2.1 Natural theories, large cardinals, and worlds

As stated in bullet point (3.), one of the purposes of Steel’s multiverse conception is precisely that of ‘representing’ all ‘natural’ theories extending $\mathsf {ZFC}$ within a unified axiomatic framework, without explicitly and directly having to deal with ‘universes’.

However, in order to show that the axioms are not semantically vacuous, there will have to be objects fixing their interpretation, and such objects are, on the one hand, sets and, on the other, worlds.Footnote ¹⁷ But notice the peculiar use of worlds in $\mathsf {MV}$ : the latter are introduced, and described, to account for the representability of different theories.Footnote ¹⁸ The strategy is, crucially, facilitated by both proof-theoretic and model-theoretic facts concerning LCs, in particular, by the following conjecture:

The main upshot of Conjecture 2.1 is that, since all theories $\mathsf {ZFC}$ +LCA considered so far are arranged in a well-ordered scale of consistency strengths,Footnote ²⁰ then also all natural theories are. Thus, the ‘invisible web’ binding together all natural theories is, finally, cast into sharp relief by the proof-theoretic connections among all LCAs.

The next step is to exploit the fact that, using LCs, one is able to construct models, in particular set-forcing extensions and inner models, which satisfy any natural theory. Using this fact, and Conjecture 2.1, Steel is, therefore, able to formulate, in full, the main meta-theoretic constraint presiding over his ‘multiverse’ (bullet point (2.)):

Meta-theoretic constraint. $\mathsf {MV}$ ’s worlds are just those models of $\mathsf {ZFC}$ +LCA which are needed to incorporate all natural theories extending $\mathsf {ZFC}$ .

Finally, since inner models may well be defined inside forcing extensions, what we just need to characterise worlds are LCs and forcing: the $\mathsf {MV}$ axioms reflect this state of affairs, by quantifying over set-forcing extensions (and their grounds) with LCs.

We will deal with the models of $\mathsf {MV}$ later on, but, first, we wish to express several concerns about Steel’s use of, and primary reliance upon, LCs, and about the notion of ‘naturalness’.

The first qualm refers to Conjecture 2.1, that all ‘natural’ extensions of $\mathsf {ZFC}$ are equiconsistent with either $\mathsf {ZFC}$ or some LCA. While no evidence against it has been found thus far, the conjecture is far from being settled, and things are compounded by the fact that we lack a general definition of ‘large cardinal’. Now, it would certainly be unfair to have the unsharpness of the notion count against Steel’s project; however, the absence of ultimate evidence that any undecidable statement will turn out to be equiconsistent with a LCA is a fact, which casts some doubts on the tenability of the conjecture.

A second, possibly more malignant, issue is that the notion of ‘natural theory’ is unclear. What we are told by Steel, at the very outset, is that a set-theoretic statement does qualify as ‘natural’ if it is consistent with $\mathsf {ZFC}$ , asserts some ‘facts’ about sets, and is not of a metamathematical or proof-theoretic nature, but this is really not much.Footnote ²¹ Moreover, $\mathsf {MV}$ ’s scope for ‘naturalness’ is too restrictive from the beginning, as it leaves out theories, such as $\mathsf {ZF}+$ AD, which unquestionably express deep set-theoretic facts.

As an interpretative option, one could define a set-theoretic statement $\varphi $ to be ‘natural’ if $\mathsf {ZFC}$ plus some LCA proves that $\varphi $ holds in a forcing extension of V, or in some definable (allowing for set parameters) inner model of $\mathsf {ZFC}$ of a forcing extension of V (including definable, with set parameters, class forcing extensions that preserve $\mathsf {ZFC}$ ). By this interpretation, all $\mathsf {ZFC}$ axioms as well as all LCAs are natural, and so are CH, V=L, V=HOD, SCH, as well as their negations (in fact, all genuinely set-theoretic statements known to be consistent with $\mathsf {ZFC}$ and asserting some facts about sets). Moreover, in this case, even theories contradicting Choice, like $\mathsf {ZF}$ +AD, would now be seen as ‘natural’, as they are also equiconsistent with $\mathsf {ZFC}$ +LCAs.Footnote ²²

This interpretation seems to fit Steel’s goals, but, surely, ‘equiconsistency with $\mathsf {ZFC}$ or $\mathsf {ZFC}$ +LCAs’ does not seem to square very well with the common-sense, intuitive meaning of ‘naturalness’. Moreover, one could easily reverse this approach, consider a theory natural if it can be expressed in models of $\mathsf {ZF}$ +LCAs, which also allow for the existence of even stronger LCs, such as Reinhardt and Berkeley Cardinals, and ‘incorporate’ theories with Choice in models of $\mathsf {ZF}$ +LCAs.Footnote ²³

2.2 Large cardinals as maximality principles

Steel’s preference for LCs is also motivated by another principle, that he calls ‘maximise interpretative power’, which might be seen as consisting of two parts:

Maximise Interpretative Power [MIP]. (A.) The $\mathsf {MV}$ axioms should be able to ‘represent’ as many theories (‘natural extensions of $\mathsf {ZFC}$ ’) as possible; (B.) all the theories represented by the $\mathsf {MV}$ axioms should be such that, for any two of them, T and S, if $Con(T) \rightarrow Con(S)$ , then $\Gamma _S \subseteq \Gamma _T$ (where, given a theory T, $\Gamma _T=\{\phi : T \vdash \phi \}$ ).Footnote ²⁴

LCAs constitute a paradigmatic case study with respect to MIP. As far as (A.) is concerned, we have seen that each ‘natural’ theory T is satisfied in a forcing extension or inner model of another ‘natural’ theory S, provided both T and S are equiconsistent with LCAs; as regards (B.), we know that the amount of mutual interpretability of natural theories rises in proportion with their consistency strength. Thus, in the end, it would be legitimate to expect that:

(*) As natural theories proceed up the large cardinal hierarchy in consistency strength, they agree on an ever-increasing class of mathematical statements.Footnote ²⁵

A few observations are in order. The first one, concerning (B.), is that, so far, MIP has been shown to be satisfied by LCAs only partially, that is, for specific kinds of sentences in the Lévy hierarchy of arithmetical sentences. The following is the most one can hope to prove so far:

Empirical fact. For any two natural theories $T, S$ whose consistency strength is at least that of the theory: $\mathsf {ZFC}$ +‘there exist infinitely many Woodin cardinals’, such that $Con(T) \rightarrow Con(S)$ , we have that $({\Pi ^{1}_\omega })_S \subseteq ({\Pi ^{1}_\omega })_T$ (where, given a theory T, $({\Pi ^{1}_\omega })_T$ is the set of $\Pi ^{1}_{\omega }$ sentences provable in T).

As a consequence, the applicability of MIP to LCAs has only been verified up to the level of second-order arithmetic.

The second observation is as follows. Steel’s purpose is that of representing a multiplicity of theories, all of which extend $\mathsf {ZFC}$ +LCs, and we wonder whether MIP is really compatible with this goal. For suppose (B.) were applicable to all sentences in the Lévy hierarchy; then, clearly, all the theories targeted by (A.), would not, just, be represented, but, for all practical purposes, they would rather be amalgamated into just one theory.Footnote ²⁶ One way out of this difficulty would be to see (A.) as being sanctioned by the presently limited range of applicability of (B.), but this wouldn’t help fully ease the tension between (A.) and (B.).

Finally, although this is very speculative, the hierarchy of LCs contradicting Choice might, potentially, be more successful at instantiating MIP than the hierarchy of LCs with Choice, but, as we have seen, none of these theories features among those targeted by $\mathsf {MV}$ .Footnote ²⁷

2.3 Models for $\mathsf {MV}$

A major asset of $\mathsf {MV}$ is that this theory is complete with respect to a specific class of models $M^{G}$ , with which we will deal in a moment, that is, one has that:

$$\begin{align*}\mathsf{MV} \vdash \phi \leftrightarrow (\forall M^{G}) \ M^{G} \models \phi. \end{align*}$$

We will not delve into the philosophical reasons for preferring a complete axiomatisation of the multiverse over one which is not, as this task has already been carried out satisfactorily.Footnote ²⁸ In this subsection, we would rather like to focus on the semantics of $\mathsf {MV}$ , in particular, on its ‘natural’ models, $M^G$ ,Footnote ²⁹ and bring to light a slightly different mathematical approach to it (Proposition 2.2).

We start with reviewing, very quickly, the axioms. The language of $\mathsf {MV}$ is the first-order language of set theory with two sorts, namely Set and World. We introduce a minor tweak to Steel’s original formulation of the axioms. As we already know, $\mathsf {MV}$ has, as its own base, the axioms of $\mathsf {ZFC}$ plus LCs (we shall mostly refer to the base theory as T). Now, let a T-theory be a theory extending $\mathsf {ZFC}$ which is preserved by set-forcing extensions, and by going to set-forcing grounds: it turns out that any theory of the form $\mathsf {ZFC}$ +‘there exists a proper class of some kind of LCs’ is a T-theory. Then, the $\mathsf {MV}$ axioms for $T (\mathsf {MV}_T)$ , which we will be mostly referring to and using throughout the paper, are the following ones:

1. 1. (Extensionality for Worlds) If two worlds have the same sets, then they are equal.

2. 2. Every world is a model of T.

3. 3. Every world is a transitive proper class. An object is a set if and only if it belongs to some world. All worlds have the same ordinals.

4. 4. If W is a world and ${\mathbb P} \in W$ is a poset, then there is a world of the form $W[G]$ , where G is ${\mathbb P}$ -generic over W.

5. 5. If U is a world and $U=W[G]$ , where G is ${\mathbb P}$ -generic over W, then W is also a world.

6. 6. (Amalgamation) If U and W are worlds, then there are posets ${\mathbb P}\in U$ and ${\mathbb Q}\in W$ , and sets G and $H\ {\mathbb P}$ -generic and ${\mathbb Q}$ -generic over U and W, respectively, such that $U[G]=W[H]$ .

If M is a countable model of T and G is $Coll(\omega , <Ord)^M$ -generic over M, let $M^G$ be the model whose sets are those in $M[G]$ (in the case M is ill-founded, then $M[G]$ is defined accordingly) and whose worlds are the grounds of models of the form $M[G\restriction \alpha ]$ , for some $\alpha \in Ord^M$ . It can be easily shown that $M^G$ is a model of $\mathsf {MV}_{T}$ , when $T=\mathsf {ZFC}$ .Footnote ³⁰ Moreover, if T is obtained by adding to $\mathsf {ZFC}$ axioms such as ‘There is a proper class of P-cardinals’, where P stands for any of the usual large-cardinal properties, then it can also be proved that $M^G$ is a model of $\mathsf {MV}_{T}$ .

The collection of all models $M^G$ provides a complete semantics, and the axiom which guarantees the completeness of $\mathsf {MV}_T$ is, as has been shown by Maddy and Meadows [Reference Maddy and Meadows21, p. 134], Amalgamation. But, as stressed by the authors, a consequence of this fact is that, in any model of the form $M^G$ , not all generic filters for posets in M may be taken into account to produce forcing extensions which act, as required by the Meta-Theoretic Constraint of Section 2.1, as the ‘worlds’ of $\mathsf {MV}$ , but only those which are produced by the $Coll(\omega , Ord^M)$ -generic filter G over M.

We now proceed to prove the following:

The following straightforward consequence of the $\mathsf {MV}_T$ axioms asserts that every set-forcing generic extension of a world is also a world.

2.4 $M^{G}$ and the translation function

As already anticipated in Section 1.1, Steel’s conception rests upon the crucial fact that the language of $\mathsf {MV}$ , $\mathcal {L}_{\mathsf {MV}}$ , may be seen as a sublanguage of $\mathcal {L}_{\mathsf {\in }}$ . In order to show this, Steel defines a recursive translation function t from $\mathcal {L}_{\mathsf {MV}}$ into $\mathcal {L}_{\in }$ , such that:

(Transl) $$ \begin{align} \mathsf{MV}_T\vdash \varphi \quad \mbox{ if and only if } \quad T\vdash t(\varphi), \end{align} $$

where T, as said, is $\mathsf {ZFC}$ +LCs. The translation function may be rendered more transparently in terms of the semantics of $\mathsf {MV}$ as follows:

Theorem 2.4 (Translation function).

For any sentence $\varphi $ of $\mathcal {L}_{\mathsf {MV}}$ , every countable model M of $\mathsf {ZFC}$ and every $G \ Coll(\omega , <Ord^M)$ -generic over M

(Transl) $$ \begin{align}M^G\models \varphi \quad \mbox{ iff } \quad M\models t(\varphi). \end{align} $$

It should be noted that, in the presence of the specified semantics of $\mathsf {MV}$ , the sentence $t(\varphi )$ asserts ‘ $\varphi $ is true in all multiverses obtained from me’.Footnote ³¹ Equivalently, ‘ $\varphi $ is true in some multiverse obtained from me’. Now, as said (section 2.3), Amalgamation is needed to show that the models $M^G$ provide a complete semantics for $\mathsf {MV}_T$ .

3.1 Prelude: the (outer) core of $\mathsf {ZFC}$

Initiated by Reitz and Hamkins a few years ago, set-theoretic geology has brought forward a very innovative approach to ‘universes of set theory’ producible using forcing.Footnote ³² Here follows a very brief review of its fundamental concepts.

Take V to be a model of the axioms $\mathsf {ZFC}$ . Hypothetically, it might be that V is a forcing extension of a ground model W, that is, that there exists a W-generic filter $G \subseteq \mathbb {P} \in W$ such that $V=W[G]$ . If this is the case, then it makes sense to explore the ‘geology’ of V, that is, the collection of grounds, the generic extensions of grounds and their grounds, and so on, which V might contain, where a ground of a model M is a model N of $\mathsf {ZFC}$ such that M has been obtained through forcing over N, that is, as an extension $M=N[G]$ , where G is a generic filter of a partial order $\mathbb {P} \in N$ . Further ‘geological’ notions will then crop up, which can be spelt out as follows.

The mantle $\mathbb {M}$ is the intersection of all the grounds of a model M of $\mathsf {ZFC}$ , and the bedrock is a class W which is a minimal ground of V. By results of Usuba Reference Usuba[27], the mantle is a model of $\mathsf {ZFC}$ . Since Reitz Reference Reitz[23], the following axiom has proved to be central to all geological investigations:

Now, if the mantle itself satisfies GA, then the mantle is a bedrock, in particular, a minimum bedrock contained in all other grounds, and the latter could legitimately be seen as the core universe of the set-generic multiverse (generated over it) that we’ll be describing in full detail in the next subsection.Footnote ³³ A fuller mathematical characterisation of the core may, thus, be attempted. However, already at this point, it emerges that the features of the core may be (over $\mathsf {ZFC}$ ) indeterminate. This is shown by the following, fundamental theorem:

In an attempt to both attain a more determinate core, and further investigate its nature, set-theoretic geologists have taken into account a different hypothesis, which can be expounded as follows. If the mantle does not satisfy GA, then it makes sense to take into account the ‘mantle of the mantle’, $\mathbb {M}^1=\mathbb {M}^{\mathbb {M}}$ , and then, always under the assumption that GA is not satisfied, the ‘mantle of the mantle of the mantle’ and so on. In other terms, it makes sense to take into account the iteration of the ‘mantle operation’. Now, iterating the mantle is not a trivial task, as there are technical aspects involved which may prevent one from even defining the nth iterates of the mantle.Footnote ³⁵ Assuming these difficulties may be overcome, one might ask whether, for an $\alpha \in Ord$ iterate of the mantle, one has that $\mathbb {M}^{\alpha }=\mathbb {M}^{\beta }$ , for all $\beta>\alpha $ , that is, whether there is a minimal $\alpha $ such that $\mathbb {M}^{\alpha } \models $ GA. If there is one, then $\mathbb {M}^{\alpha }$ is said to be the outer core of the initial model (of V, if V was such model). However, in this case, as in the previous one where $\mathbb {M} \models $ GA, Theorem 3.3 will imply that also the outer core does not have determinate features, namely, that it might satisfy a wide range of mutually incompatible properties.

Moreover, it has been shown that the outer core cannot be uniquely pinned down in the iteration process, as a recent result by Reitz and Williams has confirmed Fuchs et al. Reference Fuchs, Hamkins and Reitz[9]’s Conjecture 74,Footnote ³⁶ and shown that, for any $\alpha \in Ord$ , the outer core of a model M could be any $\mathbb {M}^{\alpha }$ , that is, any $\alpha $ th iterate of the mantle of M (possibly also including including $\mathbb {M}^{Ord}$ , although no proof is currently available for this specific case).Footnote ³⁷

To sum up, $\mathsf {ZFC}$ alone does not, in itself, guarantee the existence of a ‘core universe’ and, moreover, over $\mathsf {ZFC}$ , no full-fledged, definite mathematical characterisation of the core may arise. As we shall see, a slightly different scenario comes to light, if one takes onboard LCs.

3.2 The proof of the existence

Steel [Reference Steel and Kennedy25, p. 168] credits Woodin for observing that ‘if the multiverse has a definable world, then it has a unique definable world, and this world is included in all the others’.Footnote ³⁸ We give next a proof of this fact.Footnote ³⁹

Theorem 3.4. If the multiverse has a definable world, then it has a unique definable world. More precisely, suppose $\varphi $ and $\psi $ are formulas in the language of the multiverse with only one free variable for sets. Then,

$$ \begin{align*}\mathsf{MV}_T\vdash '\forall U,W (\forall x ((x\in U \leftrightarrow \varphi(x)) \wedge (x\in W \leftrightarrow \psi(x))\to U=W).'\end{align*} $$

Proof. Working in a model $M^G$ of $\mathsf {MV}_T$ (see Section 2.3), let U and W be worlds defined by $\varphi $ and $\psi $ , respectively. Since U and W are transitive and contain the same ordinals, it will be sufficient to show that $U\subseteq W$ by showing $V_\alpha ^U\subseteq W$ by induction on $\alpha $ . This is clear for $\alpha =0$ , and also clear for $\alpha $ a limit ordinal provided it holds for all ordinals less than $\alpha $ . So suppose $V_\alpha ^U \subseteq W$ and let us show $V_{\alpha +1}^U\subseteq W$ . Note that $V_\alpha ^U\subseteq W$ implies $V_\alpha ^U \in W$ . This is because

$$ \begin{align*}x \in V_\alpha^U \,\,\mbox{ iff } \,\,W \models t('rank(v)\!<\!\alpha \mbox{ in the unique world defined by } \varphi')[x].\end{align*} $$

Hence, $V_\alpha ^U$ is definable in W with parameter $\alpha $ .

Now let $Y\subseteq V_{\alpha }^U$ , $Y\in U$ . We must check that $Y\in W$ . As in Proposition 2.2, we can find $\gamma $ large enough so that

$$ \begin{align*}W^{Coll(\omega,\gamma)}\models '\mbox{I am a forcing extension of}\ X,\ \mbox{for some world}\ X\ \mbox{which } \\[-24pt] \end{align*} $$

$$ \begin{align*}\mbox{is definable by } \varphi \mbox{ in the multiverse generated by me}.'\end{align*} $$

It follows that $U\subset W[H]$ for all $H\ Coll(\omega ,\gamma )$ -generic over W. So $Y\in W[H]$ for all such H. Letting $\dot {Y}\in W$ be a $Coll(\omega ,\gamma )$ -name for Y, we have that Y is definable in W as the set of all $x\in V_\alpha ^U$ which are forced by $Coll(\omega , \gamma )$ to belong to $\dot {Y}$ . Hence, $Y\in W$ .  □

It follows that if there is a definable world U in the multiverse, then it is unique and, by Proposition 2.2, is contained in all of $\mathsf {MV}$ ’s worlds.Footnote ⁴⁰ Clearly, since the core, if it exists, is definable, the core exists if and only if there is a definable world.

3.3 The core is the mantle of V

We proceed to introduce the fundamental result, due to Toshimichi Usuba, which shows that, under the assumption of sufficiently strong LCs, V has a smallest ground.

Now, since all grounds are downwards-directed,Footnote ⁴² by using Theorem 3.5, we can now prove the following:

Proof. By Theorem 3.5, the mantle $\mathbb {M}_W$ of every world W of the multiverse is a ground, hence by Axiom 5 of $\mathsf {MV}$ , it is also a world. Now suppose $U_0$ and $U_1$ are worlds. By Amalgamation, they are also grounds of some world W containing them. Hence, since the grounds are downwards-directed, $\mathbb {M}_{U_0}=\mathbb {M}_{U_1}$ . It follows that all the worlds of the multiverse have the same mantle $\mathbb {M}$ , which is the intersection of all the worlds. Thus, $\mathbb {M}$ is definable by the formula

$$ \begin{align*}\forall U (U\mbox{ is a world}\to x\in U).\end{align*} $$

Hence by Theorem 3.4 the mantle $\mathbb {M}$ is the core of the multiverse.  □

3.4 The strength of the large-cardinal assumptions

By Proposition 3.6, if M satisfies that there exists an extendible cardinal, then $M^G$ has a core. It is legitimate to ask whether the large-cardinal assumption in Theorem 3.5 might be weakened to the level of a LC lower in the consistency strength hierarchy; more generally:

The following theorem shows that the existence of a proper class of supercompact cardinals is not sufficient to prove the existence of the core, thus suggesting a potential threshold for the choice of T mentioned by Question 3.7.

Proof. First, force with a class-forcing iteration with Easton support in order to make every supercompact $\kappa $ indestructible by $<\!\!\kappa $ -directed-closed forcing (see Reference Apter[2]), followed by the Jensen’s class-forcing iteration that forces the GCH. Standard arguments show that all the supercompact cardinals are preserved and no new supercompact cardinals are created. Over the generic extension, make again all supercompact cardinals $\kappa $ indestructible by $<\kappa $ -directed-closed forcing (which preserves the GCH and does not create new supercompact cardinals) and call the resulting model $V[G]$ . Now force with a class-forcing iteration $\mathbb {P}$ with Easton support that forces a version of the Continuous Coding Axiom (CCA), similarly as in Reference Reitz[23], which codes every set of ordinals proper-class-many times into the power-set function on the class S of supercompact cardinals that are not limit of supercompact cardinals. Again, standard arguments show that all supercompact cardinals in S are preserved and no new supercompact cardinals are created. Let $V[G][H_0]$ be this model. Finally, force over $V[G][H_0]$ with the class-forcing product

$$ \begin{align*}\mathbb{Q}=\prod_{\kappa \in ORD}\mathbb{Q}(\kappa),\end{align*} $$

where $\mathbb {Q}(\kappa )$ is the forcing for adding a Cohen subset of $\kappa $ , if $\kappa $ is in S, and is the trivial forcing otherwise. Let the resulting model be $V[G][H_0][H_1]$ . We claim that supercompact cardinals in S are preserved. So let $\kappa \in S$ . Working in $V[G][H_0]$ , note first that $\mathbb {Q}$ factors as $\mathbb {Q}_{\kappa } \times \mathbb {Q}_{[\kappa +1 , Ord)}$ . Also note that, since $\kappa $ is not a limit of supercompact cardinals, ${\textrm {{sup}}}(S\cap \kappa )<\kappa $ . So, $\mathbb {Q}_{\kappa }$ factors as $\mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}\times \mathbb {Q}(\kappa )$ , where $\mathbb {Q}(\kappa )$ is the forcing that adds a Cohen subset of $\kappa $ . The product $\mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}\times \mathbb {Q}(\kappa )$ is equivalent, as a forcing notion, to $\mathbb {Q}(\kappa )\times \mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}$ . Also, since $\mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}$ has cardinality less than $\kappa $ and $\mathbb {Q}(\kappa )$ does not add new bounded subsets of $\kappa $ , $\mathbb {Q}(\kappa )\times \mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}$ is equivalent to $\mathbb {Q}(\kappa )\ast \mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}$ . Now, since $\mathbb {Q}(\kappa )$ is $<\kappa $ -directed closed it preserves the supercompactness of $\kappa $ , and then so does $\mathbb {Q}_{{\textrm {{sup}}}(S\cap \kappa )}$ subsequently, since it has cardinality less than $\kappa $ . Moreover, $\mathbb {Q}_\kappa $ forces that $\mathbb {Q}_{[\kappa +1 , Ord)}$ is $<\kappa $ -directed-closed, and therefore it also preserves the supercompactness of $\kappa $ . Since $\kappa ^{<\kappa }=\kappa $ for every $\kappa \in S$ , the forcing $\mathbb {Q}$ preserves cardinals and the power-set function. Now we may argue similarly as in Reference Reitz[23] to show that every ground of $V[G][H_0][H_1]$ has a proper subground, hence there is no core.Footnote ⁴⁴ For suppose W is a ground of $V[G][H_0][H_1]$ . Let $\mathbb {P}$ be a poset in W and g a $\mathbb {P}$ -generic filter over W such that $V[G][H_0][H_1]=W[g]$ . Since the two models $V[G][H_0][H_1]$ and W agree on the values of the power-set function for a tail of elements of S, as well as on statements of the form “ $\kappa $ is the $\alpha $ th element of S” for a tail of cardinals $\kappa $ in S, every element of $V[G][H_0]$ is coded into the power-set function of W on S, hence $V[G][H_0]\subseteq W$ . Now let $\kappa $ in S be greater than $|\mathbb {P}|$ , and let $h_\kappa $ be the Cohen generic subset of $\kappa $ added by H. Since $V[G][H_0]\subseteq W$ , every condition of the forcing $\mathbb {Q}(\kappa )$ , therefore every bounded subset of $h_\kappa $ , belongs to W. Hence, since the pair $\langle W, W[h]\rangle $ satisfies the $\kappa $ -approximation property, $h_\kappa \in W$ . Thus, writing $H_1$ as the product $H_1^{\leq \kappa }\times H_1^{>\kappa }$ , we have that $V[G][H_0][H^{>\kappa }]\subseteq W$ , and $V[G][H_0][H^{>\kappa }]$ is a ground of W. Now, if $\kappa _0 <\kappa _1$ are the first two members of S greater than $|\mathbb {P}|$ , then $V[G][H_0][H^{>\kappa _1}_1]$ is a proper ground of $V[G][H_1][H_1^{>\kappa _0}]$ , hance also a proper ground of W.  □

Therefore, although no formal evidence may currently be provided, it is reasonable to conjecture, that, as a consequence of Theorem 3.8, no LCA of consistency strength lower than that of ‘there exists an extendible cardinal’ will be sufficient to prove the existence of the core. However, more work is needed to shed further light on the issue.

4.1 The core and CH

We know that, under certain assumptions on T, the core of the multiverse of $\mathsf {MV}_T$ exists, so it makes sense to find out what features it has and, based on these, what it may reveal to us about such undecidable statements as CH. In particular, we would like to address the following question:

The question may, in fact, extend to any other statement $\varphi $ which is not decided by $\mathsf {ZFC}$ . In particular, for each such $\varphi $ , one may ask whether the core implies the truth or falsity of $\varphi $ .

The results which follow provide answers to such questions, and, overall, starkly expose the indeterminacy of the core. We, first, outline the mathematical strategy for CH.

If the core exists, then it satisfies GA, namely, it does not have any proper grounds. Reitz Reference Reitz[23] proved that every model of $\mathsf {ZFC}$ has a class-forcing extension which preserves any desired $V_\alpha $ , satisfies V=HOD, and is a model of $\mathsf {ZFC}$ +GA.Footnote ⁴⁵ By results in Reference Bagaria and Poveda[5], the class-forcing used to obtain the model preserves extendible cardinals. So, starting with a model M satisfying $T=\mathsf {ZFC}+$ ‘There is a proper class of extendible cardinals’ we may class-force over it to obtain a model $M[H]$ of T satisfying GA and, e.g., $\neg $ CH. And then by forcing with $Coll(\omega , <Ord)^{M[H]}$ over $M[H]$ we obtain a model of $\mathsf {MV}_T$ whose core is $M[H]$ and satisfies $\neg $ CH.

Now we show how this strategy can be extended to all $\Sigma _2$ set-forceable statements $\varphi $ :

Besides the CH and $\neg $ CH there are many other relevant $\Sigma _2$ statements that are set-forceable, and therefore consistently hold in the core of the $\mathsf {MV}_T$ multiverse. We give next a couple more examples.

The well-known forcing axiom $\mathrm {MA}_{\aleph _1}$ is also equivalent to a $\Sigma _2$ statement, with $\aleph _1$ as a parameter. Thus, Theorem 4.2 yields the following:

4.2 The core and strong forcing axioms

Corollary 4.5 helps us to introduce, in fuller generality, the examination of the behaviour of the core with respect to forcing axioms. Now, what has neatly come to surface is that the core of $\mathsf {MV}_T$ , where T may, and even may not, have extendible cardinals, is also consistent with strong forcing axioms. Corollary 3.8 of Reference Reitz[23] already proves that the Proper Forcing Axiom (PFA) is consistent with the GA.Footnote ⁴⁷ We extend this result also to MM, MM $^{++}$ , etc.

Recall that MM $^{++}$ states that for every poset ${\mathbb P}$ that preserves stationary subsets of $\omega _1$ , every collection $\{ D_\alpha :\alpha <\omega _1\}$ of dense open subsets of ${\mathbb P}$ , and every collection $\{ \sigma _\alpha :\alpha <\omega _1\}$ of ${\mathbb P}$ -names for stationary subsets of $\omega _1$ , there exists a filter $G\subseteq {\mathbb P}$ that is generic for $\{ D_\alpha :\alpha <\omega _1\}$ (i.e., $G\cap D_\alpha \ne \varnothing $ , all $\alpha $ ), and such that $\sigma _\alpha [G]$ is a stationary subset of $\omega _1$ , all $\alpha $ . It is well-known that MM $^{++}$ can be forced, assuming the existence of a supercompact cardinal.

It follows from the theorem above that if $T=\mathsf {ZFC}$ +‘there exists a supercompact cardinal’ is consistent, then so is that the core of $\mathsf {MV}_T$ satisfies MM $^{++}$ . Also, since the class-iteration ${\mathbb P}$ that forces the GA preserves extendible cardinals,Footnote ⁴⁸ if the theory $T=\mathsf {ZFC}+$ ‘there exists a proper class of extendible cardinals’ is consistent, then so is $\mathsf {MV}_T$ plus that the core satisfies MM $^{++}$ . Moreover, in both cases, by results in Reference Asperó and Schindler[3], the core also satisfies Woodin’s $(\ast )$ axiom.

5.1 Ultimate-L and $\mathsf {MV}$

In recent years, as a consequence of the far-reaching developments of the inner model programme, the possibility of the existence of a canonical inner model for a supercompact cardinal is emerging. Woodin has called such a model Ultimate-L, insofar as this model would incorporate all features of an L-like inner model, as well as all LCs, thus leading to the completion of the inner model programme itself.Footnote ⁵⁰ If such a model exists, then one could say that it represents an ‘optimal’ approximation of V, something which would justify viewing the axiom V=Ultimate-L as the most natural extension of $\mathsf {ZFC}$ , and Steel Reference Steel and Kennedy[25] has precisely taken into account such a possibility.Footnote ⁵¹

In this subsection, we will first be concerned with mathematical results about Ultimate-L and the core and, in the next one, we will articulate the Core Universist position based on these results. We start with Woodin’s definition of the axiom V=Ultimate-L.

Woodin has shown that V=Ultimate-L implies the CH.Footnote ⁵² It also implies the Ground Axiom, i.e., that V is not a set-generic extension of any inner model, and that V=HOD.

Woodin has made the following conjecture. First, recall that an inner model N is a weak extender model for the supercompactness of $\delta $ if for every $\lambda \geq \delta $ there is a normal fine measure $\mathcal {U}$ on $\mathcal {P}_\delta \lambda $ such that:

1. 1. $\mathcal {U}\cap N\in N.$

2. 2. $\mathcal {P}_\delta \lambda \cap N \in \mathcal {U}$ .

Now, crucially for our purposes, if the Ultimate-L Conjecture holds, then letting T be the theory $\mathsf {ZFC}+$ ‘There exists a proper class of extendible cardinals’ we have that $\mathsf {MV}_T$ proves that the core has an inner model $\mathcal {V}$ which is contained in HOD, and satisfies V=Ultimate-L. Moreover, by Woodin’s Universality Theorem,Footnote ⁵³ $\mathcal {V}$ satisfies that there exists a proper class of extendible cardinals. However, $\mathcal {V}$ need not be a world itself, but, if it is so, then $\mathcal {V}$ is the core. So, here follows the first result of this subsection:

The following two propositions nail down the consistency of V=Ultimate-L with $\mathsf {MV}_T$ (where $T=\mathsf {ZFC}$ +‘there exists a class of extendible cardinals’), assuming V=Ultimate-L’s own consistency.

However, also the negation of V=Ultimate-L, as shown by the following proposition, is consistent with $\mathsf {MV}_T$ :

5.2 The role of Ultimate-L within $\mathsf {MV}$

From what Steel says in Reference Steel and Kennedy[25], it seems reasonable to assume that he would encourage the adoption of $\mathsf {ZFC}$ +LCs+V=Ultimate-L as the Core Universist’s ultimate theory. It is crucial to make clear how this could possibly happen.

Steel declares V=Ultimate-L to be the right axiom for the core for the following reasonsFootnote ⁵⁴ : (1.) it implies $V=\mathcal {C}$ ; (2.) it implies the existence of a ‘fine structure theory’ for the core; (3.) it is consistent with all LCs.Footnote ⁵⁵ But, as we know, in view of $\mathsf {MV}$ ’s own machinery and goals, these motivations may just be seen as auxiliary arguments for the acceptance of V=Ultimate-L, as our prospects to settle on V=Ultimate-L as the right extension of $\mathsf {ZFC}$ entirely depend, by $\mathsf {MV}$ ’s own lights, on detecting its presence in $\mathcal {L}_{\mathsf {MV}}$ .

So we proceed to assess these prospects by resuming the narrative about the Core Universist’s possible choices laid out at the beginning of section 5.Footnote ⁵⁶

As said there, the Universist might tentatively interpret the $\mathcal {L}_{\in }$ -translate of the $\mathcal {L}_{\mathsf {MV}}$ -sentence ‘the core exists’ as suggesting (‘practically implying’) the view that V is the core. Syntactically, this would correspond to adopting the theory $\mathsf {ZFC}+$ LCs+ $V=\mathcal {C}$ . So, our Universist could, provisionally, settle on such a theory as the ultimate extension of $\mathsf {ZFC}$ . However, as we know, the core is a highly indeterminate object, so $V=\mathcal {C}$ wouldn’t be very informative. By contrast, if our Universist interpreted the $\mathcal {L}_{\in }$ -translate of ‘the core exists’ as suggesting that V=Ultimate-L, then he would be able to determinately fix the features of the core, and would get $V=\mathcal {C}$ as a bonus. To be more precise, in the presence of such an interpretation, $\mathsf {ZFC}+$ LCs+V=Ultimate-L would prove that CH, as expressed in $\mathcal {L}_{\mathsf {\in }}$ , is equivalent to $t($ ‘CH holds in the core’), and this strategy may be extended to all $\varphi $ ’s in $\mathcal {L}_{\mathsf {\in }}$ which aren’t decided by $T= \mathsf {ZFC}+$ LCs.

So far so good, but is this strategy really workable? In fact, as we would like to point out, there may be severe hindrances in the way of its execution. First of all, if one looks at the functioning of (Transl), the $\mathcal {L}_{\mathsf {MV}}$ -sentence ‘the core exists’, doesn’t literally translate to the $\mathcal {L}_{\in }$ -sentence ‘V is the core’, let alone to ‘V=Ultimate-L’ (see Definition 2.4). Therefore, some less quick, and more subtle kind of reasoning might underlie the adoption of $V=\mathcal {C}$ on the Universist’s part, and this may have to do with the definability of the core in $\mathsf {MV}$ . As we know, if $\mathsf {MV}$ has a definable world, then it has a unique definable world (cf. Theorem 3.4). Now, we know that the core is definable, and since there is just one such object in $\mathsf {MV}$ , $\mathsf {MV}$ ’s unique definable world is the core itself. Therefore, the Core Universist might thrive on considerations about the definability of the core within $\mathsf {MV}$ , and see these as strongly suggesting that $V=\mathcal {C}$ . But even if this argument ultimately persuaded us to adopt $V=\mathcal {C}$ , it wouldn’t be sufficient to persuade us to adopt V=Ultimate-L, anyway.

For a different strategy, we could think that what we would need here to make the $\mathcal {L}_{\mathsf {\in }}$ -translate of the $\mathcal {L}_{\mathsf {MV}}$ -sentence ‘the core exists’ suggest that ‘V=Ultimate-L’, would be some strengthening of T in the theory $\mathsf {MV}_T$ . As we shall see, unfortunately, all such strengthenings will produce results which are not fully satisfactory.

Two main cases are possible. First, take T to be the theory $\mathsf {ZFC}$ +LCs+V= Ultima-te-L, and consider $\mathsf {MV}_T$ . The theory could, potentially, do the required job, but it is inconsistent. The reason is, any model of such theory would violate the Axiom 4 of $\mathsf {MV}$ , as no forcing extension of ${V}$ could be a world satisfying T. A more modest strengthening of T, on the other hand, might suit our purposes. Take T to be: $\mathsf {ZFC}$ +LCs+‘ $\mathcal {C}\models V$ =Ultimate-L’, then the $\mathcal {L}_{\mathsf {\in }}$ -translate of ‘the core is Ultimate-L’ would be a lot closer to what we would need to have at hand, but, still, the translation wouldn’t automatically imply ‘V is the core’. But there’s also another trouble with this choice of T: the axiom ‘ $\mathcal {C}\models V$ =Ultimate-L’ wouldn’t, prima facie, seem to be justified on the grounds of $\mathsf {MV}$ ’s very evidential framework, so the only reason why one would add it to T in $\mathsf {MV}_T$ would have to do with other, ‘external’, reasons, such as, (1.)–(3.) above, or our prior belief in the correctness of V=Ultimate-L, but now relying on such reasons, somehow, would beg the question of why the Core Universist should adopt V=Ultimate-L.

But maybe there are ways to make ‘ $\mathcal {C}\models $ V=Ultimate-L’ justified on the grounds of $\mathsf {MV}$ ’s very evidential framework. We proceed to examine arguments potentially to that effect in the next subsections.

5.3 Woodin’s argument for Ultimate-L

The first argument derives from Woodin’s own reflections on the philosophical aspects of Conjecture 5.2 (Ultimate-L exists).

In Reference Woodin, Baaz, Papadimitriou, Scott and Putnam[29], Woodin has proposed to construe the Inner Model Programme as the expression of the fundamentally non-formalistic character of set theory. Woodin’s Argument, as we shall call it, first introduces two different positions about set-theoretic truth, the Skeptic’s and the Set-Theorist’s.

• • Skeptic: set-theoretic theorems are truths about finitary objects (proofs).

• • Set-theorist: set-theoretic theorems are truths about an existing realm of mathematical objects.

Now, Woodin argues that the dialectic between the Skeptic and the Set-Theorist reaches its climax on the issue of the consistency of LCs: the Skeptic holds that the consistency of LCs is a purely finitistic fact (Skeptic’s Retreat), whereas the Set-Theorist believes that it depends on intuitions about universes which contain them, in particular, canonical inner models. In order to further articulate the Set-Theorist’s position, Woodin formulates the following principle:

Set-Theorist’s Cosmological Principle (SCP). A Large Cardinal Axiom is consistent if and only if there is an inner model which satisfies it; the prediction of its consistency is correct, because LCAs are true.

Woodin also indicates concrete ways in which SCP might be disconfirmed and the Skeptic’s Retreat be validated, but we shall leave aside the issue here, and exclusively focus on the potential usefulness of Woodin’s Argument for our Core Universist.Footnote ⁵⁷

The argument revolves around the idea that our belief in the correctness of all LCs is based on the belief that they are consistent, in turn, via SCP, that they have inner models. Thus, it could suggest that, if one sees LCs as the ‘right’ extensions of $\mathsf {ZFC}$ , it is because one views them as consistent via inner models. So, the reasoning goes, once one commits oneself to all LCs, as the $\mathsf {MV}$ supporter does, then, there is a sense in which one also commits oneself to Ultimate-L, that is, to Conjecture 5.2.Footnote ⁵⁸ But then, if Ultimate-L exists, it should be a definable world and, thus, based on Theorem 3.4, it would be the core; the correctness of ‘ $\mathcal {C} \models $ V=Ultimate-L’ would thus finally be vindicated, seemingly, on $\mathsf {MV}$ ’s own evidential grounds.

However, the argument has two main problems for our $\mathsf {MV}$ -based Core Universist.

The first one is that the evidential resources it invokes (truth, consistency predictions, etc.) are not, in fact, available to him. Steel’s conception prides itself on not being dependent on any ‘metaphysics’ of universes. Therefore, intuitions about the structure of models with LCs, which underlie Woodin’s Argument and SCP, if argumentatively efficacious, may not bear on our Core Universist’s acceptance of ‘ $\mathcal {C}\models $ Ultimate-L’.

The other one is that, by using Woodin’s Argument in the present context, then the Core Universist would become practically indistinguishable from the Classic Universist who supports V=Ultimate-L: even coming from slightly different backgrounds, both would, indeed, agree on the fact that there is one universe of set theory (Ultimate-L) which is preferable to all others from the beginning, and it does not seem that the $\mathsf {MV}$ supporter’s evidential framework would commit him to such a view.

Let us now move on to explore another potential justificatory strategy.

5.4 An extrinsic argument

A different argument is based on resuming Steel’s reasons to adopt V=Ultimate-L (1.)–(3.) stated in Section 5.2: these, overall, suggest to the Core Universist that V=Ultimate-L is a very successful axiom, which could be seen as being already justified ‘extrinsically’ – insofar as practically all undecidable statements are settled by it.

More precisely, the Core Universist might use a form of ‘regressive’ reasoning here: he will require $\mathsf {MV}_T$ to incorporate ‘ $\mathcal {C} \models $ V=Ultimate-L’ in T, because he takes ‘V=Ultimate-L’ to be an already independently justified axiom.

However, in Section 5.2, we have already hinted at the inherent limitations of this strategy: $\mathsf {MV}$ would no longer be used as a means to indicate what bits of $\mathcal {L}_{\in }$ ought to be believed to be ‘meaningful’ (to legitimately hold); on the contrary, it would be $\mathcal {L}_{\in }$ to provide us information about what specific T should be chosen in $\mathsf {MV}_T$ . In other terms, the Core Universist wanting to use this ‘extrinsic’ argument would, ultimately, violate his own unbiasedness about all different theories expressed by $\mathsf {MV}$ .

Another, more general, concern is that V=Ultimate-L isn’t the only successful axiom he has at hand. In particular, since he knows that the core need not be Ultimate-L, and that T in $\mathsf {MV}_T$ is consistent with ‘ $\mathcal {C} \not \models $ V=Ultimate-L’ (Theorem 5.6), the Core Universist could ultimately settle on other, equally successful, axioms for the core. For instance, he might want to adopt ‘ $\mathcal {C} \models $ MM $^{++}$ ’, as MM $^{++}$ implies that $c=\aleph _2$ , and is clearly able to settle many other undecidable statements.

One final worry about the argument, which is worth mentioning, is that ‘success’ wasn’t really part of Steel’s narrative concerning $\mathsf {MV}$ from the beginning, although, clearly, LCs may be interpreted as being very ‘successful’ axioms. We do not want to delve into the full intricacies of the topic, but ‘success’ may really be a very volatile criterion, which, although helpful, may not lead the Core Universist to make ultimate choices about the nature of $\mathsf {MV}$ .

5.5 Summary

Let’s take stock. The progression from $\mathsf {ZFC}$ to $T=\mathsf {ZFC}$ +LCs+‘ $\mathcal {C} \models $ V=Ultimate-L’, summarised in Table 1, shows that ascending through interpretative power (and consistency strength) of theories, is, presently, insufficient to suggest to the Core Universist that V is the core, or that V is any specific ‘world’, for instance, Ultimate-L. As we have seen, on the one hand, by adding further hypotheses to T in $\mathsf {MV}_T$ , one may get an inconsistent theory; on the other, one could make choices, such as the addition of ‘ $\mathcal {C} \models $ V=Ultimate-L’, which, however, on $\mathsf {MV}$ ’s own evidential grounds, do not seem to be much justified. This, overall, leaves us with the following, somewhat unpalatable, dilemma: either to stay with $T=\mathsf {ZFC}$ +LCs in $\mathsf {MV}_T$ , and thus view the core as fundamentally indeterminate, or move to a, globally, less justified strengthening of T, but finally get a determinate core.

Table 1 Behaviour of $\mathsf {MV}_T$ with respect to the core for different choices of T.