFC.

But if we pick Aκ to be a short cofinal set in κ, which is possible because κ is singular, then κ will not be worldly in the inner model V[GA], since in Vκ[GA] we will be able to identify that sequence as the places where the GCH fails. So κ is not worldly in V[GA].

In summary, κ was worldly in V[G], but not in the transitive inner model W=V[GA], and so worldliness is not downward absolute. QED