Internal Categoricity and the Generic Multiverse

TL;DR

This paper explores whether John Steel's generic multiverse theory is internally categorical, aiming to understand if the theory's subject matter is uniquely determined within its framework.

Contribution

It investigates the internal categoricity of the generic multiverse theory, connecting forcing, categoricity, and the foundations of mathematics.

Findings

MV may be internally categorical under certain conditions

Internal categoricity arguments can shed light on the determinacy of the multiverse

The results impact the understanding of mathematical foundations and multiverse theories

Abstract

John Steel's theory, MV, of the generic multiverse provides a foundation for mathematics that aims to neutralize the effects of incompleteness brought on by forcing arguments. Jouko V\"a\"an\"anen's development of internal categoricity arguments provides opportunities to argue that the subject matter of some theory is, in some sense, determined. This paper investigates whether MV is internally categorical.

Equations

$$\sigma :⟨M,{\in }_0^{\mathcal{M}}⟩\cong ⟨M,{\in }_1^{\mathcal{M}}⟩$$

$$U=\{x|\psi (x,r)\}.$$

$$\displaystyle\exists\mathbb{P}_{0}\in W_{0}\exists\mathbb{P}_{1}\in W_{1}\exists U\exists g_{0},g_{1}\in U\$$

$$(g_0\in gen({\mathbb{P}}_0,W_0)\wedge g_1\in gen({\mathbb{P}}_1,W_1)\wedge$$

$$W_0[g_0]=U=W_1[g_1]{)}^U.$$

$$\mathcal{W}={\mathcal{M}}^G.$$

$${\mathcal{D}}_X=\{D\in M[\prod _{\gamma \in X}{G}_{\gamma }]|D\text{ is dense in }{\mathbb{C}}_{Or{d}^M}\}.$$

$$\dot{A}=\{⟨\check{\alpha },p⟩\in {\check{\omega }}_1\times {\mathbb{C}}_{{\omega }_1}|p(0,\alpha )=1\}$$

$$|\{z\in M|z{\in }^{\mathcal{W}}x\}|=|\{z\in M|z{\in }^{\mathcal{W}}y\}|.$$

$$f_x^{\ast }:{\omega }^{\mathcal{W}}\cong \{z\in M|z{\in }^{\mathcal{W}}x\}.$$

$$\{y\in M|y{\in }^{\mathcal{W}}x\}$$

$$M⊊M[H]⊊M[H\times J].$$

$$D=\{g\in {\mathbb{P}}^{\ast }|g\text{ is }{\mathbb{C}}_{\alpha }\text{-generic}/M\}$$

$$E=\{g\in {\mathbb{P}}^{\ast }|x\in M[g]\}$$

$${\mathcal{V}}_0\subseteq {\mathcal{M}}_1\subseteq {\mathcal{V}}_0$$

$$\phi (x,y_0,...,y_n,U_0,...,U_m)$$

$$x\in Z\iff V[H]\models {\phi }_Z(x)$$

$$p⊩{\phi }_Z(\dot{y})\wedge \dot{y}\notin \check{V}\wedge |\{x\subseteq \lambda |{\phi }_Z(x)\}|\le \omega .$$

$$\mathcal{W}\models \phi (\bar{x})\iff ⟨N,M⟩\models e(\phi )(\bar{x})$$

$$\mathcal{W}\models {\phi }_M(x,\bar{y},U)\iff ⟨U[G],U⟩\models e(\phi )(x,\bar{y}).$$

Full text

I’m very grateful to Gabe Goldberg for helping with some of the crucial claims in this paper. I would like to thank Pen Maddy for helpful conversations around the underlying philosophy. I also want to thank Joan Bagaria, Jeremy Heis and John Steel.

It is well known that second order $ZFC$ is almost categorical. Given a pair of full models of second order $ZFC$, one of them will be isomorphic to a rank initial segment of the other and the rank of that initial segment will be an inaccessible cardinal. It is a pleasing uniqueness theorem that sits naturally beside the existence theorems delivered by consistency proofs. Moreover, the moths of philosophy have been drawn to its tantalizing flame since Zermelo (1976) and Kreisel (1969). One might hope to use this result to conclusively demonstrate that a large chunk of the subject matter of set theory, including the continuum hypothesis, is definitively fixed. These hopes have largely been dashed on the rocks of circularity, however, more recently Jouko Väänänen has formulated an intriguing notion of *internal categoricity *that seems to avoid these problems and perhaps deliver opportunities for more promising philosophical arguments (Väänänen, 2021; Maddy and Väänänen, 2023). Internal categoricity will be the focal concept of this paper.

As to this paper’s target, we consider a theory that looks to be motivated by ideas that oppose the fixity of subject matter suggested by categoricity. Since the advent of Cohen’s forcing technique and the realization that the continuum hypothesis is independent of $ZFC$, there has been a recurring thought that at least some of the incompleteness of $ZFC$ is more of a feature than a bug (Mostowski, 1967). This motivates the idea that there is a plurality of different universes wrought through forcing that might all be taken seriously as alternatives upon a somewhat level playing field. W. Hugh Woodin brought this idea to life, albeit in order to bury it, along with its provocative name: the generic multiverse (Woodin, 2012, 2011, 2004). This is a multiverse of universes linked together by sequences of generic extensions and refinements. More recently, John Steel gave this idea a more solid foundational footing in the form of a computably axiomatizable theory, $MV$ (Steel, 2014). $MV$ will be the main target of this paper.

Putting these ideas together, we have an interesting tension. On one hand, we have an intriguing theorem template that might rigidify the subject matter of set theory; and on the other, we have a theory that appears to embrace a radically pluralistic vision of what set theory is about. To bring this contrast to a point, the categoricity arguments seem to tell us that the continuum hypothesis has a definitive answer, while the generic multiverse appears to embrace its indeterminacy. In spite of all these appearances, this paper has a simple, but – I think – surprising goal. We are going to prove that $MV$ is internally categorical!

Before we get too carried away, I’ll say something even stranger and then I’ll try to clarify the underlying picture. So first, We are also going to prove that $MV$ is not internally categorical. Now it sounds like we have a contradiction, but the devil is in the details; and in particular, the axioms we use. Just as we have with $PA$ and $ZFC$, the internal categoricity of $MV$ can be made to hold or not hold depending on the axioms we use to generate that theory.111See, for example, Theorems 47 and 52 in (Enayat and Łełyk, 2024). Where does this leave the philosophical preambling above? We’ll defer some discussion of that until then end, but I will say that these results nicely illustrate that internal categoricity is a subtle matter. Moreover, this subtlety warrants further investigation and $MV$ presents an intriguing case study for that project.

The paper is organized as follows. In Section 1, we provide a brief overview of internal categoricity and some of the philosophical arguments it is entangled with. Section 2 provides an overview of Steel’s $MV$ that includes a rough classification of $MV$’s models. In Section 3, we prove that $MV$ is not internally categorical and then in Section 4, we prove that an alternative axiomatization of $MV$ is internally categorical. The paper then closes in Section 5 with some discussion of these results.

1. What is internal categoricity?

We know from Dedekind that second order $PA$ is categorical and from Zermelo and Shepherdson that second order $ZFC$ is quasi-categorical. More formally,

Theorem 1**.**

(1) (Dedekind, 1963) If $\mathcal{M}$ and $\mathcal{N}$ are full models of second order $PA$, then $\mathcal{M}\cong\mathcal{N}$.

(2) (Zermelo, 1976; Shepherdson, 1951) If $\mathcal{M}$ and $\mathcal{N}$ are full models of second order $ZFC$, then either: $\mathcal{M}$ is isomorphic to $V_{\kappa}^{\mathcal{N}}$ where $\kappa$ is inaccessible in $\mathcal{N}$; or $\mathcal{N}$ is isomorphic to $V_{\lambda}^{\mathcal{M}}$ where $\lambda$ is inaccessible in $\mathcal{M}$.

With regard to (1), we have an intriguing uniqueness result. To put some context around this, let’s note that for most us, it’s obviously desirable that, despite the weaknesses of first order logic in pinning down structures, there should be just one intended model of arithmetic (up to isomorphism).222As we progress further in this paper, I’ll frequently omit the phrase “up to isomorphism” in the hopes that this will keep the words flowing smoothly and that the reader will charitably discern my true intentions. It seems natural then to wonder if Dedekind’s result provides evidence that this is the case. There is a lot of philosophy to unpack in such a conjecture, but fortunately this has been thoroughly discussed before (Kreisel, 1969; Parsons, 1990; Button and Walsh, 2016; Maddy and Väänänen, 2023). As such, we’ll satisfy ourselves by briefly considering just one compelling objection, which I’ll motivate with a question: “If I were worried about whether there was a unique intended model of arithmetic, then surely I should be even more worried about full second order models of $PA$ which presuppose that it makes sense to talk about the unique powerset of the natural numbers?” It has been suggested many times that the conjecture above makes use of a kind of circular reasoning (Weston, 1976; Hamkins, 2012; Meadows, 2013; Koellner, 2013; Maddy and Väänänen, 2023). In order justify our thought that there is a unique model of arithmetic, we appeal to the uniqueness of a much more complex structure. As such, the idea that Dedekind’s result provides evidence for our intuitions about arithmetic, looks to be on shaky ground. None of this is to say that there isn’t a unique model of arithmetic, just that Dedekind’s result doesn’t provide compelling evidence for that claim.

With regard to (2), we get closer to the target of this paper and analogous remarks apply. Of course, we don’t always get a genuine isomorphism between full models of second order $ZFC$. So there is less temptation to think we have evidence that the subject matter of set theory has been pinned down. But any pair of such models still have a great deal in common. In particular and saliently, any two full models of second order $ZFC$ must agree on their evaluation of the continuum hypothesis. Thus, we might be tempted to think that Zermelo’s result provide us with some evidence that the continuum hypothesis has a definitive answer. As with arguments making use of Dedekind’s theorem, worries about circularity also emerge in this arena. Once again, the problem comes from the use of full second order models and their free wheeling use of the powerset operation.

Much more recently, Väänänen has offered a new perspective on categoricity theorems that makes no use of second order logic and thus, defuses the circularity worries discussed above. He calls this internal categoricity (Väänänen, 2021, 2019).333Strictly speaking, Väänänen considers two kinds of internal categoricity. The first is still rooted in second order logic, although uses Henkin models instead of full models of second order logic. The second only uses first order logic and will be our focus in this paper. This is also the usage taken up in (Enayat and Łełyk, 2024). We start by considering an expansion of the ordinary language of set theory where we let $\mathcal{L}(\in_{0},\in_{1})$ be the language with a pair of 2-place relation symbols. We then let $ZFC(\in_{0},\in_{1})$ be the theory that has: an $\in_{0}$ and $\in_{1}$ version of every axiom of $ZFC$; and has $\in_{0}$ and $\in_{1}$ versions of the Replacement and Separation Schemata where the formulae used may make use of both $\in_{0}$ and $\in_{1}$.444My notation is cosmetically different from that of Väänänen (2021). I’ve done this so that I can notate things a little more compactly on the page, but do refer to Väänänen (2021, 2019); Maddy and Väänänen (2023) for the official notation. Moreover, Enayat and Łełyk (2024) provide an excellent notation system for generalizing these ideas to other theories. Given that our focus is quite narrow in this paper, we’ll avoid this and just remind the reader of our meaning when required. We then obtain the following.

Theorem 2**.**

(Väänänen, 2019)* If $\mathcal{M}=\langle M,\in_{0}^{\mathcal{M}},\in_{1}^{\mathcal{M}}\rangle$ is a model of $ZFC(\in_{0},\in_{1})$, then there is an isomorphism*

[TABLE]

where $\sigma$ is uniformly definable over $ZFC(\in_{0},\in_{1})$.

Some remarks about the relationship with Zermelo’s theorem are warranted. With Zermelo, we learn something about a pair of models, while with Väänänen we are just considering a single model. The interesting thing is that this model is – in a sense – two models in disguise. Moreover, those two models are deeply entangled through the mashup theory $ZFC(\in_{0},\in_{1})$. In particular, they are allowed to make us of each other’s languages in their respective Replacement and Separation Schemata. Thus, and helpfully for the proof of the theorem above, each model can deliver the transitive collapse of the other.

But why should we allow this sharing of vocabulary between their Replacement and Separation Schemata? From a mathematical point of view, there is nothing stopping us. But from a more philosophical perspective, there is an intriguing argument in support of this move. If we consider the intention behind our use of a schema to articulate the idea behind Replacement, we might say that it is intended to be a first order approximation of the second order idea that the pointwise image of any class function through a set should itself be a set of outputs.555See Parsons (1990) and Field (1999) for thorough arguments for this attitude toward schemata in the context of arithmetic. See Section 4.3 of (Maddy and Väänänen, 2023) for a nuanced discussion of the relationship Parson’s thought experiment and internal categoricity. One might worry that this talk of approximating a second order concept with a schema doesn’t take us far enough away from second order logic. See page 30 of (Maddy and Väänänen, 2023) for further discussion of this point. We shall raise this again in the Discussion section. Thus, if we expand our language in such a way that could allow us to define new class functions, these class functions ought to also be subject to the closure conditions required by Replacement.

What should we make of this? To motivate the milieu behind such results, we might consider a thought experiment based on ideas from interpretability that is similar in spirit if not letter to Parsons (1990). Suppose that you use a theory $DLO(<)$ of dense linear orders that is based on a strict $<$ relation, and I use a theory $DLO(\leq)$ based on non-strict relation $\leq$. There is, of course, no substantive debate to be had between us. But how might we demonstrate this? Given that you are working in $\mathcal{L}(<)$, you might move to an expansion $\mathcal{L}(<,\leq)$ that adds my non-strict $\leq$. You might then accommodate my thoughts about the $\leq$ by extending your theory to be $DLO(<)\cup DLO(\leq)$. This puts us in a position to model how you might think about my $\leq$ in terms of your $<$. At this point, we know nothing about that, but you might be suddenly struck by inspiration and observe that you can add an axiom to $DLO(<)\cup DLO(\leq)$ that defines $\leq$ in terms of $<$ by saying that $x\leq y$ iff $x<y$ or $x=y$. You might then prove that the resulting theory remains consistent. Moreover, I might then do the analogous thing but in the other direction. If we both succeed then it can be seen that our initial theories were very closely related; in particular $DLO(<)$ and $DLO(\leq)$ are definitionally equivalent.666See (Lefever and Székely, 2019) for an excellent overview of different characterizations of definitional equivalence. This is a simple illustration of a pivotal concept based on the theory of interpretability. Results like this are a cornerstone of contemporary set theory.

I think there is a analogy to be had between the example above and a way of applying Väänänen’s theorem. Suppose you and I are both users of $ZFC$. In this situation, we have no worries about whether our theories are definitionally equivalent since they are identical. However, we might wonder if your version of $\in$ has the same meaning as mine. To explore this question, we might tag our membership relations to differentiate them and thus, use $\in_{0}$ and $\in_{1}$. We then might say that you are working $ZFC(\in_{0})$ articulated in $\mathcal{L}(\in_{0})$ and I am working in $ZFC(\in_{1})$ articulated in $\mathcal{L}(\in_{1})$. In order to explore the relationship between our theories you might then expand your language from $\mathcal{L}(\in_{0})$ to $\mathcal{L}(\in_{0},\in_{1})$ and accommodate my ideas about set theory by working in the theory $ZFC(\in_{0})\cup ZFC(\in_{1})$. This puts us in a good position to model how you might think about my $\in_{1}$ in terms of your $\in_{0}$. But at this point, our setup has nothing to say on this matter. Then, as before, you might be struck by inspiration and adopt a Parsonian attitude to the Replacement and Separation Schemata by permitting formulae of the total language in both versions of those schemata. This leaves us in $ZFC(\in_{0},\in_{1})$ where we may finally use Väänänen’s theorem to define an isomorphism between my version of the membership relation and yours. This might give you some reason to think that what I mean by $\in_{1}$ is very closely related to you mean by $\in_{0}$. Moreover, it might give you some reason to think that whatever you think about the truth value of the continuum hypothesis, I should think the same. The analogy with the dense linear orders is certainly not perfect, but I think it nicely illustrates the fact that we are dealing with a common or, at least, overlapping methodology. If we want to compare a pair of theories, then it makes good sense to consider ways of combining them. Finally, we note that second order logic has disappeared from view and so the circularity worries we had above no longer have a target. Putting all this together, it could seem that we have good reason to think that you and I should agree on the truth value of the continuum hypothesis using a methodology common in set theory that does not fall prey to worries about circularity.

I don’t plan to precisely formulate arguments about the speculative reasoning above. Nor do I plan to evaluate their force. I have two reasons for this. First, I think there is value in merely illustrating the intuitive pull that surrounds Väänänen’s theorem in an effort to flag it for more intense philosophical scrutiny. Second, I think the results that follow below place a sufficiently large question mark over such argument sketches that the effort required to understand what is going on will be substantial and well beyond the scope of this paper.

2. What is $MV$?

The incompleteness of $ZFC$ as evidenced by forcing arguments is sometimes thought of as particularly deep. By contrast, shallow forms of incompleteness can be resolved by adopting a stronger theory that we think is true. For example, although $ZFC$ cannot prove that it is consistent, if we adopt an axiom saying that there are inaccessible cardinals then the consistency of $ZFC$ follows almost immediately. But over at the deep end, the continuum hypothesis can be both forced to hold and forced to fail. Moreover, no large cardinal axiom can come to the rescue and decide the matter. The generic multiverse was introduced as a means of taking such deep incompleteness seriously as something we might just have to live with. In its simplest form, a generic multiverse $\mathbb{V}_{M}$ is formed by taking a countable transitive model $M$ of $ZFC$ and then considering all the models that can be obtained from $M$ by finite sequences of generic extensions and refinements (Woodin, 2012). We might then live with the incompleteness of $ZFC$ by only taking seriously those statements that are true in every world $N$ in $\mathbb{V}_{M}$. In particular, the continuum hypothesis is relegated to the liminal space between always true and always false. Woodin describes the motivation behind the generic multiverse particularly clearly:

Is the generic-multiverse position a reasonable one? The refinements of Cohen’s method of forcing in the decades since his initial discovery of the method and the resulting plethora of problems shown to be unsolvable, have in a practical sense almost compelled one to adopt the generic-multiverse position.777I should mention at this point that although Woodin clearly describes why embracing the generic multiverse might be attractive and understandable, his goal in that paper is to argue against its adoption. (Woodin, 2012)

If our goal is to take deep incompleteness seriously, then Woodin’s $\mathbb{V}_{M}$ has some limitations. $\mathbb{V}_{M}$ is a model that we describe, so to speak, from the outside rather than foundational framework like $ZFC$ than we can use, so to speak, from the inside. It’s still a valuable tool for understanding how the generic multiverse position works, but it doesn’t offer us a way of truly inhabiting its multiverse. To address this, Steel introduced a computably axiomatizable theory of the generic multiverse that is designed to be an autonomous foundation for mathematics, which is on a par with $ZFC$, and within which we can live with deep incompleteness (Steel, 2014).

Steel’s theory has received a healthy dose of high quality philosophical attention since its inception (Maddy and Meadows, 2020; Meadows, 2021; Bagaria and Ternullo, 2023; Blue, 2024). My goals in this section and the rest of this paper are more logical in spirit. I want to give an overview of: what $MV$ looks like from the outside, through a rough classification of its models; and what $MV$ feels like on the inside, by highlighting some significant consequences of the theory. My thought is that by delivering a more comprehensive overview of $MV$, the internal categoricity results below will make more sense to the reader and perhaps over-optimistically that these results might make $MV$ more accessible to a wider class of readers.

2.1. Notation

Before we launch into the theory, let’s first make some remarks about notation. In general, I am aiming to follow the notation conventions used by contemporary set theorists as used in, for example (Jech, 2003; Kunen, 2011) and (Foreman and Kanamori, 2009). However, I’ll highlight a few local idiosyncrasies that I’ve taken up in order to make things look a little smoother on the page.

First and contrary to the usual conventions in mathematical logic, a theory will just be a set of sentences that we usually call axioms. We shall not demand, unless stated otherwise, that a theory is closed under logical consequence. Given that internal categoricity is sensitive to the axioms used to generate a theory, this is particularly important.888We should note that this approach to theories is quite common in the theory of interpretability. See, for example, (Lindström, 2003) and (Enayat and Łełyk, 2024).

We shall, below, be frequently concerned with models of $\mathcal{L}_{\in}=\{\in\}$. Most often these will be transitive models and for those, we adopt the usual convention of denoting the model $\langle M,\in\rangle$ by its domain $M$. However, when we come to denote models that have a non-standard membership relation we shall use the calligraphic font and write $\mathcal{M}=\langle M,\in_{\mathcal{M}}\rangle$.

In almost all of the many simple forcing arguments that follow we shall make great use of Lévy’s collapse forcing. Recall that $Col(\omega,X)$ for $X\subseteq Ord$ is the partial order whose conditions are finite partial functions $p:\omega\times X\rightharpoondown X$ where for all $n\in\omega$ and $\alpha\in X$, $p(n,\alpha)<\alpha$ if it is defined. We then order $Col(\omega,X)$ be reverse inclusion. Rather than write $Col(\omega,X)$ again and again, we shall instead write $\mathbb{C}_{X}$ where the $\mathbb{C}$ is intended to be mnemonic for collapse. We shall also use interval notation for sets of ordinals so, for example $[\alpha,\beta)=\{\gamma\in\beta\ |\ \gamma\geq\alpha\}$. If $H$ is $\mathbb{C}_{\beta}$-generic over $V$ for some $\beta\in Ord\cup\{Ord\}$ and $\alpha<\beta$, we write $H\restriction\alpha$ to denote $\{p\in\mathbb{C}_{\alpha}\ |\ p\in H\}$.

The following theorem of Laver and Woodin plays a pivotal role in the theory of the generic multiverse. We shall also use the terms “generic refinement” and “ground” as synonyms below. $U$ is a generic refinement of $W$ if there is some $U$-generic $G$ such that $W=U[G]$.

Fact 3**.**

*(Laver, Woodin) Every generic refinement $U$ of $V$ is uniformly definable in $V$ using a parameter from $U$. That is, there is a formula $\psi(x,y)$ of $\mathcal{L}_{\in}$ such that if $V=U[G]$ is a forcing extension of $U$ by a $U$-generic filter $G\subseteq\mathbb{P}\in U$, then there is some $r\in U$ such that999See (Reitz, 2007) for a gentle proof and (Woodin, 2012) for something more general. *

[TABLE]

Let us fix a formula $\psi(y,x)$ as in the statement of the fact above. We shall write $x\in\mathsf{W}_{r}$ to mean that $\psi(x,r)$ holds. This is usually just written as $W_{r}$ rather than $\mathsf{W}_{r}$. However, when talking about the generic multiverse and using the theory, $MV$, upper case variables like $W$ are in high demand. Thus, we use the sans serif font $\mathsf{W}$ to avoid confusion.

2.2. The $MV$ theory

We are now ready to describe $MV$. The theory is articulated in the language $\mathcal{L}_{MV}$ which is a two sorted language extending $\mathcal{L}_{\in}$ with a new sort for worlds beyond the existing sort for sets. Sets can be members of both sets and worlds, but worlds can be members of neither sets nor worlds.101010Strictly we should have two membership relations: one for when sets are members of sets; and the other for when sets are members of worlds. However, using just one symbol lightens our notation load in a helpful way so we adopt this abuse of notation. The axioms of $MV$ can then be stated somewhat formally as follows. We shall write $gen(\mathbb{P},W)$ to denote the set of $\mathbb{P}$-generic filters of a world $W$.

• (World-extensionality)

$\forall x(x\in W\leftrightarrow x\in U)\to W=U$

• (World-transitivity)

$x\in W\to x\subseteq W$

• (World-ordinals)

$\alpha\in Ord^{W}\leftrightarrow\alpha\in Ord^{U}$

• ($ZFC$-worlds)

$\varphi^{W}$ where $\varphi$ is an axiom of $ZFC$.

• (Set-capture)

$\exists W\ x\in V$

• (Extension)

$\mathbb{P}\in W\exists U\exists g\in U\ (g\in gen(\mathbb{P},W)\wedge U=W[g])^{U}$

• (Refinement)

$r\in W\exists U\ (U=(\mathsf{W}_{r})^{W}$

• (Amalgamation)

[TABLE]

The first five axioms essentially say that worlds are inner models of $ZFC$ and that every set $x$ is captured by a world $W$ in the sense that $x\in W$. The final three axioms breathe life into the generic aspect of this multiverse. Extension tells us that any generic extension of a world is itself a world. Refinement tells us that any generic refinement of a world is a world. Amalgamation tells us that any pair of worlds have a common generic extension.

Extension and Refinement are straightforwardly motivated. They tell us that the multiverse is closed under forcing and its inverse. Amalgamation is a little more complicated.111111See Maddy and Meadows (2020) for some extensive discussion of this. One quite simple-minded motivation for including it is that it ensures that our multiverse doesn’t contain, so to speak, isolated galaxies since any pair of worlds are subsumed into a larger world. However, we shall see very soon that Amalgamation also allows us to obtain particularly elegant models of $MV$ that have a structure which is very familiar to set theorists.

It’s easy enough to offer these informal explanations but they only get us so far. In the next section, we’ll start thinking more seriously about models of $MV$, but before we do this, let’s ask a natural question about the consequences of this theory:

Given that $MV$ is a theory of sets and worlds, we can consider its reduction to the set domain, which gives us a theory in $\mathcal{L}_{\in}$. What is that theory like?

It turns out that it has a very nice theory indeed. Informally, we might say that it satisfies countable set theory. More precisely, let $ZFC^{-}$ be $ZFC$ articulated using the Collection schema (rather than Replacement) and a Set Induction schema (rather than Foundation) where we remove the Powerset axiom.121212Nothing important swings on using Set Induction rather than Foundation. Like Collection, it just behaves better in weakenings of $ZFC$. Let $ZFC_{count}^{-}$ be $ZFC^{-}$ with the following additional axiom: $\forall x\ |x|\leq\omega$. This is the natural theory of the hereditarily countable sets and it is implied by $MV$.131313Note that Choice is already implied by $\forall x\ |x|\leq\omega$ but we retain the $C$ just to emphasize that Choice is available.

Theorem 4**.**

$MV\vdash ZFC_{count}^{-}$.

Before we remark on the proof of this theorem, this is a good moment to return our attention to the continuum hypothesis and consider how it should be understood in the context of $MV$. We remarked above that $MV$ was developed with an eye to taking the independence of the continuum hypothesis seriously. But how is this achieved? It turns out that there are two ways of thinking about this. On the first way of understanding things, we treat each world in the multiverse as a genuine alternative and we see then that there will be worlds where the continuum hypothesis is true and worlds where it is false. This is a theorem of $MV$ that is essentially a consequence of Cohen’s initial work on forcing (Cohen, 1963). We might call this the local perspective. From this point of view, the cardinality of the continuum is not determined since it has many possible values across the multiverse. By contrast, the global perspective takes up the point of view considered in Theorem 4. Rather than worrying about whether the continuum hypothesis is true in one world or false in another, we ask whether it is true in the underlying, global, universe of sets that sits behind each of the worlds in the multiverse. The theorem above then tells us that every set is countable. Unless we wish to contradict Cantor’s famous theorem, this means that there cannot be a set of all subsets of the natural numbers: $\mathcal{P}(\omega)$ does not exist. Thus, from the global perspective the continuum hypothesis is simply inexpressible and thus, arguably enjoys an even greater degree of indeterminacy than is exhibited locally. I do not wish to put forward a claim that one perspective is better than the other or that a choice must be made. After all, they both fit the formal theory quite well. Regardless, it is clear that – either way – the continuum hypothesis has no definitive answer in $MV$.

2.3. The models of $MV$

Let return to Theorem 4 and its proof. It shouldn’t be too surprising to see that $MV$ implies every set is countable: Extension ensures that for any cardinal, there will be a world where it has been collapsed. Since Cantor’s theorem tells us that any set and its powerset have different cardinalities, we also see that the Powerset Axiom must fail. To show that $ZFC^{-}$ is satisfied, it will be helpful to consider some existing results about models of $MV$. Let’s begin with some technical preliminaries.

A model of $MV$ is of the form $\mathcal{W}=\langle\mathbb{W}^{\mathcal{W}},\mathbb{D}^{\mathcal{W}},\in^{\mathcal{W}}\rangle$ where $\mathbb{W}^{\mathcal{W}}$ is the world domain, $\mathbb{D}^{\mathcal{W}}$ is the world domain and $\in^{\mathcal{W}}$ is the membership relation. In (Steel, 2014) and (Maddy and Meadows, 2020), we are given a particularly nice class of models for $MV$ that we might regard as its standard models. Let $\mathcal{M}=\langle M,\in^{\mathcal{M}}\rangle$ be a model of $ZFC$ and suppose that $G$ is $(\mathbb{C}_{Ord})^{\mathcal{M}}$-generic over $\mathcal{M}$. Then let $\mathcal{M}[G]\supseteq\mathcal{M}$ be the (class) generic extension of $\mathcal{M}$ via $G$.141414If $\mathcal{M}$ isn’t transitive, it’s a little fiddly to ensure that $\mathcal{M}\subseteq\mathcal{M}[G]$. A way of doing this is described in Appendix A.1 of (Maddy and Meadows, 2020), where it is denoted as $\mathcal{M}_{ult}[G]$, however, we’ll gloss over that subtlety here. Note that since $(\mathbb{C}_{Ord})^{\mathcal{M}}$ is a pretame forcing over $\mathcal{M}$, we see that $\mathcal{M}[G]$ satisfies $ZFC^{-}$. A standard model $\mathcal{M}^{G}$ is then formed from the set $\mathbb{W}^{\mathcal{M}^{G}}$ of generic refinements of $\mathcal{M}[G\restriction\alpha]$ for each $\alpha\in Ord^{\mathcal{M}}$.151515Or a little more precisely, $\mathcal{N}=\langle N,\in^{\mathcal{N}}\rangle$ is a world in $\mathcal{M}^{G}$ is there is some $\alpha\in Ord^{\mathcal{M}}$ and $r\in\mathcal{M}[G\restriction\alpha]$ such that $x\in N$ iff $\langle\mathcal{M}[G],\mathcal{M},\in^{\mathcal{M}[G]}\rangle\models x\in(\mathbb{W}_{r})^{\mathcal{M}[G\restriction\alpha]}$. Thus, $\mathcal{M}^{G}=\langle\mathbb{W}^{\mathcal{M}^{G}},\mathcal{M}[G],\in^{\mathcal{M}[G]}$. We then recall that:

Fact 5**.**

(Steel, Theorem 26 and 27 in (Maddy and Meadows, 2020)) (1) If $\mathcal{M}$ is a model of $ZFC$ and $G$ is $(\mathbb{C}_{Ord})^{\mathcal{M}}$-generic over $\mathcal{M}$, then $\mathcal{M}^{G}\models MV$.

(2) If $\mathcal{W}$ is a countable model of $MV$ and $\mathcal{M}$ is a world in $\mathcal{W}$, then for some $G$ that is $(\mathbb{C}_{Ord})^{\mathcal{M}}$-generic over $\mathcal{M}$, we have

[TABLE]

Informally speaking, given any model $\mathcal{M}$ of $ZFC$ and a generic to collapse its ordinals, we may use them to obtain a standard model of $MV$. And in the other direction, we see that if $\mathcal{W}$ is a *countable *model of $MV$ then it must be a standard model. We can use this to prove the theorem about the set part of $MV$.

Proof.

(of Theorem 4). Let $\mathcal{W}$ be a model of $MV$. We’d like to apply Fact 5(2), but for that we need $\mathcal{W}$ to be countable. But if $\mathcal{W}$ is not countable, we can simply take a countable Skolem hull. Now let $\mathcal{M}$ be an arbitrary world in $\mathcal{W}$. Fact 5(2) then tells us that we may fix some $(\mathbb{C}_{Ord})^{\mathcal{M}}$-generic over $\mathcal{M}$ such that $\mathcal{W}=\mathcal{M}^{G}$. Then since $(\mathbb{C}_{Ord})^{\mathcal{M}}$ is pretame, the set domain $\mathcal{M}[G]$ of $\mathcal{M}^{G}$ satisfies $ZFC^{-}$. Moreover, Extension ensures that every set is countable. ∎

The Fact above is stated quite generally in that it accommodates the worlds that are nonstandard in the sense that their membership relation is not well-founded. But most (although not all) of the time, we’ll be dealing with models in which every world is transitive. In those cases, we know that $\in^{\mathcal{W}}$ is just the *real *$\in$ restricted to $\mathbb{D}^{\mathcal{W}}$. Moreover, in these case we can, without loss of generality, think of $\mathbb{W}^{\mathcal{W}}$ as literally being the set of those transitive sets. If we do this then we see that $\mathbb{D}^{\mathcal{W}}=\bigcup\mathcal{W}$.161616We’ll see later that the converse does not hold: we cannot recover the world domain from the set domain. Thus, when we deal with transitive models of $MV$, we see that $\mathbb{D}^{\mathcal{W}}$ and $\in^{\mathcal{W}}$ are redundant and so in such cases, we’ll often abuse notation and let $\mathcal{W}$ just be its world domain $\mathbb{W}^{\mathcal{W}}$. Now if we restrict our attention countable transitive models we get a particularly nice model-theoretic representation $MV$.

Corollary 6**.**

For countable $\mathcal{L}_{MV}$-structures $\mathcal{W}$ containing transitive worlds, the following are equivalent:

1. (1)

$\mathcal{W}\models MV$; and 2. (2)

$\mathcal{W}=M^{G}$* for some $G$ that is $Col(\omega,<Ord^{M})$-generic over $M.$*

For a helpful way to think about standard models of $MV$, it is helpful to recall the proof behind Solovay’s famous model where the reals are Lebesgue measurable, have the property of Baire and the perfect set property (Solovay, 1970). We begin that proof by supposing that there is an inaccessible cardinal $\kappa$ and then consider a model where all the ordinals below $\kappa$ have been collapsed using a generic $G$ for $\mathbb{C}_{\kappa}$. This was the first of many similar models obtained by collapsing large cardinals. It is a familiar playground for set theorists and – as we’ll see many of the techniques for working there translate quite directly to models of $MV$. The main difference is that to make a standard model of MV we don’t just collapse some of the ordinals, we collapse all of them. As we’ve seen this gives us our set domain. We might then regard the worlds of the multiverse are recording the history of all possible forcings along the road to this great collapse.

Standard models arguably provide the most natural tool for understanding $MV$ and how it is connected to $ZFC$. Moreover, this bridge provides a helpful and well understood toolbox for proving theorems about $MV$. However, not all models of $MV$ are standard. Given that we concerned in this paper with questions around categoricity, it will be helpful to survey the landscape beyond the realm of standard models. Of course, $MV$ is just a two-sorted theory in first order logic and so, by the upward Löwenheim-Skolem theorem, it is obvious that there are models of $MV$ of every possible cardinality. But if we place some natural constraints on the models and the worlds they contain a more interesting picture emerges. For example, given Corollary 6 we might wonder if every model $\mathcal{W}$ of $MV$ containing countable transitive models must itself be countable. This is not the case.

Proposition 7**.**

If $MA(\kappa)$ holds for $\kappa\geq\omega$, then there are models $\mathcal{W}$ of $MV$ with $|\mathcal{W}|=\kappa^{+}$ where every world $M$ in $\mathcal{W}$ is countable and transitive (if there is a countable transitive model of $ZFC$). Moreover, any such $\mathcal{W}$ will be nonstandard.

Proof.

We start by observing that such models cannot be standard. To see this, suppose $M$ is a world in $\mathcal{W}$ and $G$ is $\mathbb{C}_{Ord}^{M}$-generic over $M$. Then for all $\alpha\in Ord^{M}<\omega_{1}$, $M[G\restriction\alpha]$ is countable and so it has countably many generic refinements, $\mathsf{W}_{r}^{M[G\restriction\alpha]}$, indexed by elements $r\in M[G\restriction\alpha$]. Thus, the worlds of $M^{G}$ are a countable union of countable sets, while $\mathcal{W}$ is uncountable.

Now we prove that such models exist. Let $M$ be a countable transitive model of $ZFC$. Before we build our target model, first observe that if $G_{0}$ and $G_{1}$ are $\mathbb{C}_{Ord^{M}}$-generic over $M$, then we can easily find $G_{2}$ that is $\mathbb{C}_{Ord^{M}}$-generic over $M$ where $M[G_{2}]=M[G_{0}\times G_{1}]$. Thus, we obtain a standard model $M^{G_{2}}$ extending both $M^{G_{0}}$ and $M^{G_{1}}$. Clearly, this can be generalized to any finite set of $\mathbb{C}_{Ord^{M}}$-generics. We might say that any finite collection standard models based on $M$ can be amalgamated into another standard model. Similar remarks apply to countable collections of standard models, but as we’ve seen no amalgamation fact can hold for uncountable collections of standard models. We address this problem by inductively defining a sequence $\langle G_{\alpha}\rangle_{\alpha<\kappa^{+}}$ of filters that are mutually $\mathbb{C}_{Ord^{M}}$-generic over $M$. Let $\alpha<\kappa^{+}$ and suppose that we’ve already defined $\langle G_{\beta}\rangle_{\beta<\alpha}$ such that for all finite $X\subseteq\alpha$, $\prod_{\gamma\in X}G_{\gamma}$ is $\prod_{\gamma\in X}\mathbb{C}_{Ord^{M}}$-generic over $M$.171717Note that $\prod_{\gamma\in X}\mathbb{C}_{Ord^{M}}$ is just $(\mathbb{C}_{Ord^{M}})^{|X|}$. We wish to define $G_{\alpha}$ so that $\langle G_{\beta}\rangle_{\beta<\alpha+1}$ has the analogous property at $\alpha+1$. For finite $X\subseteq\alpha$, let

[TABLE]

Then let $\mathcal{D}=\bigcup_{X\in[\alpha]^{<\omega}}\mathcal{D}_{X}$. Since $\alpha<\kappa^{+}$, we see $|\mathcal{D}|\leq\kappa$ and since $\mathbb{C}_{Ord^{M}}$ has the ccc (in $V$), we may use $MA(\kappa)$ we may fix a filter $G_{\alpha}$ where $G_{\alpha}\cap D\neq\emptyset$ for all $D\in\mathcal{D}$. This ensures that for all finite $X\subseteq\alpha$, $G_{\alpha}$ is $\mathbb{C}_{Ord^{M}}$-generic over $M[\prod_{\gamma\in X}G_{\gamma}]$ and application of the product lemma completes our inductive argument. Finally, we then let $\mathcal{W}$ be the set of generic refinements of models of the form $M[\prod_{\gamma\in X}(G_{\gamma}\restriction\alpha)]$ where $X\in[\kappa^{+}]^{<\omega}$ and $\alpha\in Ord^{M}$. ∎

Thus, we see that in contexts where some form of Martin’s axiom holds, there will be large nonstandard models of $MV$ in which every world is countable and transitive. Moreover, since $MA(\omega)$ is a theorem of $ZFC$, we see that such nonstandard models are unavoidable.

Corollary 8**.**

There are models $\mathcal{W}$ of $MV$ with $|\mathcal{W}|=\aleph_{1}$ where every world $\mathcal{M}$ in $\mathcal{W}$ is countable and transitive.

Given that the delivery of generic sets for models of $ZFC$ is often dependent on those models being countable, we might also wonder whether standard models of $MV$ can only contain countable models. This is not the case.

Proposition 9**.**

If $0^{\#}$ exists, there there are standard models $\mathcal{W}$ of $MV$ that contain uncountable transitive worlds.

Proof.

Since $0^{\#}$ exists, we know that $L_{\omega_{1}}$ satisfies $ZFC$. To deliver a standard model of $MV$ as desired, it will suffice to show that there is some $G$ that is $\mathbb{C}_{\omega_{1}}$-generic over $L_{\omega_{1}}$. Since $\mathbb{C}_{\omega_{1}}$ is not countable, the usual Baire category argument will not work. We address this by defining the required generic inductively.181818A version of this argument can be found in the first claim within the proof of Theorem 3.5 in (Hjorth, 1995). I learned it from a MathOverflow response by Farmer Schlutzenberg.

First, we observe that for all $\alpha<\omega_{1}$, $\mathcal{P}(\mathbb{C}_{\alpha})^{L}$ is countable and so we may obtain a $\mathbb{C}_{\alpha}$-generic filter over $L_{\omega_{1}}$ for any such $\alpha$. To set up the induction, fix the club sequence $\langle\kappa_{\alpha}\rangle_{\alpha\leq\omega_{1}}$ of indiscernibles below $\omega_{1}$ witnessing that $0^{\#}$ exists and where $\kappa_{\omega_{1}}=\omega_{1}$. We aim to deliver a sequence $\langle G_{\alpha}\rangle_{\alpha\leq\omega_{1}}$ such that:

1. (1)

$G_{\alpha}$ is $\mathbb{C}_{\kappa_{\alpha}}$-generic over $L$; and 2. (2)

$G_{\alpha}\subseteq G_{\beta}$ for all $\alpha\leq\beta\leq\omega_{1}$.

For successor ordinals, $\alpha+1$, we just find $H$ that is $\mathbb{C}_{[\kappa_{\alpha},\kappa_{\alpha+1})}$-generic over $L[G_{\alpha}]$ and then $G_{\alpha+1}$ can be what is essentially the product of $G_{\alpha}$ and $H$. At limit ordinals $\beta$, we have to be a little more careful. Note first that each $\kappa_{\beta}$ is inaccessible in $L$. Thus, $L$ thinks that $\mathbb{C}_{\kappa_{\beta}}$ has the $\kappa_{\beta}$-cc property and so any maximal antichain $A$ in $\mathbb{C}_{\kappa_{\beta}}$ will be a subset of $\mathbb{C}_{\kappa_{\alpha}}$ for some $\kappa_{\alpha}<\kappa_{\beta}$.191919See the first lemma on page 15 of (Solovay, 1970) for the canonical proof of this fact. Given this, we can let $G_{\beta}=\bigcup_{\alpha<\beta}G_{\alpha}$. ∎

This result depends on a strong assumption beyond $ZFC$. Without it, it is possible to find models $\mathcal{W}$ of $MV$ where every world is transitive, but $\mathcal{W}$ is not standard.

Proposition 10**.**

If $L_{\omega_{1}}\models ZFC$, then it is consistent that there is a model $\mathcal{W}$ of $MV$ where every world $M$ in $\mathcal{W}$ is uncountable and transitive, but $\mathcal{W}$ is not standard.

Proof.

Again, we are going to build a multiverse from $L_{\omega_{1}}$. However, we know from Proposition 9 that this cannot be done within a model where $0^{\#}$ exists. More precisely, we need to find a model where there is a $\mathbb{C}_{\alpha}$-generic set for every $\alpha<\omega_{1}$, but no $\mathbb{C}_{\omega_{1}}$-generic set exists. To obtain such a model, we use Silver’s collapse forcing. For $X\subseteq Ord$, we let $\mathbb{S}_{X}$ be the poset whose domain is formed from functions $p:s\times Y\to Ord$ where we have: the first domain, $s\in[\omega]^{<\omega}$; the second domain,* *$Y\in[X]^{\leq\omega}$; and for all $n\in s$ and $\alpha\in Y$, $p(n,\alpha)<\alpha$.202020See Section 20 in Cummings’ chapter in (Foreman and Kanamori, 2009) for more information on this and a more general definition. Silver’s collapse forcing can be obtained from Lévy’s collapse by weakening the constraint on the second domain so that it merely has to be countable rather than finite. This gives $\mathbb{S}_{\omega_{1}}$ quite different properties from $\mathbb{C}_{\omega_{1}}$. In particular, $\mathbb{S}_{\omega_{1}}$ has, so to speak, a certain amount of countable completeness in its second domain.

Now let $H$ be $\mathbb{S}_{\omega_{1}}$-generic over $L$.212121Strictly speaking, there is no reason to think such an $H$ exists. However following usual conventions, we can think of ourselves as now working virtually within the context of $\Vdash_{\mathbb{S}_{\omega_{1}}}$ as calculated in $L$. This is sufficient for our consistency claim. Ordinarily, such metamathematical subtleties can be largely ignored. But given that this paper is so concerned with generic sets and whether they exist, it is worth taking the time to take a little care around this. Since $\mathcal{P}(\mathbb{C}_{\alpha})^{L}$ is countable for all $\alpha<\omega_{1}$, we see that there is a $\mathbb{C}_{\alpha}$ -generic over $L$ for each such $\alpha$. It will thus suffice to show that $L[H]$ contains no sets $G$ that are $\mathbb{C}_{\omega_{1}}$-generic over $L$.222222Our proof is an elaboration of a sketch by Mohammad Golshani on MathOverflow. Nothing in the argument relies on us using $L$ rather than $V$. It just lines up more smoothly with the story of the proof. I’m grateful to Jason Chen for drawing my attention to it. To demonstrate this, we prove a claim that suffices for the proposition.

Claim*.*

(1) If $A\in[\omega_{1}]^{\omega_{1}}\cap V[H]$, then there is a countably infinite $X\subseteq A$ in $L$.

(2) If $G$ is $\mathbb{C}_{\omega_{1}}$-generic over $L$, then for some $A\in[\omega_{1}]^{\omega_{1}}\cap L[G]$ there is no countably infinite $X\subseteq A$ in $L$.

To see that this is sufficient, suppose toward a contradiction that $G\in V[H]$ is $\mathbb{C}_{\omega_{1}}$-generic over $L$. Then by (1) we may fix $A\in[\kappa]^{\kappa}\cap V[G]$ which has no countably infinite subset in $L$. But then we see that $A\in V[G]\subseteq V[H]$ and so by (2) $A$ must have a countable infinite subset, which is impossible.

To prove (1), we make an argument that exploits the version of countable completeness possessed by the second domain of $\mathbb{S}_{\omega_{1}}$.232323The argument is a simple generalization of the standard argument that forcings closed under $\omega$-sequences don’t add any new $\omega$-sequences. See, for example, Theorem VII.6.14 in (Kunen, 2006). We suppose toward a contradiction that $L$ contains no injection from $\omega$ into $A$. We continue by letting $\dot{A}\in L^{\mathbb{S}_{\omega_{1}}}$ be a name such that $\dot{A}_{H}=A$ and fixing $p\in H$ that forces the statement: there is is no injection $f:\omega\to\dot{A}$ with $f\in\check{V}$. Working in $L$, we then define sequences $\langle p_{n}\rangle_{n\in\omega}$ and $\langle\alpha_{n}\rangle_{n\in\omega}$ such that $p_{0}\leq p$ and for all $n\in\omega$:

• •

$p_{n}\Vdash\alpha_{n}\in\dot{A}$;

• •

$\alpha_{n+1}>\alpha_{n}$;

• •

$p_{n+1}\leq p_{n}$; and

• •

$dom(p_{n})=m\times Y$ where $m\in\omega$ is least among the second domains of conditions satisfying the first three bullet points.

We claim that there is some $k\in\omega$ such that for all $n\in\omega$, if $m$ is the second domain of $p_{n}$, then $m\leq k$. To see this, we move back to $V$ and observe that since $|A|=\kappa$ and there are only countably many possibilities for the first domains of conditions, there must be some $k\in\omega$ such that there are infinitely many $\alpha\in A$ with some $p\in H$ with $p\Vdash\alpha\in\dot{A}$ where the second domain of $p$ is $<\kappa$. Since we have a bound on the first domain, it can be seen that $q=\bigcup_{n\in\omega}p_{n}\in\mathbb{S}_{\omega_{1}}$. Finally, let $J$ be $\mathbb{S}_{\omega_{1}}$-generic over $L$ with $q\in J$. Then we see that $\langle\alpha_{n}\rangle_{n\in\omega}$ is an injection from $\omega$ into $A$ that is an element of $L$, which gives a contradiction completing our proof of (1).

To prove (2), we provide a $\mathbb{C}_{\omega_{1}}$-name $\dot{A}$ for a set $A$ whose realization has no countably infinite subset in $L$. More specifically, we let

[TABLE]

and $A=\dot{A}_{G}$. The see the idea behind this name, let $g=\bigcup G:\omega\times\omega_{1}\to\omega_{1}$ and note that $\alpha\in\dot{A}_{G}$ iff $g(0,\alpha)=1$. So we are restricting our attention to how conditions $p\in\mathbb{C}_{\omega_{1}}$ act on [math] in the first domain and we only care about whether the output is $1$ or not. This gives us a name for an element of $[\omega_{1}]^{\omega_{1}}\cap L[G]$.

Next we claim that $A$ is $Add(\omega_{1},1)$-generic over $V$. To see this, we recall a basic, abstract forcing fact. Let us say that $\sigma:\mathbb{P}\to\mathbb{Q}$ is weakly dense if $\sigma$ is order preserving and for all $p\in\mathbb{P}$ and $q\in\mathbb{Q}$ with $q\leq\sigma(p)$, there is some $p^{*}\in\mathbb{P}$ such that $p\not\perp p^{*}$ and $\sigma^{*}(p)\leq q$. It is well-known then that if $J$ is $\mathbb{P}$-generic over $V$, then the upward closure of $\sigma``J$ is $\mathbb{Q}$-generic over $V$.242424See Lemma 15.45 in (Jech, 2003) for a proof. Now let us construe $Add(\omega_{1},1)$ as the poset whose conditions $q$ are finite partial functions from $\omega_{1}$ to $2$; and let $\sigma:\mathbb{C}_{\omega_{1}}\to Add(\omega_{1},1)$ be such that for all $p\in\mathbb{C}_{\omega_{1}}$, $\sigma(p)(\alpha)=1$ iff $p(0,\alpha)=1$.252525This is not strictly correct. Since $p(0,0)$ has no possible output and $p(0,1)$ can only output [math], we can’t just use $\alpha$ on both sides. Fixing this makes a simple idea ugly on the page, so I’ll stick with the sloppiness. It is, however, easily addressed and we leave that to the reader. It can be seen that $\sigma$ is a weakly dense embedding and so $J=\sigma``G$ is $Add(\omega_{1},1)$-generic over $L$. Moreover, it is not difficult to see that $A=\bigcup J$. Finally, suppose toward a contradiction that $X\in L$ and $X\subseteq A$. Now let $\dot{J}$ be the canonical name for a $Add(\omega_{1},1)$-generic set.262626Here we mean that $\dot{J}=\{\langle\check{q},q\rangle\ |\ q\in\mathbb{Q}\}$. Then we may fix $q\in J$ such that $q\Vdash X\subseteq\bigcup\dot{J}$. But this means that we have a finite condition $q\in Add(\omega_{1},1)$ determining infinitely many membership facts about $\bigcup\dot{J}$, which is impossible. ∎

Taking a little stock, we’ve seen that while standard models of $MV$ are common, they are not ubiquitous. We should also note that in the models of $MV$ above that contained uncountable transitive worlds, $W$, the ordinals of those worlds were identical to $\omega_{1}$. We might then wonder if we can have models of $MV$ containing worlds with a longer sequence of ordinals. This is not the case.272727This was observed by Usuba in the remarks following Definition 7.16 of (Usuba, 2017). To see this, we start by noting that the “external” cardinality of any pair of sets in some $\mathcal{W}$ must be the same.

Proposition 11**.**

If $\mathcal{W}$ is a model of $MV$ then for all $x,y$ in the set domain of $\mathcal{W}$

[TABLE]

Proof.

First note that since $MV$ implies $ZFC_{count}^{-}$, if $x$ is any element of the set domain of $\mathcal{W}$, then there is some $f_{x}$, also in $\mathcal{W}$’s set domain, such that $\mathcal{W}\models f_{x}:\omega\cong x$. It is then clear that $f$ can be pulled back into $V$ to obtain a bijection

[TABLE]

Thus, $f_{y}^{*}\circ f_{x}^{*-1}$ witnesses the required claim. ∎

The nonstandardness of the worlds of such a model then follows easily.

Corollary 12**.**

If $\mathcal{W}$ is a model of $MV$ in which there is a world $\mathcal{M}$ and some $x\in\mathcal{M}$ such that

[TABLE]

is uncountable, then $\mathcal{W}$ is not an $\omega$-model.

Proof.

Let $x$ be a member of the set domain of $\mathcal{W}$ be such that its $\in^{\mathcal{W}}$-members are uncountable. Then using the proof of Proposition 11, we may fix a bijection $f_{x}^{*}$ between $\omega^{\mathcal{W}}$ and the $\in^{\mathcal{W}}$-members of $x$. This mean $\omega^{\mathcal{W}}$ is uncountable and thus $\mathcal{W}$ cannot be an $\omega$-model. ∎

3. When $MV$ is not internally categorical

In this section, we show $MV$ in its standard axiomatization is not internally categorical. The idea behind the proof is very simple. We are going to take a standard model $\mathcal{W}$ of $MV$ and some world $U$ from $\mathcal{W}$. We are then going to take a class generic extension $U[G]$ of $\mathcal{W}$ that cannot be accessed from $U$ by set forcing but which is also a world in a multiverse with the same set domain as $\mathcal{W}$.

Theorem 13**.**

There exist distinct models $\mathcal{W}_{0},\mathcal{W}_{1}$ of $MV$ whose set domains are identical, assuming there is a countable transitive model of $ZFC$.

Proof.

Our goal is to describe a pair of models of $MV$ whose set domains are identical. Let us start with a countable transitive model $M$ of $ZFC$. Let $\mathbb{Q}\in M$ be $M$’s version of the Easton support product of forcings that add a Cohen subset to every regular cardinal.282828This is also the forcing used by Reitz to show that there is a model of $ZFC$ that doesn’t satisfy the Bedrock axiom. See Theorem 5.3 in (Reitz, 2007). Now let $H$ be $\mathbb{Q}$-generic over $M$ and observe that this is a class generic extension of $M$ for which there can be no set generic extension $M[g]$ of $M$ with $M[g]=M[H]$. Next, we let $J$ be $\mathbb{C}_{Ord^{M}}$-generic over $M[H]$. Thus, we have

[TABLE]

We plan to show that there is some $G$ that is $\mathbb{C}_{Ord^{M}}$-generic over $M$ with $M[G]=M[H\times J]$. To see that this will suffice, note that this will give us two standard models $M^{G}$ and $M[H]^{J}$ of $MV$ with the same set domain. However, their world domains cannot be the same since $M[H]$ is not a world in $M^{G}$.

To prove our target claim, we shall obtain the required set $G$ from a generic filter on a class poset $\mathbb{P}^{*}$ over $M[H\times J]$ that will assemble $G$ through a sequence of approximations. More specifically, we let $\mathbb{P}^{*}$ consist of sets $g\in M[H\times J]$ that are $\mathbb{C}_{\alpha}$-generic over $M$ for some $\alpha\in Ord^{M}$, and we order $\mathbb{P}^{*}$ by reverse inclusion.292929A similar forcing is used in the proof of Theorem 3.1.5 in (Larson, 2004). Indeed, it can be used to give a more elegant proof of Theorem 27 in (Maddy and Meadows, 2020). I learned this from Gabe Goldberg. Now let $G^{*}$ be $\mathbb{P}^{*}$-generic over $M[H\times J]$ and let $G=\bigcup G^{*}$. First, we claim that $G$ is $\mathbb{C}_{Ord^{M}}$-generic over $M$. To see this let $A\subseteq\mathbb{C}_{Ord^{M}}$ be a maximal anti-chain that is definable in $M$. It can be seen that $\mathbb{C}_{Ord^{M}}$ is $Ord$-cc in $M$ and so there must be some $\alpha\in Ord^{M}$ such that $A\subseteq\mathbb{C}_{\alpha}$.303030This can be established with a trivial generalization of the proof of first lemma on page 15 of (Solovay, 1970). Thus, it will suffice to show that there is some $g\in G^{*}$ where $g\cap\mathbb{C}_{\alpha}\neq\emptyset$. To see this, we exploit the genericity of $G^{*}$ and make a density argument. In particular, we let

[TABLE]

and show that $D$ is pre-dense in $\mathbb{P}^{*}$. To see this, let $h\in\mathbb{P}^{*}$ and suppose $h$ is $\mathbb{C}_{\beta}$-generic over $M$ where $\beta\in Ord^{M}$. If $\beta\geq\alpha$, we would be done, so suppose $\beta<\alpha$. It will suffice to show that for some regular cardinal $\kappa>\alpha$ there is some $h^{*}\in M[H\times J]$ that is $\mathbb{C}_{[\beta,\kappa+1)}$-generic over $M[h]$. Let us start by fixing a $(\mathbb{Q}\times\mathbb{C}_{Ord^{M}})$-name, $\dot{h}$, such that $\dot{h}_{H\times J}=h$. Now since $\dot{h}$ is just a set it is clear that its transitive closure can only contain a set of conditions from $\mathbb{C}\times\mathbb{C}_{Ord^{M}}$. Thus, we may fix some $\gamma\in Ord^{M}$ such that $h=\dot{h}_{(H\times J)\restriction\gamma}$ and so $h\in M[(H\times J)\restriction\gamma]$. Now let $\kappa$ be a regular $M$-cardinal $>\gamma,\alpha$. It is clear that $J_{\{\kappa\}}$ is $\mathbb{C}_{\{\kappa\}}$-generic over $M[(H\times J)\restriction\gamma]$.313131We write $H\restriction\alpha$ to denote those $p\in H$ with $dom(p)\subseteq\alpha$. Moreover, it can be seen that $\mathbb{C}_{\{\kappa\}}$ is forcing equivalent323232See Proposition 10.20 and and the proof of Proposition 10.21 in (Kanamori, 2003) for more details. to $\mathbb{C}_{[\beta,\kappa+1)}$ and so we may fix $h^{*}$ that is $\mathbb{C}_{[\beta,\kappa+1)}$-generic over $M[(H\times J)\restriction\gamma]\supseteq M[h]$ as required.

Finally, we claim that $M[G]=M[H\times J]$. Clearly, $M[G]\subseteq M[H\times J]$ so we want to show that every $x\in M[H\times J]$ is in $M[G]$. To do this, we again exploit the genericity of $G^{*}$ by showing that if $x\in M[H\times J]$ then

[TABLE]

is dense in $\mathbb{P}^{*}$. To see this, let $\dot{x}$ be a $(\mathbb{Q}\times\mathbb{C}_{Ord^{M}})$-name in $M$ such that $\dot{x}_{(H\times J)}=x$. Since $\dot{x}$ is just a set, we may fix, as in the previous claim, some $\alpha\in Ord^{M}$ such that $\dot{x}_{(H\times J)\restriction\alpha}=x$. Moreover, we may assume without loss of generality that $\alpha=\kappa+1$ for some regular $M$-cardinal. Then it can be seen that $\mathbb{C}_{\{\kappa\}}$, $\mathbb{C}_{\kappa+1}$ and $(\mathbb{Q}\times\mathbb{C}_{Ord^{M}})\restriction(\kappa+1)$ are all forcing equivalent to each other. Thus, there is some $g$ that is $\mathbb{C}_{\kappa+1}$-generic over $M$ with $x\in M[g]$. ∎

We shall now use this fact to establish that $MV$ is not internally categorical. However, the underlying idea is quite obvious. The models given in the theorem above are not isomorphic, even though they share a set domain. Before we state the claim, let us first described our target theory. Let $\mathcal{L}_{MV}(\in_{0},\in_{1})$ be the language with a set sort and a world sort, but which has two membership relations that behave as in $\mathcal{L}_{MV}$. Let $MV(\in_{0},\in_{1})$ be the theory consisting of $\in_{0}$ and $\in_{1}$ versions of every axiom of $MV$ where we allow formulae involving both $\in_{0}$ and $\in_{1}$ into both $\in_{0}$ and $\in_{1}$ versions of all schemata.

Corollary 14**.**

$MV(\in_{0},\in_{1})$* is not internally categorical, if there is a countable transitive model of $ZFC$.*

Proof.

Using the proof of Theorem 13, let us start with a countable transitive model $M$ of $ZFC$, but this time suppose that $M$ satisfies $V=L$. We may then obtain standard models $\mathcal{W}_{0}\neq\mathcal{W}_{1}$ of $MV$ with the same world domain whose worlds are countable transitive models satisfying $ZFC$. In particular, $\mathcal{W}_{1}$ is generated from a world $N$ that is obtained from $M$ by adding a Cohen subset to every $M$-regular cardinal. It can be seen that no world in $\mathcal{W}_{1}$ satisfies the Ground Axiom.333333See Theorem 5.3 in (Reitz, 2007) for a proof.

We now define a model $\langle\mathbb{W},\mathbb{D},\in_{0},\in_{1}\rangle$ of $MV(\in_{0},\in_{1})$. We let the set domain $\mathbb{D}=\bigcup\mathcal{W}_{0}=\mathcal{W}_{1}$. It is clear that $\mathcal{W}_{0}$ and $\mathcal{W}_{1}$ are both countable, so we may fix a bijection $\sigma:\mathcal{W}_{1}\to\mathcal{W}_{0}$. Let $\mathbb{W}=\mathcal{W}_{0}$. We then let $\in_{0}$ be $\in$ restricted to $\bigcup\mathcal{W}_{0}$. And for $\in_{1}$, we let: $x\in_{1}y$ iff $x\in y$ where $x$ and $y$ are both sets; and we let $x\in_{1}M$ iff $x\in\sigma(M)$ when $x$ is a set and $M$ is a world. Since $\in_{0}$ and $\in_{1}$ are the same on the set domain, it is easy to see that having both $\in_{0}$ and $\in_{1}$ available to the (relativized) Collection and Separation schemas, used in $MV$, makes no difference. Thus, $\langle\mathbb{W},\mathbb{D},\in_{0},\in_{1}\rangle$ satisfies $MV(\in_{0},\in_{1})$. Finally, to see that there is no definable isomorphism between $\mathcal{W}_{0}=\langle\mathbb{W},\mathbb{D},\in_{0}\rangle$ and $\mathcal{W}_{1}=\langle\mathbb{W},\mathbb{D},\in_{1}\rangle$ it suffices to observe that they are not even elementary equivalent. To see this we note that some world in $\mathcal{W}_{0}$ satisfies the Ground Axiom since $\mathcal{W}_{0}$ is generated from a model $M$ satisfying $V=L$. But we’ve already seen that no world in $\mathcal{W}_{1}$ satisfies the Ground Axiom. ∎

Thus, we see that $MV$ is not internally categorical, but it is natural to wonder what would happen if we augmented $MV$ with axioms demanding that certain large cardinals exist at some, or perhaps, all worlds.343434See Section 5 of (Steel, 2014) for some discussion of this. This can be quite attractive since, if every world contains a Woodin cardinal, then it can be seen that every world must contain a proper class of them. Moreover, if there is a proper class of Woodin cardinals in every world then the theory of $L(\mathbb{R})$ and analysis will be the same in every world.353535See Chapter 3 of (Larson, 2004) for a proof of this. Beyond the issue of whether it is interesting to add large cardinals to $MV$, it is also known that large cardinals can exert a powerful influence on the structure of the models of $MV$. In particular, Usuba (2017) has shown that if a world has an extendible cardinal, then the multiverse has a core, or bedrock; i.e., there is a world that is a submodel of every other world and every other world is a generic extension of it. This could give us reason to doubt that the failure internal categoricity will be preserved into strengthenings of $MV$. These doubts appear to be ill-founded although some work is required to show this.

When we argued that $MV$ is not internally categorical, it sufficed to prove that we could have distinct models of $MV$ with the same world domain. However, the proof of this relied on a class forcing argument. We started with a model of $ZFC$ and added a class of Cohen sets. This forcing trivially preserves $ZFC$, but some remarks are warranted with regard to the preservation of large cardinal assumptions. This is not a small question, but rather overlaps an entire subfield of set theory concerned with the preservation and indestructibility of large cardinal axioms through forcing.363636See Cummings’ chapter in (Foreman and Kanamori, 2009) for an introduction to this topic. As such, we shall content ourselves with small sample of large cardinal axioms that are intended to cut across a wide swathe of consistency strengths.

Lemma 15**.**

Consider the theories obtained by expanding $MV$ with each of the following axioms:

1. (1)

There is a world with an inaccessible cardinal; 2. (2)

There is a world with a measurable cardinal; 3. (3)

There is a world with a supercompact cardinal; and 4. (4)

There is a world with an extendible cardinal.

Assuming sufficient large cardinals, each of these theories has a pair of distinct models $\mathcal{W}_{0}\neq\mathcal{W}_{1}$whose set domains are identical.

For the last two of these theories, we make use of the following facts about supercompact and extendible cardinals.

Fact 16**.**

(Reitz, 2007)* If $\kappa$ is a supercompact cardinal, then373737This can be proved using two moves from (Reitz, 2007). Starting in $V$ with a supercompact cardinal, $\kappa$, we use the proof of Theorem 3.9 to class force to a model $V[G]$ where the continuum coding axiom holds and $\kappa$ remains supercompact and is indestructible by $<\kappa$-directed forcing. Then we the proof of Theorem 5.5 to add a Cohen subset to every regular $\lambda$ such that $2^{<\lambda}=\lambda$. It can then be seen via the indestructibility that the supercompactness of $\kappa$ is preserved by this extension. there is a class forcing extension $V[G]$ in which $\kappa$ is supercompact and if we class force over $V[G]$ to add a Cohen subset to every regular cardinal $\lambda$ in $V[G]$ where $2^{<\lambda}=\lambda$, then $\kappa$ remains supercompact in that extension.*

Fact 17**.**

(Bagaria and Poveda, 2023)* Suppose Vopenka’s principle holds. Then there is an extendible cardinal $\kappa$. Moreover,383838This is essentially a suboptimal version Corollary 5.13 in (Bagaria and Poveda, 2023). The assumptions of the Corollary follow from Theorems 6.1-3 as well as Theorem 2.4 and Corollary 2.6 in (Bagaria and Poveda, 2023). if $G$ is generic for the Easton support forcing adding a Cohen subset to every regular cardinal, then $\kappa$ remains extendible in $V[G]$.*

Proof.

(1) The argument used in Theorem 13 suffices since the class forcing adding Cohen subsets to each regular cardinal preserves inaccessible cardinals.

(2) The argument from Theorem 13 can be adapted by only adding Cohen subsets of regular cardinals greater than the least measurable cardinal of our initial model $M$.

(3) Suppose that $\kappa<\lambda$ are supercompact and inaccessible respectively. Then take the collapse of a countable Skolem hull to obtain a countable transitive model $M$ that things some $\bar{\kappa}$ is supercompact. Now using Fact 37, we may adapt the argument of Theorem 13 to deal with a forcing that only adds Cohen sets to regular cardinals $\delta$ such that $2^{<\delta}=\delta$.

(4) Suppose that $\lambda$ is a Vopenka cardinal. Then take the collapse of a countable Skolem hull of $V_{\lambda}$ to obtain a countable transitive model $M$ satisfying Vopenka’s principle. Let $\kappa\in M$ be such that $M$ thinks $\kappa$ is extendible. Fact 38, then ensures that the class forcing used in Theorem 13 preserves the extendibility of $\kappa$. The rest of that argument remains the same.393939This argument will also work for the theory where every world has an extendible cardinal and thus every world has a proper class of extendible cardinals. ∎

Corollary 18**.**

None of the theories considered in Lemma 15 are internally categorical.

Thus, in its standard axiomatization we have seen that $MV$ and some its large cardinal variants are not internally categorical. Intuitively, this probably doesn’t come as a surprise, however, there is value in thinking about how we proved this. In a nutshell, the trick was to find a pair of countable transitive models of $ZFC$ that share a $\mathbb{C}_{Ord}$-collapse but where one is a (proper) class forcing extension of the other. Reflecting on this issue motivates a modified axiomatization of $MV$ that avoids this problem.

4. When $MV$ is internally categorical

We are not reading to prove the more surprising result. There is an alternative axiomatization $MV^{*}$ of $MV$ that *is *internally categorical. Let us begin by introducing the new axiom schemata.

• (Global-$ZFC_{count}^{-}$)

$\varphi$ for every axiom of $ZFC_{count}^{-}$.

• (World-domination)

If $N$ is a definable inner model then there is a world $W$, such that $N\subseteq W$.

We already saw above that Global-$ZFC_{count}^{-}$ follows from $MV$. So – at least from a mathematical perspective – there is no harm in adding it to $MV$. World-domination is more complicated. Informally, it tells us that if you can define an inner model $N$ of $ZFC$ using $\mathcal{L}_{MV}$, then there will be a world $W$ that dominates in the sense that $W\supseteq N$. Let us call the theory obtained by augmenting $MV$ with the schemata above $MV^{*}$. We shall show first that $MV^{*}$ is internally categorical and then we will show that $MV^{*}$ is an alternative axiomatization of $MV$ by demonstrating that $MV^{*}$ follows from $MV$.

We shall begin by establishing that $ZFC_{count}^{-}$ is internally categorical. This may be of independent interest to those with countabilist sympathies. However, it will also be helpful for our main proof since we have the Global-$ZFC_{count}^{-}$ schema. We let $ZFC_{count}^{-}(\in_{0},\in_{1})$ be the theory articulated in $\mathcal{L}(\in_{0},\in_{1})$ that is obtained by adding $\in_{0}$ and $\in_{1}$ versions of every axiom of $ZFC_{count}^{-}$ and by allowing formulae involving both $\in_{0}$ and $\in_{1}$ into both $\in_{0}$ and $\in_{1}$ versions of the Separation and Collection schemata.

Theorem 19**.**

$ZFC_{count}^{-}(\in_{0},\in_{1})$* is internally categorical; i.e., we can prove in $ZFC_{count}^{-}(\in_{0},\in_{1})$ that there is a definable isomorphism between $\in_{0}$ and $\in_{1}$.*

Proof.

We work informally in $ZFC_{count}^{-}(\in_{0},\in_{1})$. Roughly speaking, our plan is to mutually collapse $\in_{0}$ and $\in_{1}$ within each other, then repeat the collapses and observe that there is no room in between for difference. Let us write $\mathcal{V}_{0}$ for $\langle V,\in_{0}\rangle$ and $\mathcal{V}_{1}$ to denote $\langle V,\in_{1}\rangle$ where $V$ is the underlying universe.404040Strictly speaking, I cannot refer to order $\langle V,\in_{0}\rangle$ within the framework of $ZFC_{count}^{-}(\in_{0},\in_{1})$ since it is an ordered pair of proper classes. It is easy enough to eliminate this abuse of notation, but it makes things less pleasant on the page, so we shall continue to talk in this way. Note that using the Set Induction schema, it is easy to see that $\mathcal{V}_{0}$ thinks $\in_{1}$ is well-founded and that $\mathcal{V}_{1}$ thinks that $\in_{0}$ is well-founded. Given this, we may fix collapse maps:

• •

$\pi_{1}:\mathcal{V}_{1}\cong\mathcal{M}_{1}=\langle M_{1},\in_{0}\rangle$; and

• •

$\pi_{0}:\mathcal{V}_{0}\cong\mathcal{M}_{0}=\langle M_{0},\in_{1}\rangle$.

Thus, $\mathcal{M}_{1}$ is the result of collapsing $\mathcal{V}_{1}$ in $\mathcal{V}_{0}$ and $\mathcal{M}_{0}$ is the result of collapsing $\mathcal{V}_{0}$ in $\mathcal{V}_{1}$. Since these are isomorphisms, we can then, so to speak, find a copy of $\mathcal{V}_{1}$ inside $\mathcal{M}_{0}$. More formally, we have:

• •

$\pi_{1}\circ\pi_{0}``\mathcal{V}_{0}=\pi_{1}``\mathcal{M}_{0}\subseteq\mathcal{M}_{1}\subseteq\mathcal{V}_{0}$; and

• •

$\pi_{0}\circ\pi_{1}``\mathcal{V}_{1}=\pi_{0}``\mathcal{M}_{1}\subseteq\mathcal{M}_{0}\subseteq\mathcal{V}_{1}$.

Now if things worked out particularly nicely, we’d see that $\pi_{1}\circ\pi_{0}``\mathcal{V}_{0}$ was the collapse of $\mathcal{V}_{0}$ and thus was $\mathcal{V}_{0}$ itself. This would essentially complete the argument. However, for all we know one of $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ is, so to speak, shorter than the other and this means that the collapse might not reach every element of the other model. We get around this by noting that, according to both $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$, since every set is countable, every set can be coded by a set of natural numbers. It will, thus suffice to show that the $\pi_{0}\circ\pi_{1}``\mathcal{V}_{0}$ and $\mathcal{V}_{0}$ have the same powerset of the naturals.

To show this, we start by reducing our notational burden by letting $\mathcal{N}_{0}=\pi_{0}\circ\pi_{1}``\mathcal{V}_{0}$. It will then suffice show that $\mathcal{P}(\omega)^{\mathcal{V}_{0}}\subseteq\mathcal{N}_{0}$ since the other direction is trivial. Let’s start with some $x\in\mathcal{P}(\omega)^{\mathcal{V}_{0}}$.414141Note that it is not strictly correct to speak of $\mathcal{P}(\omega)^{\mathcal{V}_{0}}$ since $\mathcal{V}_{0}$ contains no set of all subsets of its $\omega$. However, we can instead regard $\mathcal{P}(\omega)^{\mathcal{V}_{0}}$ as the definable class of subsets of $\mathcal{V}_{0}$’s $\omega$. It should be clear that $\langle trcl(\{x\})^{\mathcal{V}_{0}},\in\rangle$ is set-like according to $\mathcal{V}_{1}$ and so $\pi_{0}(x)$ is defined. From here it is can be seen that $\pi_{1}\circ\pi_{0}(x)=x\in\mathcal{N}_{0}$ as required. To close things out, note that – after unpacking the coding – this implies that $\mathcal{N}_{0}=\mathcal{V}_{0}$ and so

[TABLE]

which means that $\mathcal{V}_{0}=\mathcal{M}_{1}$ and thus, $\pi_{1}:\mathcal{V}_{1}\cong\mathcal{V}_{0}$ as required. ∎

4.1. $MV^{*}$ is internally categorical

We are now ready for the main result. Let $MV^{*}(\in_{0},\in_{1})$ be the theory articulated in $\mathcal{L}_{MV}(\in_{0},\in_{1})$ that consists of $\in_{0}$ and $\in_{1}$ versions of every axiom of $MV^{*}$ where we allow formulae involving both $\in_{0}$ and $\in_{1}$ into both $\in_{0}$ and $\in_{1}$ versions of all schemata.

Theorem 20**.**

$MV^{*}(\in_{0},\in_{1})$* is internally categorical; i.e., we can prove in $MV^{*}(\in_{0},\in_{1})$ that there is a definable bijection on the set domain $\sigma$ witnessing an isomorphism between $\in_{0}$ and $\in_{1}$ where $\sigma$ also determines an isomorphism on the world domain.*

Proof.

We work in $MV^{*}(\in_{0},\in_{1})$. Let $\mathcal{V}_{0}=\langle V,\in_{0}\rangle$ and $\mathcal{V}_{1}=\langle V,\in_{1}\rangle$. By Fact 4, we see that both $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$ satisfy $ZFC_{count}^{-}$. Thus, we may use Theorem 19, to obtain a definable isomorphism $\sigma:\mathcal{V}_{0}\cong\mathcal{V}_{1}$ as we require. This leaves the world domain. For worlds $\in_{0}$-worlds, $W$, we let $\sigma_{+}(W)=\sigma``W$. It will suffice to show that $\sigma_{+}$ is a bijection. There are two things that might go wrong: (1) $\sigma_{+}(W)$ might not be a world with respect to $\in_{1}$; and (2) $\sigma_{+}$ might not be a surjection.

Regarding (1), we see using ($*$) for $\in_{1}$ that there an $\in_{1}$-world $U$ such that $\sigma_{+}(W)\subseteq U$. Moreover, using ($*$) for $\in_{0}$, we also see that there is an $\in_{0}$-world $W^{\dagger}$ such that $U\subseteq\sigma_{+}(W^{\dagger})$. By the intermediate submodel theorem, in $\sigma_{+}(W^{\dagger})$, we see that $\sigma_{+}(W)\subseteq U\subseteq\sigma_{+}(W^{\dagger})$ is a sequence of generic extensions. This means $\sigma_{+}(W)$ is a generic refinement of $U$ and so $\sigma_{+}(W)$ is an $\in_{1}$-world as we required.

Regarding (2), we use a similar argument. Given an arbitrary $\in_{1}$-world $U$, it will suffices to show that $\sigma_{+}^{-1}(U)$ is an $\in_{0}$-world. To see this, we use ($*$) for $\in_{0}$, to fix an $\in_{0}$-world $W$ such that $\sigma_{+}^{-1}(U)\subseteq W$ and so $U\subseteq\sigma_{+}(W)$. Then using ($*$) for $\in_{1}$, we may fix an $\in_{1}$-world $U^{\dagger}$ such that $\sigma_{+}(W)\subseteq U^{\dagger}$. As in the previous case, this implies that $\sigma_{+}(W)$ is a generic extension of $U$ and thus, that $W$ is a generic extension of $\sigma_{+}^{-1}(U)$ and so $\sigma_{+}^{-1}(U)$ is an $\in_{0}$-world as we require. ∎

4.2. $MV$ implies $MV^{*}$

To complete our demonstration that an axiomatization of $MV$ is internally categorical, we need to prove that $MV$ implies $MV^{*}$. To do this, we need to show that:

1. (1)

World-domination is expressible in $\mathcal{L}_{MV}$; and 2. (2)

World-domination follows from $MV$.

4.2.1. World-domination is expressible

To motivate why there might even be a problem here, let us recall that the naive statement of the ordinary axioms of $MV$ is not obviously expressible in $\mathcal{L}_{MV}$. For example, the Refinement axiom is often stated as follows: “If $U$ is a world, and $U=W[G]$, where $G$ is $\mathbb{P}$-generic over $W$, then $W$ is a world” (Steel, 2014). While this conveys the idea very clearly, it’s not so clear where $W$ came from. We appear to be quantifying over it like a world before we even know that it is a world. The problem is resolved by using Fact 3, which allows for the “official” formulation given in Section 2.2. As stated above, our formulation of World-domination has similar problems. It appears to quantify over inner models, but in fact, it is more correctly and much more formally stated as a schema as follows:

• (World-domination)

For all sets $y_{0},...,y_{n}$ and for all worlds $U_{0},...,U_{m}$, If

[TABLE]

is a formula of $\mathcal{L}_{MV}$ that defines an inner model of $ZFC$, then there is some world $W$ such that $\forall x(\varphi(x)\to x\in W)$.

By referring to formulae we are able to remove any quantification over inner models. However, we still need to know when a particular formula defines an inner model of $ZFC$. This can be addressed with a classic fact.

Fact 21**.**

*($ZFC^{-})$ A transitive class $M$ is an inner model of $ZF$ iff it is closed under the Gödel operations and is almost universal; i.e., for every subset $X\subseteq M$ there is some $Y\in M$ with $Y\supseteq X$.424242For a proof see Theorem 13.9 in (Jech, 2003). The proof there is conducted in $ZFC$ rather than $ZFC^{-}$, however, the proof is the same. The only (cosmetic) modification occurs in establishing almost universality of an inner model, $M$ of $ZFC$. We start with $X\subseteq M$, and then obtain that $X\subseteq V_{\alpha}^{M}\in M$ rather than $X\subseteq V_{\alpha}\cap M$ since, in general, we cannot guarantee that $V_{\alpha}$ exists in $ZFC^{-}$. *

It is easy to see that being almost universal and closed under the Gödel operations is expressible in $\mathcal{L}_{\in}$. Then using this and the fact that $MV$ implies $ZFC^{-}$, we can reformulate World-domination by demanding that the class of $x$ such that $\varphi(x,y_{0},...,y_{n},U_{0},...,U_{m})$ is closed under the Gödel operations and is almost universal. This is easily expressed in $\mathcal{L}_{MV}$ and so World-domination is expressible as a schema in $\mathcal{L}_{MV}$.

4.2.2. World-domination follows

Now we move onto our final task by showing that World-domination follows from $MV$. Our strategy is to exploit the fact, explored earlier, that models of $MV$ are closely related to models like Solovay’s where an inaccessible cardinals is collapsed to be $\omega_{1}$. The key idea of this proof was provided by Gabe Goldberg and I’m grateful to him for letting me use it here. We work our way gently toward our target by starting with a warmup lemma, which contains the main idea of the proof. We then use this and some lemmas from Steel and Goldberg to push the result into less familiar setting of the generic multiverse, which then gives us that result we require.

Lemma 22**.**

Suppose $G$ is $\mathbb{C}_{Ord}$-generic over $V$ and that $M\subseteq V[G]$ is an inner model of $ZFC$ that is definable in $V[G]$ using $V$ as a parameter and parameters from $V$. Then $M\subseteq V$.

Proof.

It will suffice to show that every set of ordinals in $M$ is also in $V$. To show this, we let $\lambda$ be an arbitrary ordinal and let $Z=\mathcal{P}(\lambda)^{M}$. Since $\lambda$ is arbitrary, it will suffice to show that $Z\subseteq V$. Our plan is to exploit the homogeneity of $\mathbb{C}_{Ord}$ and chase a contradiction regarding the cardinality of $Z$. The proof closely follows the argument used by Solovay to establish that perfect set property in his well-known model. Suppose toward a contradiction that $Z\nsubseteq V$ and fix some $y\in V\backslash Z$ witnessing this.434343A similar argument can be found in the proof of Theorem 1.1 in (Kanovie and Lyubetsky, 2017). By standard facts about collapse forcing, we may fix $\alpha<\kappa$ sufficiently large such that:

1. (1)

$y\in Z\in V[G\restriction\alpha]$; and 2. (2)

$Z$ is countable in $V[G\restriction\alpha]$.

Using the homogeneity and absorption properties of $\mathbb{C}_{Ord}$ we may fix a formula $\varphi_{Z}(x)$ using $\lambda$ and elements of $V$ as parameters, which is such that for any $H$ that is $\mathbb{C}_{\alpha}$-generic over $V$

[TABLE]

for all $x\in V[H]$. Next we let $\dot{y}$ be a $\mathbb{C}_{\alpha}$-name such that $\dot{y}_{G\restriction\alpha}=y$. Then we may put things together and fix $p\in\mathbb{C}_{\alpha}$ such that:

[TABLE]

Using the first two conjuncts and noting that $y\subseteq\lambda\subseteq V$, it can be seen – using Solovay’s argument – that in $V[G]$, $Z$ contains a perfect set. But using the final conjunct, we see that $Z$ is countable in $V[G]$, which gives us our desired contradiction. ∎

We now recall an important fact about Steel’s generic multiverse.

Fact 23**.**

(Goldberg, Lemma 38 in (Maddy and Meadows, 2020)) There is a total computable function $e:\mathcal{L}_{MV}\to\mathcal{L}_{\in}(\dot{M})$ with the following property. Suppose $\mathcal{W}\models MV$, $M$ is a world of $\mathcal{W}$, and $N$ is the collection of sets of $\mathcal{W}$. Then

[TABLE]

where $\langle N,M\rangle$ denote a model of $\mathcal{L}_{\in}$ expanded by a one-place relation symbol.

We now use this fact to prove a special case of World-domination where only one world parameter is involved.

Lemma 24**.**

$MV$* proves the following. Suppose $M$ is an inner model of $ZFC$ that is definable using a single world parameter $U$ and parameters from $W$. Then $M\subseteq U$.*

Proof.

Let $\mathcal{W}$ be a model of $MV$ and without loss of generality, we suppose that $\mathcal{W}$ is countable. We show that the claim holds in $\mathcal{W}$. We start by letting $\varphi_{M}(x,\bar{y},U)$ be the formula defining $M$ in $\mathcal{W}$ where $U$ is a world parameter $\bar{y}$ is a finite sequence of set parameters from $U$. Now using Fact 5, we may fix a $\mathbb{C}_{Ord^{U}}$-generic $G$ such that $U[G]$ is identical to the set domain of $\mathcal{W}$. Then using Fact 23, we see that

[TABLE]

Thus, we see that $M$ is defined in $U[G]$ by a formula of $\mathcal{L}_{\in}$ using $U$ as a parameter and parameters from $U$. Finally, we use Lemma 22 to see that $M\subseteq U$, as required. ∎

Next, we use Amalgamation to prove the general case where multiple world parameters are involved.

Theorem 25**.**

$MV$* proves the following. Suppose $M$ is an inner model of $ZFC$ that is definable using set and world parameters. Then there is some world $W$ such that $M\subseteq W$.*

Proof.

Again, we show that the claim holds in arbitrary models of $MV$. We start by fixing a model $\mathcal{W}$ of $MV$ and we shall now work inside $\mathcal{W}$. Our plan is to use the Amalgamation axiom to reduce our problem to that addressed in the previous lemma. We suppose that $M$ is definable by the $\mathcal{L}_{MV}$-formula $\varphi(x,y_{0},...,y_{m},W_{0},...,W_{n},)$ where $y_{0},...,y_{m}$ are set parameters and $W_{0},...,W_{n}$ are world parameters. First, we let $W_{y_{i}}$ be a world with $y_{i}\in W_{y_{i}}$ for all $i\leq m$,. Then by repeated applications of the Amalgamation axiom, we may fix a world $W$ such that for all $j\leq n$, $W_{j}\subseteq W$ and for all $i\leq m$, $W_{y_{i}}\in W$. The observe by the Laver-Woodin theorem, each of these worlds is definable in $W$ from a parameter in $W$. Thus, $M$ is definable by a formula using $W$ and elements of $W$. Thus by Lemma 24, we see that $M\subseteq W$ as required. ∎

Finally, our main claim is complete.

Corollary 26**.**

$MV\vdash\text{World-domination}$.

4.3. A couple of applications in relative interpretablity

Before we move on to discuss these results, we first take a slight detour into the world of interpretation. As has been noted by Enayat and Łełyk (2024), categoricity properties are often closely related to important equivalence relations between theories. We shall see that $MV$ bears further witness to this close relationship. The following quite technical definition will allow us to set things up. For models $\mathcal{A}$ and $\mathcal{B}$ of some language $\mathcal{L}$ we shall write $\mathcal{A}\triangleright_{par}\mathcal{B}$ to indicate that $\mathcal{A}$ interprets $\mathcal{B}$ with parameters.444444See (Enayat and Łełyk, 2024) for a proper definition, but the essential idea is that the domain and relations on $\mathcal{B}$ are all definable in $\mathcal{A}$.

Definition 27**.**

Let $T$ be a theory articulated in some language $\mathcal{L}$. We say:454545I believe this terminology with the exception of (3) originates with (Enayat, 2016). I include (3) as a weakening since it provides an interesting warmup for the main claims.

1. (1)

$T$ is solid if whenever $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ are models of $T$ where $\mathcal{A}\triangleright_{par}\mathcal{B}\triangleright_{par}\mathcal{C}$ and $\sigma:\mathcal{A}\cong\mathcal{C}$ is definable in $\mathcal{A}$ from parameters, then there is $\tau:\mathcal{A}\cong\mathcal{B}$ that is also definable in $\mathcal{A}$ from parameters. 2. (2)

$T$ is tight if whenever $T_{0}$ and $T_{1}$ are bi-interpretable extensions of $T$, then $T_{0}$ and $T_{1}$ are the same theory in the sense that they have the same deductive closure. 3. (3)

$T$ is *weakly tight *if whenever $T_{0}$ and $T_{1}$ are definitionally equivalent extensions of $T$, then $T_{0}$ and $T_{1}$ are the same theory in the sense that they have the same deductive closure.

(1) implies (2) and (2) implies (3). 464646See Remark 9 in (Enayat and Łełyk, 2024). (1) is a technical, but very useful property. (2) and (3) seem to tell us something about the “stability” of a theory. Bi-interpretability and definitional equivalence are canonical equivalence relations that give us plausible accounts of when two theories might considered intuitively equivalent for some purposes. (2) and (3) then tell us that – something like – any pair of intuitively equivalent theories are actually identical. For some examples, we’ve already seen above that both $ZFC$ and $PA$ are internally categorical. It can be seen using very similar arguments that they are also both solid. With regard to $MV$ we can use what we already have above to show that it is weakly tight.

Corollary 28**.**

$MV$* is weakly tight; i.e., If $T$ and $S$ are definitionally equivalent extensions of $MV$, then $T$ and $S$ are the same theory in the sense that $T\dashv\vdash S$.*

Proof.

Suppose that $T$ and $S$ are extensions of $MV$ that are definitionally equivalent. To make things simpler, suppose that $T$ is articulated using $\in_{0}$ and $S$ is articulated using $\in_{1}$. Then there are definitional expansions $T^{+}$ and $S^{+}$ that add definitions of $\in_{1}$ and $\in_{0}$ respectively to $T$ and $S$ such that $T^{+}$ and $S^{+}$ have the same deductive closure.474747Here we are using the characterization of definitional equivalence that Lefever and Székely (2019) call definitional mergeability. While this is not always the same as definitional equivalence, it is the same when the languages of the theories being compared are disjoint. Hence, our move to $\in_{0}$ and $\in_{1}$. Call the deductive closure of these theories $U$. Since we can define $\in_{0}$ in terms of $\in_{1}$ and vice versa, this means that $U$ extends $MV^{*}(\in_{0},\in_{1})$. Thus, by Theorem 20, there is a definable isomorphism between the multiverses associated with $\in_{0}$ and $\in_{1}$. This means that in any model $\mathcal{A}$ of $U$, the reducts $\mathcal{A}\restriction\in_{0}$ and $\mathcal{A}\restriction\in_{1}$ of $\mathcal{A}$ to $\in_{0}$ and $\in_{1}$ respectively are elementary equivalent. Thus, since $T^{+}$ and $S^{+}$ are conservative extensions of $T$ and $S$, we see that they have the same theorems. ∎

In many interesting cases, tightness follows from weak-tightness. For some examples, Visser and Friedman (2014) have shown that for a wide class of theories, including $ZFC$, bi-interpretability implies definitional equivalence. More generally, Enayat and Łełyk (2024) have shown that for sequential theories that eliminate imaginaries, internal categoricity implies tightness.484848See Proposition 50 in (Enayat and Łełyk, 2024). See Section 4.4 of (Hodges, 1997) for a discussion of imaginary elimination. A sequential theory is defined in (Visser and Friedman, 2014), however, one might think of it as a particularly low bar for a foundational theory to meet and one that every theory in this paper traverses. $ZFC$ is such a theory. When a theory eliminates imaginaries, this means (roughly) that it can deliver representatives for any definable equivalence relation. In $ZFC$ this is easily achieved by using Scott’s trick by taking the set of elements of minimal rank from each cell of the equivalence relation. This technique crucially relies on the powerset axiom to deliver the relevant $V_{\alpha}$ to intersect. A such, it is not available in $ZFC_{count}^{-}$. Nonetheless, it is possible to show that $ZFC_{count}^{-}$ is solid as we now do. The original proof of this claim achieves is goal by drawing it as a corollary of the facts that: second order arithmetic is bi-interpretable with $ZFC_{count}^{-}$; second order arithmetic is tight; and tightness is preserved through bi-interpretations. The proof below is a little more direct and also harks back to our proof of Theorem m19. Thus, it seems worthwhile to include it.

Theorem 29**.**

(Enayat, 2016)* $ZFC_{count}^{-}$ is solid.*

Proof.

Suppose that $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are models of $ZFC_{count}^{-}$ such that $\mathcal{A}\triangleright_{par}\mathcal{B}\triangleright_{par}\mathcal{C}$ and there is some $\sigma:\mathcal{A}\cong\mathcal{C}$ that is definable in $\mathcal{A}$ from parameters. We claim that $\mathcal{A}$ can define with parameters some $\tau:\mathcal{A}\cong\mathcal{B}$. The argument is similar to our proof that $ZFC_{count}^{-}$ is internally categorical. Since $\mathcal{C}$ is definable in $\mathcal{B}$ we can take the transitive collapse $\mathcal{C}_{1}$ of $\mathcal{B}$ in $\mathcal{C}$ via the possibly partial function $\pi_{0}:\mathcal{C}\rightharpoondown\mathcal{C}_{1}$. At present, we don’t know that $\mathcal{C}$ and $\mathcal{C}_{1}$ are isomorphic since we don’t know that $\mathcal{C}$ is set-like in $\mathcal{B}$. Similarly, we can take the transitive collapse $\mathcal{B}_{1}$ of $\mathcal{B}$ in $\mathcal{A}$ by the possibly partial function $\pi_{1}:\mathcal{B}\to\mathcal{B}_{1}$. Finally, wee may also use $\pi_{1}$ to collapse $\mathcal{C}_{1}$ in $\mathcal{A}$ to give us $\mathcal{C}_{2}$. The following diagram summarizes the situation.

\mathcal{A}$$\mathcal{B}$$\mathcal{C}$$\triangleright_{\text{par}}$$\triangleright_{\text{par}}$$\mathcal{B}_{1}$$\mathcal{C}_{1}$$\mathcal{C}_{2}$$\pi_{1}$$\pi_{0}$$\pi_{1}$$\sigma$$\subseteq$$\subseteq$$\subseteq

We claim that $\mathcal{A}=\mathcal{B}_{1}=\mathcal{C}_{2}$. To see this, we show that it suffices – as in the proof of Theorem 19 – to observe that $\mathcal{A}$ and $\mathcal{C}_{2}$ have the same sets of natural numbers. We already can see that $\mathcal{C}_{2}\subseteq\mathcal{A}$, so suppose that $\mathcal{A}\models x\subseteq\omega$. Then we see that $\mathcal{C}\models\sigma(x)\subseteq\omega$ since $\sigma(\omega^{\mathcal{A}})=\omega^{\mathcal{C}}$. It is also clear that the transitive closure of $\{\sigma(x)\}$ in $\mathcal{C}$ is set-like in $\mathcal{C}$ and so we have $\mathcal{B}\models\pi_{0}\circ\sigma(x)\subseteq\omega$. By similar reasoning we see that $\mathcal{A}\models\pi_{1}\circ\pi_{0}\circ\sigma(x)\subseteq\omega$. Moreover, it can be seen that $\pi_{1}\circ\pi_{0}\circ\sigma(z)=z$ for any $z$ where the left hand side is defined. Thus, $x=\pi_{1}\circ\pi_{0}\circ\sigma(x)\in\mathcal{C}_{2}$ as required. Now since $\mathcal{A}=\mathcal{B}_{1}$, we see that $\pi_{1}:\mathcal{B}\cong\mathcal{B}_{1}=\mathcal{A}$ and so $\tau=\pi_{1}^{-1}$ is the isomorphism we’ve been looking for. ∎

As in Section 3, we can then piggyback on this argument to show that $MV$ is also solid.

Theorem 30**.**

$MV$* is solid.*

Proof.

Suppose that $\mathcal{W}$, $\mathcal{U}$ and $\mathcal{V}$ are models of $ZFC_{count}^{-}$ such that $\mathcal{W}\triangleright_{par}\mathcal{U}\triangleright_{par}\mathcal{V}$ and there is some $\sigma:\mathcal{W}\cong\mathcal{V}$ that is definable in $\mathcal{W}$ from parameters. We claim that $\mathcal{W}$ can define with parameters some $\tau:\mathcal{W}\cong\mathcal{U}$. As in the previous proof we take collapses of $\mathcal{V}$ in $\mathcal{U}$ and $\mathcal{U}$ in $\mathcal{V}$ to give us models $\mathcal{V}_{1}$, $\mathcal{U}_{1}$ and $\mathcal{V}_{2}$ as depicted in the following diagram.

\mathcal{W}$$\mathcal{U}$$\mathcal{V}$$\triangleright_{\text{par}}$$\triangleright_{\text{par}}$$\mathcal{U}_{1}$$\mathcal{V}_{1}$$\mathcal{V}_{2}$$\pi_{1}$$\pi_{0}$$\pi_{1}$$\sigma$$\subseteq$$\subseteq$$\subseteq

Given that $MV$ implies $ZFC_{count}^{-}$, we see that $\pi_{1}:\langle\mathbb{D}^{\mathcal{U}},\in^{\mathcal{U}}\rangle\cong\langle\mathbb{D}^{\mathcal{W}},\in^{\mathcal{W}}\rangle$ is an isomorphism on the set domains of $\mathcal{U}$ and $\mathcal{W}$ that is definable in $\mathcal{U}$. To address the worlds, it will suffices to show that $\mathcal{W}$ and $\mathcal{U}_{1}$ have the same worlds. Our argument is similar to the one we used to prove Theorem 20. First, suppose that $W$ is a $\mathcal{W}$-world. Then since $\pi_{1}\circ\pi_{0}\circ\sigma$ is the identity function on sets in $\mathcal{W}$, we see that $\pi_{1}\circ\pi_{0}\circ\sigma``W=W\in\mathcal{V}_{2}\subseteq\mathcal{U}_{1}$. Thus by World-domination in $\mathcal{U}_{1}$, we may fix a $\mathcal{U}_{1}$-world $U\supseteq W$. And then by World-domination in $\mathcal{W}$ we may fix a $\mathcal{W}$-world $W^{*}\supseteq U$. Thus, we have $W\subseteq U\subseteq W^{*}$ where $W^{*}$is a generic extension of $W$ and so $W$ is a generic refinement of a $\mathcal{U}_{1}$-world meaning that $W$ is also a $\mathcal{U}_{1}$-world. A similar argument establishes that every $\mathcal{U}_{1}$ world is also a $\mathcal{W}$ world. ∎

With this in hand, we then see (roughly) that equivalent extensions of $MV$ (and $ZFC_{count}^{-}$) are, in fact, identical.

Corollary 31**.**

$ZFC_{count}^{-}$* and $MV$ are both tight; i.e., any pair of theories extending $MV$ (or $ZFC_{count}^{-}$) that are bi-interpretable have the same deductive closure.*

5. Discussion

I think the results above are sufficiently surprising, and perhaps confusing, that some discussion is warranted. However, I’ll concede at the get-go that while I hope to clarify matters, I don’t have anything definitive to say on the philosophy: just more questions. Nonetheless, from a logical point of view I think these results are of obvious interest and this justifies further investigation. Categoricity results related to second order logic (very broadly construed) tend to hold for theories whose models have an obvious “inductive structure.”494949I’m mentioning second order logic here to distinguish these results from other forms of categoricity, like the categoricity of the theory of dense linear orders over the space of countable models. The natural examples thus come from arithmetic and set theory, where we start from a basic object like [math] or $\emptyset$ and then construct the rest of the universe inductively from there. The key point is that one level of the construction determines the next. But $MV$ is only partially like this. Since the set domain of a model of $MV$ satisfies $ZFC_{count}^{-}$, we have some inductive structure from which something like a standard categoricity argument can be conducted. But nothing like this comes along with the world domain. The worlds are generated using generic extension (and refinement) through a process that is not inductive. Given a particular world, it simply makes no sense to talk about the next world. They are not ordered in an interesting, well-founded manner. To prove the internal categoricity of $MV^{*}$ we take up a different strategy that relies on the ways in which the worlds of the respective multiverse overlap each other when collapsed. This makes the proof of the internal categoricity of $MV^{*}$ quite different from other proofs of its kind. At present, I know of no other example like it. It would be good to know if other examples are out there.

Let us now move onto the philosophy. I’ll start by considering how the internal categoricity result above might affect our interpretation of similar claims. Then I’ll consider some reasons we might have for disregarding the significance of those considerations.

In the introduction to this paper we observed that traditional, external categoricity results for set theory rely on the use of full models of second order $ZFC$, which presuppose that it makes sense to talk of the “true” powerset operation in order to define those models. Given that large chunk of the burden in the categoricity claim is obtaining a unique powerset operation inside those models, we appear to be presupposing what we set out to obtain. This has been a classic reason to disregard the evidential value of these results when considering questions about the determinateness of the set theory’s subject matter. However, internal categoricity makes no use of such second order resources. The theorem and the proof are all articulated in first order logic, within a theory $ZFC(\in_{0},\in_{1})$ that shares its membership relations evenly amongst its axiom schemata. Moreover loosely following Parsons (1990), we can justify the use of such a theory using a linguistically motivated thought experiment that is closely related to the way in which we use translation to interpret different extensions of $ZFC$. The theorem then shows that the universes associated with $\in_{0}$ and $\in_{1}$ are isomorphic and that this isomorphism is visible within the context of the thought experiment since it is definable. If we then consider any statement articulated in $\mathcal{L}_{\in}$, like the continuum hypothesis, we see that both the $\in_{0}$-user and the $\in_{1}$ -user must agree on its truth value (even if they are not in a position to decide it). Thus, we seem to have some reason to think that the continuum hypothesis has a determinate truth value using an approach that is not exposed to circularity problems but is based in a methodology common to the practice of set theorists. Now this is hardly a watertight argument, but it does look like a promising sketch.

Unfortunately, I think the internal categoricity of $MV^{*}$ suggests that this sketch cannot be filled in successfully. Or at least, any effort to fill it in will need to explain what is happening with $MV^{*}$ and this will likely make the completed argument subtle and complex. This seems unattractive as one of the draws of using categoricity arguments for these purposes is the promise of extracting philosophy straightforwardly from the math. While this kind of strategy rarely works, when it does, the resulting claim is difficult to challenge when it is closely related to something incontrovertible: a theorem. But once the philosophy creeps in, the wiggle room quickly expands and the opportunities to disagree multiply.

So why can’t the argument sketch be filled in? The essential idea is that people using $MV$ have access to an analogous argument that would appear to lead to a conclusion that contradicts the one drawn above. This suggests that something is wrong with the argument sketch. Let’s try to step through this analogy patiently. We saw above that $MV$ was introduced as an alternative to $ZFC$ that takes the incompleteness wrought by forcing as a feature and not a bug. It was designed to be an alternative foundation for mathematics that is more conservative in the questions it permits us to ask. Regarding $ZFC$ and its canonical extensions, Steel remarks, “the mathematical theory based on large cardinal hypotheses that we have produced to date can naturally be expressed in the multiverse language” (Steel, 2014). Moreover, it has been shown that $ZFC$ and $MV$ are not only equiconsistent, but there are interpretations witnessing this that display a degree of invertibility akin to bi-intepretability, although somewhat weaker.505050See Theorem 18 in (Meadows, 2021) and Proposition 6.1.1 in (Steel, 2024) for more detail. In particular, Meadows shows that $ZFC+V=HOD+GA$ is sententially equivalent to $MV+\exists W(V=HOD+GA)^{W}$. This means that there are interpretations going back and forth between these theories that take models of one theory and return a model that is elementary equivalent to the original. For the purposes of providing a foundation for mathematics and, indeed, doing contemporary set theory $ZFC$ and $MV$ are functionally identical. But there is, of course, a salient difference in their treatments of the continuum hypothesis. While in the framework of $ZFC$, the continuum hypothesis is naturally articulated as a simple statement about a putative bijection between two sets, there is no corresponding formulation in $MV$. From the local perspective, it is true in some worlds and false in others. From the global perspective, it simply cannot be articulated. This is good reason to think that $MV$ is successful in satisfying the goal of providing a foundation for mathematics that takes deep incompleteness seriously.

Nonetheless, the users of $MV$ (like those of $ZFC$) might still have some qualms about the the fixedness of the content of their new theory. Perhaps $MV$ is an algebraic theory like group theory that has many very different models. I do not want to make any claims about whether this would affect $MV$’s ability to provide a foundation for mathematics, but I think we can set that problem aside. Regardless of how it is settled, it would still be interesting and informative for users of $MV$ to understand whether it is algebraic or not. This is a natural time for internal categoricity to enter our story. We might follow something like the Parsonian thought experiment described above for $ZFC$, with $MV^{*}$ in its place.515151The reader will note that I’m moving quite freely between $MV$ and $MV^{*}$ here. This will be discussed further below. We might imagine two users of $MV^{*}$ who wonder whether they are talking about the same thing. Following the template, we might then move to a combined language with two membership relations $\in_{0}$ and $\in_{1}$. Regarding the theory used to govern those relations, we might agree to use the combined language in each of the axiom schemata, thus, giving us $MV^{*}(\in_{0},\in_{1})$. The results above then tell us that there will be a definable isomorphism between the multiverse associated with $\in_{0}$ and that of $\in_{1}$. This seems to provide a helpful response to qualms about the fixedness of $MV$’s content. For example, it is obvious that group theory cannot be axiomatized in a way that is is internally categorical.525252Perhaps the easiest way to see this is to observe that group theory has, and is intended to have, distinct models with different cardinalities. Thus, we see something about $MV$ that makes it more like $ZFC$ than group theory. So far so good. But can we say more? I think that a more extensive analysis of the thought experiment sketched above could provide us with further insights that would shed light on both $MV$ and internal categoricity. Let us, however, leave that for future work and return to our analogy.

Earlier we saw our, arguably straw, $ZFC$ users take the solace drawn from the internal categoricity of $ZFC$ and use that to sketch an argument toward a bold conclusion about the determinateness of the continuum hypothesis. Could incautious $MV$ users do something similar? Given that we just have a coarse-grained sketch, I don’t see why they couldn’t. The $MV$ users have a theory that is compatible with the idea that the continuum hypothesis is making something like a category error. Internal categoricity then appears to tell us something positive about the semantic stability of the perspective they have adopted. Any pair of $MV^{*}$ users who commit to the combined theory $MV^{*}(\in_{0},\in_{1})$ will also be committed to the continuum hypothesis being indeterminate from both the local and global perspectives. At this point, I’m reluctant to follow through on the analogy and say that this puts them in a position to argue that the continuum hypothesis is therefore, indeterminate. But I think this is a matter of hindsight since I now feel the same way about the $ZFC$ users described above. Whatever conclusions can be drawn from the internal categoricity of these theories, it cannot tell us that the continuum hypothesis is both determinate and indeterminate. Nonetheless, it might be put to work to answer a coherence challenge. We might say that the internal categoricity of $MV$ does put us in a position to sustain an attitude that continuum hypothesis is indeterminate.

With that little argument out of the way, let’s close by considering a couple of concerns one might have about this line of reasoning. As we go, I’ll also offer brief responses to each of them. For our first worry, the reader may have observed that I moved quite freely from talking about $MV^{*}$ to $MV$ in the previous paragraphs. These axiomatizations have an important difference. $MV^{*}$ is internally categorical, while $MV$ is not. Moreover, given that $MV$ is the official axiomatization, we might wonder whether we should care about what $MV^{*}$ and its peculiar properties. This might give us reason to doubt the robustness of the analogy described above. Our response is relatively straightforward. While $ZFC$ is internally categorical, it also has an alternative axiomatization $ZFC^{*}$ that is not internally categorical.535353See Theorem 52 in (Enayat and Łełyk, 2024) for a proof of this. Thus, exactly the same problem emerges in the context of $ZFC$. So which axiomatization is the right one? This is an interesting question that has become more pressing in recent years.545454See the conclusion of (Enayat and Łełyk, 2024) for some discussion of this. But it is not so easy to answer. In contrast to the convention adopted in this paper, it is more common to regard a theory as the deductive closure of some set of axioms rather than the axioms themselves. Moreover, this accords well with the ordinary practice of set theorists. We are generally just interested in what follows from $ZFC$ regardless of how it is axiomatized. Evidence for this can be found in the fact that different textbooks frequently give quite different axiomatizations of $ZFC$. Some use Collection rather than Replacement. Some prefer Foundation over Set Induction.555555For another classic difference, Kunen delivers a Pairing Axiom by saying that given any sets $x$ and $y$ there is a set $z$ that contains $x$ and $y$, while Jech says more by saying that there is a set $w$ that contains *exactly *$x$ and $y$ (Kunen, 2006; Jech, 2003). Of course, Separation and Extensionality wipe away any worries of inequivalence, but the point is just that there are many distinct axiomatizations of $ZFC$ out there and we rarely have need to worry about this. I should also note that when we consider weakenings of $ZFC$ then the question of axiomatization can become very important. For example, it is well known that in the absence of Extensionality or Powerset, Replacement behaves very oddly while Collection does not (Scott, 1961; Gitman et al., 2016). But none of this changes the theorems of $ZFC$, so the mathematical practice of users of $ZFC$ remains generally unaffected. Perhaps the right question to ask is not: is this axiomatization of theory $T$ internally categorical; but rather, is there an axiomatization of theory $T$ that is categorical?565656Of course, another option would be to consign the internal categoricity of both $ZFC$ and $MV^{*}$ to the flames. Perhaps the axiom sensitivity of internal categoricity makes it insignificant. I think it would be extremely premature to draw such a conclusion when so much remains unknown. Moreover, we’ve seen, with Corollary 28, that despite the axiom sensitivity, the internal categoricity of $MV^{*}$ can still be used to draw significant conclusions about $MV$ even when it is construed as a deductively closed theory. If so, $MV^{*}$’s internal categoricity seems to be just a significant as that of $ZFC$. If not, then much more work needs to be done to understand when one axiomatization of a theory is preferable to another. Cards on the table, I think this is a fertile problem to explore that is likely to deliver significant foundational insights that could be pertinent to our current problem. But at present, this is a field in its infancy.

For our second and final concern, let’s consider the World-domination schema. In $MV(\in_{0},\in_{1})$ both $\in_{0}$ and $\in_{1}$ are permitted into both the $\in_{0}$ and $\in_{1}$ versions of the World-domination schema. This is directly analogous to what we did with $ZFC(\in_{0},\in_{1})$. However, in the $ZFC(\in_{0},\in_{1})$ case we also spent some time discussing why we should allow the both membership relations into both versions of, say, the Replacement Schema. We are yet to do this for World-domination. And there is an important hitch. We motivated our use of the shared version of Replacement on the grounds that it provided a natural approximation of the full second order version of Replacement. No such motivation can be offered for World-domination. To see this recall that World-domination says that every definable inner model is covered by a world. What would it mean to say that this was an approximation of the full second order version of World-domination? To answer this question, we need to know what the full second order version of World-domination would look like. The obvious suggestion is the following: every inner model is covered by a world. With Replacement we moved from definable class functions to arbitrary class functions; here we move from definable inner models to arbitrary inner models. But second order World-domination is inconsistent.575757It is also worth bearing in mind that the other axiom schemata also face similar problems. For example, if we were to treat the Replacement Schemata relativized to worlds as an approximation of a second order version this also leads to inconsistency. To see this suppose $W$ is a world and $W[c]$ is a generic extension of $W$ by a Cohen real $c\subseteq\omega$. Now let $f:\omega\to\omega$ be the injection that enumerates $c$. This is a(n improper) class function over $W$, but a second order Replacement Axiom would tell us that $f``\omega=c\in W$, which is impossible. To see this consider, the models $\mathcal{W}_{0}$ and $\mathcal{W}_{1}$ described in the proof of Theorem 13 (showing that $MV$ is not internally categorical). We claim that $\mathcal{W}_{0}$ doesn’t satisfy second order World-domination. To see this recall that while $\mathcal{W}_{0}$ is generated from a world $M$, $\mathcal{W}_{1}$ is generated from a class generic $N$ extension of $M$ that no world in $\mathcal{W}_{0}$ can cover. As such, we have in $N$ an uncovered inner model of $\mathcal{W}_{0}$. So how do we motivate World-domination and how does this affect the analogy above? Our response is to concede that World-domination is not an approximation of a second order axiom, but rather, it is about definability. Theorem 25 not only tells us that World-domination is a consequence of $MV$, its proof also gives us some explanation as to why it is true and some expectation of when it will continue to hold. In particular, we see that World-domination reveals deep limitations in what one can expect to define over a model of $MV$. In allowing both membership relations into the World-domination Schemata, we can be understood as merely flagging our expectation that these limitations will continue to hold. Of course, this is not as simple as saying that we are approximating a second order axiom, but this may be more of a blessing than a curse. When we motivated shared Replacement with the approximation story, we might wonder how far we’ve removed ourselves from second order logic and our worries about circularity. When motivating World-domination through definability rather than second order logic, there is no implicit appeal to the collection of all subclasses of the domain. Perhaps this is a benefit. I think it’s fair to say that more work is required here, and that this work will tell much more about internal categoricity and foundational theories in mathematics. But even at just the beginning of this road, it is interesting to see an axiom schema in a foundational setting that cannot be explained as an approximation of a second order cousin.

Conclusion

In this paper, we’ve shown that Steel’s multiverse theory can be axiomatized in alternative manners such that it is both internally categorical and not internally categorical. This makes Steel’s theory an interesting example to consider in the theory of relative interpretation. Given that internal categoricity is sometimes thought to have significance with regard to the determinateness of the continuum hypothesis and that $MV$ is designed to, in a sense, ignore it, this also makes the internal categoricity of (a version of) $MV$ interesting in the philosophy and foundations of mathematics. We have argued that this result suggests that arguments from internal categoricity to the fixedness of subject matter in a theory are unlikely to work or, at least, will be much more complicated than one might have expected.