1.1 The domain for both $\in $ and $\in ^{*}$ should be the same

In the setup of $ZFC(\in ,\in ^{*})$ , our quantifiers range over a single domain object. This assumption is crucial for us to be able to obtain the genuine isomorphism between the $\in $ and $\in ^{*}$ relations rather than a mere embedding of one relation into a rank initial segment of the other. I’m not aware of a convincing philosophical argument that this is either a good or a bad assumption, however, I’ll now record a couple of observations that may give some reason for caution. First, we might think that some simple consequences of this theory are counterintuitive. To see this, let’s consider an obvious example. In $ZFC(\in ,\in ^{*}),$ there will be an emptyset, $\emptyset $ , associated with $\in $ and an emptyset, $\emptyset ^{*}$ , associated with $\in ^{*}$ . In general, they will be distinct, and in these cases, there could be some x such that $x\in ^{*}\emptyset $ and $y,$ where $y\in \emptyset ^{*}$ . Thus, we have objects that are empty according to one relation and nonempty according to another. Of course, this is not a mathematical issue: Väänänen’s extension of Zermelo’s theorem is both elegant and intriguing. However, this issue may give us cause to wonder how well it fits with our thought experiment above.Footnote ⁷

This brings us to the second observation: perhaps there is a theory better suited to our argument than $ZFC(\in ,\in ^{*})$ . We motivated our thought experiment by imagining that you and I were concerned that our discourse about set theory might not—as we might have assumed—have been focused on a particular subject matter. In an attempt to formalize this situation, we proposed to expand our languages to accommodate two membership relations. Väänänen’s theorem then appeared to resolve the situation. However, we also noted that the use of a shared domain led to some counterintuitive consequences. With that in mind, we might think that a more natural theory for our thought experiment would be one that separated the domains associated with the two membership relations. Thus it might be more natural to incorporate predicate symbols V and $V^{*}$ that we treat as domains for $\in $ and $\in ^{*}$ in the obvious way. We might then add an axiom demanding that V and $V^{*}$ partition the domain of quantification. This will remove the counterintuitive features we’ve just discussed, but it will also mean that the isomorphism is lost. As with Zermelo’s Theorem 1, we’ll only get an embedding of one structure into an initial segment of the other. So perhaps less ground has been gained than appeared at first sight.

1.2 Both membership relations should be permitted into formulae used in the Separation and Replacement schemas

$ZFC(\in ,\in ^{*})$ has two Separation schemas; one for $\in $ and one for $\in ^{*}$ . For example on the $\in $ -side, we have it that for any x there is a set y such that for all z

$$\begin{align*}z\in y\leftrightarrow z\in x\wedge\varphi(z), \end{align*}$$

where crucially $\varphi (z)$ is any formula of $\mathcal {L}_{\in ,\in ^{*}}$ . Thus, we can use both $\in $ and $\in ^{*}$ to provide conditions for separation. This is a powerful assumption that is also crucial to obtaining Väänänen’s result. In particular, we use the shared version of Replacement to obtain a definable collapse of one membership relation within the other. But what reason do we have for thinking that this assumption is a reasonable one within the context of our thought experiment?

This question has a more canonical response that originated in the context of the induction schema in arithmetic [Reference Field17, Reference Parsons53]. To apply this to set theory, rather than merely taking the axioms of $ZFC$ at face value, we are implored to consider the intentions behind them. With regard to the Replacement Schema, it literally says the following:Footnote ⁸ Suppose x is a set and that $\boldsymbol {F}$ is a (class) function that is definable in $ZFC$ by a formula of $\mathcal {L}_{\in }$ . Then the pointwise image of x via $\boldsymbol {F}$ is itself a set; i.e., $\boldsymbol {F}"x$ is a set. We might argue that the restriction to (class) functions that are expressible in $\mathcal {L}_{\in }$ is merely forced upon us by our choice of language: this is as much as we can do with the tools we have available. But really the motivation for Replacement is that given any set x and any (class) function $\boldsymbol {G}$ whatsoever, $\boldsymbol {G}"x$ should also be a set regardless of whether or not $\boldsymbol {G}$ is definable. So the story goes. To articulate this enhanced version of Replacement, we need to move to a second-order version of set theory or one that allows quantification over classes. We then regard the Replacement schema of $ZFC$ as a mere approximation (among many) of the true second-order version.Footnote ⁹ But for our purposes in $ZFC(\in ,\in ^{*}),$ this is not required. In order to stay faithful to this intention, the natural thing do to is to admit—not only those (class) functions definable using a single membership relation—but rather (class) functions that can be defined using both of them. If we are taking seriously the idea that any (class) function should enjoy Replacement, then those defined using $\mathcal {L}_{\in ,\in ^{*}}$ are certainly within our scope.

1.3 We only need to agree on $ZFC$

Our use of Väänänen’s theorem in the thought experiment aimed to show that if you and I both agree on $ZFC$ then we have good reason to think that we are talking about the same subject matter. However, our motivation coming into that example was a little more ambitious. We were concerned about the possibility that you think $\varphi $ is true while I do not, where $\varphi $ is some sentence that is not decidable by $ZFC$ . But this puts some pressure on our decision to use $ZFC(\in ,\in ^{*})$ to model our debate. To see this, let’s make things more specific and suppose that you think that the continuum hypothesis, $CH$ , is true while I think it is not. Then it would seem natural to represent this situation by letting, say, $\in $ have $CH$ satisfied while $\in ^{*}$ has $\neg CH$ . We might record these statements as $CH^{\in }$ and $\neg CH^{\in ^{*}}$ , respectively. Following through on this, we should then use the theory

$$\begin{align*}T=ZFC(\in,\in^{*})+CH^{\in}+\neg CH^{\in^{*}} \end{align*}$$

to model our debate. If we do this, then Väänänen’s theorem still holds since it just requires $ZFC(\in ,\in ^{*})$ . Thus, we get an isomorphism between the $\in $ and $\in ^{*}$ relations and this entails that they satisfy the same statements. But this is, of course, impossible since $\in $ and $\in ^{*}$ disagree on $CH$ . So T is inconsistent. What should we make of this? The first and perhaps most obvious response is that this reveals that one of us is wrong. Indeed, this little argument is really just another way of saying—as we did with arithmetic—that there is a right answer. It gives us no clues regarding how to identify it, but we are given reason to believe that there is one.

In this article, I want to open the door to a second response and provide a critical investigation of its prospects and boundaries. Rather than taking the road above, I would like to retain some doubts around the assumptions that motivate our thought experiment and see where this can lead us. For one reason to explore this territory, we note that a non-trivial section of the set-theory community appears to retain doubts as to the fixedness of the subject matter of set theory. One example of this occurs in the debate between John Steel and Hugh Woodin regarding what is known as the generic multiverse [Reference Steel63, Reference Steel and Kennedy65, Reference Hugh Woodin and Link74, Reference Hugh Woodin, Heller and Hugh Woodin75, Reference Hugh Woodin77]. Far too briefly, this is a view inspired by the ubiquity of undecidability in set theory.Footnote ¹⁰ Rather than accepting incompleteness as a problem to be solved, we are encouraged to think of it as evidence of a failure to pose these questions correctly. In particular, we take it that generic extensions of models of set theory are, so to speak, just as good as their ground models.Footnote ¹¹ Other examples of roughly this attitude can be found in Joel David Hamkins’ multiverse approach to set theory and Saharon Shehah’s probabilistic attitude to set theory [Reference Hamkins23, Reference Shelah57]. In the sections that follow, we are going to investigate this territory by exploring two perspectives through which one might not take all debates in set theory seriously: that some debates are merely verbal; and that there is sufficient common ground that we need not worry.

2.1 Too weak

Let us start by observing that mere interpretability—as instantiated in Example 5—is generally asymmetric. While one theory can interpret another, this does not mean that the other can interpret the original. For example, it is easy to see that $ZFC+V=L$ cannot interpret $ZFC+MC$ . This is because $ZFC+MC$ says there is a measurable cardinal $\kappa $ , and so it can prove that there isa model of $ZFC+V=L$ . This means that $ZFC+MC$ proves outright—and not merely relatively—that $ZFC+V=L$ is consistent. Thus, if $ZFC+MC$ were also consistent relative to $ZFC+V=L$ , then $ZFC+MC$ could prove its own consistency, thus, contradicting Gödel’s second incompleteness theorem. Thus, there is an interpretative asymmetry.

But what does this tell us about our debate scenario? While $ZFC+MC$ is in a position to interpret $ZFC+V=L$ in such a way that it can provide a complete simulation of $ZFC+V=L$ via translation, $ZFC+V=L$ cannot do the same. Only one party to the dispute is able to make a move that could put them in position to translate the debate away. Indeed, the debate ensuing out of Example 5 is in an even worse position. Not only is the $ZFC+V=L$ user unable to interpret $ZFC+MC$ and thus, simulate $ZFC+MC$ via translation; the $ZFC+V=L$ user proves that there are no measurable cardinals. As such, I think it could be reasonable for the $ZFC+V=L$ user to conclude that $ZFC+MC$ is just plain false. Perhaps $ZFC+MC$ does not have an interpretation since it simply doesn’t deserve one. On the other side of this dialectic, the $ZFC+MC$ user may draw different conclusions. The $ZFC+MC$ user may see the ${ZFC+V=L}$ user’s inability to simulate $ZFC+MC$ as indicative of a pathological weakness in $ZFC+V=L$ . They might claim that $ZFC+V=L$ is unduly restrictive as theory purporting to provide a foundation for mathematics.Footnote ¹⁷ Regardless of who is right, it seems clear that they are not merely arguing past each other. This asymmetry motivates an obvious improvement of mere interpretation.

Definition 6. T and S are mutually interpretable if there are translations, t and s such that

$$\begin{align*}t:mod(T)\leftrightarrow mod(S):s. \end{align*}$$

It is no longer enough that T interprets S, they must be able to interpret each other.

Mutual interpretability is clearly an equivalence relation on theories. Does it give us a more plausible relation to represent our motivating idea of arguing past each other? Clearly, the asymmetry of mere interpretability has been removed. However, I now want to discuss three reasons to be reserved about whether mutual interpretability is really sufficient to deflate debates between alternative foundations of mathematics.

1. (1) Practical triviality.

2. (2) Poor fit.

3. (3) Loss in translation.

2.1.2 Poor fit

Another reason for reservation is the weakness of the simulation offered by an interpretation. When T interprets S via some translation t, we know that every theorem of S is provable—modulo the translation—in T. We might say that the simulation is complete. But is it also sound? Returning to the characters of Example 5, we find a situation where this is not the case. Recall that we had a translation t such that for all sentences $\varphi $ of $\mathcal {L}_{\in }$

$$\begin{align*}ZFC+V=L\vdash\varphi\ \Rightarrow\ ZFC+MC\vdash t(\varphi). \end{align*}$$

However, the converse fails. For example, $ZFC+MC$ proves that there is a countable transitive model satisfying $ZFC$ and the statement that $0^{\#}$ exists, which we abbreviate as $0^{\#}\exists $ . The existence of such a model is a $\Sigma _{2}^{1}$ proposition and thus, by the Lévy–Shoenfield theorem,Footnote ¹⁹ it doesn’t change its meaning when its quantifiers are restricted to L. Thus, $ZFC+MC$ also proves that there is a transitive model in L that satisfies $ZFC+0^{\#}\exists $ . But from Gödel’s second theorem, we know that $ZFC+V=L$ cannot prove its own consistency let alone that there is a transitive model of $ZFC$ , so the arrow above cannot go in both directions. Regarding soundness, we see that $ZFC+MC$ proves more than $ZFC+V=L$ even through the translation. Moreover, the excess statements are ones with regard to which a reasonable advocate of $ZFC+V=L$ might demur. Of course, this example involves interpretative asymmetry, however, this problem also occurs between $ZFC$ and $ZFC+\neg Con(ZFC)$ . In particular, when we interpret $ZFC$ in $ZFC+\neg Con(ZFC)$ using the identity translation, we end up with $\neg Con(ZFC)$ being an obvious theorem of $ZFC+\neg Con(ZFC)$ but not of $ZFC$ .

What should we make of this? In the context of our thought experiment, we see that mere interpretability doesn’t provide a perfect simulation. When a theory T interprets a theory S, we see that rather than proving exactly what the S proves, T proves everything S proves and possibly more. Suppose I am the S user and you are using T to interpret me via some interpretation t that overshoots in this way. You might claim that you are providing a simulation that deflates any potential debate, but I may object that you are proving too much. I might think that among your extra theorems there is a false statement. My reason for using T could be that it represents the very limit of my present knowledge and therefore that the excess given by T goes too far. This lack of fit motivates our next equivalence relation, which is based on a refinement of ordinary interpretation.

Definition 10. Let us say that T faithfully interprets S if there is a translation t such that

$$\begin{align*}S\vdash\varphi\ \Leftrightarrow\ T\vdash t(\varphi). \end{align*}$$

We thus solve the fit problem by fiat. We demand that the translation proves exactly the translations of the theorems of S. Such an interpretation seems to have a better claim to providing a genuine simulation. Faithful interpretation also has a pleasing semantic version: T faithfully interprets S via t if and only if the associated mod-functor $t:mod(T)\to mod(S)$ is a surjection up to elementary equivalence.Footnote ²⁰ The use of faithful interpretation gives a more plausible claim to providing a proper simulation and thus we might wonder if it is a suitable model for debates where parties argue past each other. Disappointingly, we see that for a wide class of theories, faithful interpretation is almost as easy to obtain as ordinary interpretation.

Theorem 11 [Reference Lindström37].

Let T and S be essentially reflective theories that can interpret $PA$ . Then the following are equivalent:

1. (1) T faithfully interprets S; and

2. (2) Every $\Pi _{1}^{0}$ theorem of S is a theorem of T and every $\Sigma _{1}^{0}$ theorem of T is a theorem of S.

Roughly speaking, this is a generalization of Theorem 8 in which we use the completeness theorem in T to obtain a model of every complete theory extending S. This then gives us every model of S up to elementary equivalence. To do this, we use the tree of different completions of S as given by Henkin’s completeness theorem.Footnote ²¹ We then use some inherent incompleteness in T to chose a path through that tree depending on how certain undecidable sentences turn out in a model.Footnote ²² The conditions of (2) in the theorem ensure this construction is not derailed. With this in hand, we can easily obtain the following result.

Fact 12. $ZFC$ and $ZFC$ plus the statement “there is no $\omega $ -model of $ZFC$ ” are mutually faithfully interpretable.

This follows since these theories have exactly the same $\Sigma _{1}^{0}$ and $\Pi _{1}^{0}$ consequences; and indeed, this is the necessary and sufficient condition for mutual faithful interpretation. The requirements are obviously stricter than those for mere mutual interpretation, but as the Fact above shows, we still seem to end up with something that is far too easy to obtain.Footnote ²³ In particular, the claim that there is no $\omega $ -model of $ZFC$ seems like a substantive point of disagreement.

It’s also possible to obtain a little insight into why faithful interpretation doesn’t deliver what we want. In the proof of Theorem 11, we need to make a bunch of random decisions to ensure that we capture all the models of the interpreted theory up to elementary equivalence. To do this, we make use of the undecidability of the interpreting theory. This allows us to make infinitely many choices through the tree of theory completions.Footnote ²⁴ However, it is not difficult to see that this proof strategy is not revealing any kind of deep connection between the theories. Rather, we are using a clever trick to connect the incompleteness of one theory to that of another. This does the job, but it is not telling us that these theories provide two different ways to talk about the same subject matter.

2.1.3 Loss in translation

If we are to think of our interlocutors as arguing past each other, then we want to have some sense that they are really talking about the same subject matter and the source of disagreement is merely a matter of semantics. If we’re really using language differently to talk about the same content, then we should expect that if I translate my language into your language and then back again, I should end up saying exactly what I said in the first place. Or from the semantic perspective of Theorem 4, we should expect that if we start with a model of my theory transform and it into one of yours and then back into mine, I should end up exactly where I started. This does not occur with our example of mutual interpretability above. To see this, let

$$\begin{align*}i:mod(ZFC+V=L)\leftrightarrow mod(ZFC):t \end{align*}$$

be the mod-functors determined by the interpretations described in Example 7. Now suppose that $\mathcal {M}$ is a countable model of $ZFC$ and let $\mathcal {M}[c]$ be a generic extension of $\mathcal {M}$ by a Cohen real.Footnote ²⁵ Then $\mathcal {M}[c]$ satisfies $ZFC$ and that $V\neq L$ . Then note that $i\circ t(\mathcal {M}[c])$ is just $\mathcal {M}[c]$ ’s version of L and so it satisfies $V=L$ . Thus $i\circ t(\mathcal {M}[c])\neq \mathcal {M}[c]$ .

Thus, the mutual interpretations do not return us to exactly where we started: information has been lost in translation and it cannot be recovered. In particular, if we have a model of $ZFC$ with non-constructible sets, then once those sets are gone we have no way to get them back through interpretation. Thus, it seems wrong to think of a dispute over whether to include $V=L$ is one where the participants are merely arguing past each other. We need a stronger relation.

2.2 Too strong

We now have number of failed approaches to modeling a debate between users of alternative foundations that are practically identical. We started from mere interpretation and worked our way up to mutual faithful interpretation. However, in each case, we found reason to think that the fit between the equivalence relation and the idea of merely verbal dispute was flawed. In this section, we are going to start from the other end of the spectrum and work our way down. In particular, we are going to begin with the strongest natural equivalence relation on theories aside from identity. We start from the semantic perspective.

Definition 13. Let T and S be theories articulated in $\mathcal {L}_{T}$ and $\mathcal {L}_{S,}$ respectively. We say that T and S are definitionally equivalent if they are mutually interpretable as witnessed by mod-functors

$$\begin{align*}t:mod(T)\leftrightarrow mod(S):s \end{align*}$$

in such a way that:

1. (1) whenever $\mathcal {M}\models T$ , then $\mathcal {M}=s\circ t(\mathcal {M})$ ; and

2. (2) whenever $\mathcal {N}\models S$ , then $\mathcal {N}=t\circ s(\mathcal {N})$ .

Intuitively speaking, this tells us that if we start with a model of T and then transform it into a model of S and back again using t and s, then we end up exactly where started again. Similarly, if we start with a model of S. Thus, these translations do not suffer any of the information loss we discussed in Section 2.1.3.

For a trivial example, let $sLO$ be the theory of stricty linear orders using the $<$ reation and let $nsLO$ be the theory of non-strict linear orders using the $\leq $ relation. We then translate $x<y$ as $x\leq y$ and $x\neq y$ ; and in the other direction, we translate $x\leq y$ as $x<y$ or $x=y$ . We make no alterations to the domains.Footnote ²⁶ It is then easy to see that if we start with a model $\mathcal {A}$ of $sLO$ and use these translations to make a model of $nsLO$ and then a model of $sLO$ again, we get back to $\mathcal {A}$ . Similarly, if we start with a model of $nsLO$ .Footnote ²⁷ If we imagined a debate between an $sLO$ user and an $nsLO$ user, then intuitively we’d probably think of this as being a pointless debate. The fact that these theories are definitionally equivalent reveals why. It gives us a precise way of understanding the way in which the parties to such a debate would be merely arguing past each other.

For a more interesting example, recall Aczel’s proposal for a theory of sets that intentionally admits ill-founded sets. The theory is formulated by removing Foundation and replacing it with Aczel’s anti-foundation axiom.Footnote ²⁸ Let us call this $AZFC$ . This theory had been proposed as a means of addressing an apparent weakness of ordinary $ZFC$ : it’s inability to deal with an ill-founded membership relation. For example, according to $ZFC,$ there can be no set x such that x is itself its only member. $AFZC,$ on the other hand, proves that such sets exist. Nonetheless, we have the following result.

If we brought this to bear on our debate between alternative theories, then this result shows that $ZFC$ and $AZFC$ can be understood as different ways of talking about the same subject matter. If we imagined a debate between the $ZFC$ user and the $AZFC$ user, then this result tells that there is not so much at stake. Or perhaps better, while we haven’t ruled out the possibility that the $ZFC$ and $AZFC$ users are talking about two different abstract structures, it is clear that $ZFC$ and $AZFC$ could be used to talk about the same abstract structure. This doesn’t mean that there cannot still be some kind of disagreement. In particular, the $AZFC$ user might claim that—although the $ZFC$ user can simulate them perfectly— $AZFC$ is simply a more natural tool for the investigation of ill-founded structures. Such a debate seems comparable to debates between programming languages when all such languages are Turing complete. There’s something to debate, but it’s on a very different level to the discussion in this article.

While I believe the semantic perspective on definitional equivalence gives us a very concrete way of understanding what is happening, we might worry that it relies too much on toy models and so we might only be able to make use of this relationship from an—so to speak—external perspective. It turns out that an internal approach can be offered which is quite similar in spirit to Theorem 2. To motivate this, first recall that given a theory T articulated in $\mathcal {L}_{T}$ , we may form a simple definitional expansion $T^{*}$ of T by adding a new relation symbol P to $\mathcal {L}_{S}$ and an axiom of the form

$$\begin{align*}\forall\bar{x}(P\bar{x}\leftrightarrow\varphi(\bar{x})), \end{align*}$$

where $\varphi (\bar {x})$ is a formula of $\mathcal {L}_{T}$ .Footnote ²⁹ In effect, we are defining the extension of P using the formula $\varphi (\bar {x})$ . This means that the new symbol, P, is redundant in the sense that it can always be replaced by $\varphi (\bar {x})$ wherever it occurs.Footnote ³⁰ Let us say that a definitional expansion of T is formed by a series of simple definitional expansions. We then have the following result linking definitional equivalence with a more internal perspective.

Informally speaking, this tells us that S and T can each be definitionally expanded to the same language $\mathcal {L}_{S}\cup \mathcal {L}_{T}$ to give the same theory. Let’s apply this to our comparison between $ZFC$ and $AZFC$ by supposing that you think we should use $ZFC$ while I think we should use $AZFC$ . Both theories are articulated in $\mathcal {L}_{\in }$ , but say different things about the membership relation. Following the same course as we did in relation to Väänänen’s theorem, it seems appropriate to move to a language with two membership relations, $\mathcal {L}_{\in ,\in ^{*}}$ . We let $ZFC$ use $\in $ and $AZFC$ use $\in ^{*}$ . Then you can use $ZFC$ in $\mathcal {L}_{\in ,\in ^{*}}$ to provide a definition for $\in ^{*}$ . And similarly I can use $AZFC$ in $\mathcal {L}_{\in ,\in ^{*}}$ to provide a definition for $\in $ . The resultant theories then have exactly the same consequences. As in our discussion of Väänänen’s theorem, we move to a shared language and obtain a kind of collapse between the theories in question.

Definitional equivalence addresses all the problems of the previous section and gives us a plausible way to model disputes where participants are arguing past each other. But how well does it work on our original project to compare alternative extensions of $ZFC$ ? Very interestingly, we see something like categoricity rears its head once more.

Proof Suppose that we have mod-functors $t:mod(T)\leftrightarrow mod(S):s$ witnessing the definitional equivalence of T and S. We claim that T and S have the same consequences. To see this, let $T^{*}$ be a definitional expansion of T into $\mathcal {L}_{\in ,\in ^{*}}$ by adding the axiom

$$\begin{align*}\forall x\forall y(x\in^{*}y\leftrightarrow\varepsilon_{t}(x,y)) \end{align*}$$

defining $\in ^{*}$ using $\varepsilon _{t}$ . Then note that since t and s witness definitional equivalence we also see that $T^{*}$ implies that $\in $ can be defined in terms of $\in ^{*}$ ; i.e., we have

$$\begin{align*}\forall x\forall y(x\in y\leftrightarrow\varepsilon_{s}(x,y)^{\in^{*}}), \end{align*}$$

where $\varepsilon _{s}(x,y)^{\in ^{*}}$ is the same formula as $\varepsilon _{s}(x,y)$ except that instances of $\in $ have been replaced by $\in ^{*}$ . Given that T and S both extend $ZFC$ , this means that $T^{*}$ is an extension of $ZFC(\in ,\in ^{*})$ and so by Väänänen’s theorem, $\in $ and $\in ^{*}$ are always isomorphic and so T and S must have the same logical consequences.

This seems like bad news. Our goal has been to develop a precise means of modeling disputes between adherents of distinct set theories extending $ZFC$ in such a way that we could think of the parties as arguing past each other. We’d hoped to translate the disagreement away and having investigated a number of equivalence relations that were too weak, we used this discussion to motivate the relation of definitional equivalence. This provided the, so to speak, gold standard of equivalence through which no information is lost. However, in the case of set theories extending $ZFC$ , we see that definitional equivalence is trivial in that it reduces straight back to identity. For some, this could be the end of the road. An argument very like the categority argument we started with also appears to block the use of interpretability for our project. This much is certainly true: we’ll need to lower our standards if we are to retain our goal of adequately modeling such disputes as instances of arguing past each other.

2.2.1 The interpretability hierarchy

Fortunately, there are a number of lower standards that remain philosophically appealing. So with the problems of the previous section in mind, we now introduce a series of natural weakenings of definitional equivalence and investigate their prospects for deflating debates between strong set theories.

Definition 17. Let T and S be theories articulated in $\mathcal {L}_{T}$ and $\mathcal {L}_{S,}$ respectively. Suppose that T and S are mutually interpretable as witnessed by mod-functors

$$\begin{align*}t:mod(T)\leftrightarrow mod(S):s. \end{align*}$$

We say that t and s witness that T and S are:

• • Bi-intepretable if there are definable functions f and g over T and S:

  1. (1) whenever $\mathcal {M}\models T$ , then $f^{\mathcal {M}}:\mathcal {M}\cong s\circ t(\mathcal {M})$ ; and

  2. (2) whenever $\mathcal {N}\models S$ , then $g^{\mathcal {M}}:\mathcal {N}\cong t\circ s(\mathcal {N})$ .

• • Iso-congruent if:

  1. (1) whenever $\mathcal {M}\models T$ , then $\mathcal {M}\cong s\circ t(\mathcal {M})$ ; and

  2. (2) whenever $\mathcal {N}\models S$ , then $\mathcal {N}\cong t\circ s(\mathcal {N})$ .

• • Sententially equivalent if:

  1. (1) whenever $\mathcal {M}\models T$ , then $\mathcal {M}\equiv s\circ t(\mathcal {M})$ ; and

  2. (2) whenever $\mathcal {N}\models S$ , then $\mathcal {N}\equiv t\circ s(\mathcal {N})$ .

So less formally, we see that definitional equivalence takes us back and forth to exactly where we started. Sentential equivalence takes us back and forth to a model with the same complete theory. With isocongruence, we return to an isomorphic structure. And with bi-interpretability, we return to an isomorphic structure and we can actually define the isomorphism. Taking little stock, we see that we have the following chain of implications among the equivalence relations we’ve discussed so far:

Definitional equivalence $\to $ Bi-interpretability $\to $ Iso-congruence $\to $ Sentential equivalence $\to $ Mutual faithful interpretability $\to $ Mutual interpretability $\to $ Equiconsistency.

I don’t believe it’s known whether the first two arrows reverse, however the rest of the sequence is a strict hierarchy.Footnote ³³

We’ve seen some reason to think that definitionally equivalent theories just provide two different ways of talking about the same thing. But given that the new relations are weaker we might wonder whether they could adequately model disputes about alternative set theories. We can provide some philosophical motivations for using these relations for this purpose. For example, suppose that we adhere to some form of structuralism about mathematics. There are many ways to formulate such positions, however, at their core is the maxim that we should not be concerned with what a structure is made from, we should only be concerned with the structure itself.Footnote ³⁴ More succinctly, we should only care about a mathematical structure up to isomorphism. Thus in general, given two isomorphic structures, we think there is no good reason to prefer on over the other. Then both bi-interpretability and iso-congruence seem to provide a good fit with this position. Both of these relations tell us that if we go back and forth, we end up with a model that is—by structuralist lights—just as good as the one we started with. Bi-interpretability provides an extra feature in that we are—so to speak—able to see the isomorphism, while this is not required for mere isocongruence. We might say that bi-interpretability gives an internalperspective to the structuralism involved, while mere iso-congruence provides an externalpoint of view.Footnote ³⁵ With respect to sentential equivalence, we might turn to some species of formalism in which the enterprise of mathematics is understood to be—at its root—a syntactic process of proving theorems from axioms. Our talk about infinite mathematical structures is to be thought of as a fiction or heuristic crutch that supports the real work that is fundamentally syntactic. From this perspective, our talk of moving back and forth between models is merely a congenial way of getting at the important fact, the back and forth move preserves the real syntactic information since it preserves the complete theory.

Returning to our motivating problem, we must now ask whether going lower in the hierarchy of equivalence relations helps. The first step down is disappointing. Visser and Friedman have shown that for theories that are able to provide a reasonable foundation for mathematics, if they are bi-interpretable, then they are definitionally equivalent. Recall that a sequential theoryis a theory that interprets a very weak set theory that is intended to capture the minimal requirements for coding sequences and thus, syntax.Footnote ³⁶

Theorem 18 [Reference Visser and Friedman72].

If T and S are bi-interpretable theories that are sequential, then T and S are definitionally equivalent.Footnote ³⁷

The proof takes a few steps, however, a crucial move is to take the internally defined isomorphisms to deliver an internal version of the Cantor–Bernstein theorem which allows us to turn the injections into a bijection. It is then easy to see that $ZFC$ and its extensions are sequential theories and so we have the following result.Footnote ³⁸

Corollary 19 [Reference Enayat, Eijck, Iemhoff and Joosten11].

If S and T are bi-interpretable extensions of $ZFC$ , then S and T are the same theory.Footnote ³⁹

Thus, it seems that our internal structuralist perspective cannot support the possibility of disagreement between adherents of mutually inconsistent extensions of $ZFC$ . This could be another stopping off point for some readers. It tells us that the only times we can translate away apparent disagreement by translating back and forth to an isomorphic structure—where that isomorphism is visible—the disagreement was illusory in the first place.

2.3 The middle ground

We appear to be running out of room in our hierarchy with only iso-congruence and sentential equivalence remaining. These are relative newcomers and as such, less is known about them. Nonetheless, the spacebetween iso-congruence and bi-interpretability seems very small. The only example distinguishing them that I know relies on restricting our attention to countable models to ensure certain isomorphisms exist but remain undefinable. More work is required here, but given this we shall also leave iso-congruence aside. Fortunately, it is possible to separate sentential equivalence from iso-congruence. This also gives us a good example of sententially equivalent extensions of $ZFC$ .

Morally speaking, it’s not difficult to see why these theories aren’t isocongruent. If we start with a transitive model of $ZFC+V=L$ and define a model of $ZFC+V=L[c]$ it will want to be well-founded and have the same order type for its ordinals if it has any hope of defining a model of $ZFC=V=L$ that is isomorphic to the original. But then it can be collapsed and so—with a little work—we end up having a Cohen real in the original model, which is impossible.

To show sentential equivalence, the obvious thing to do here is take a countable model M of $ZFC+V=L$ and then add a Cohen real to obtain a model of $ZFC+V=L[c]$ . And in the other direction taking a model N of $ZFC+V=L[c]$ , we can simply define N’s version of L to obtain a model of $ZFC+V=L$ . The problem with this strategy is that the first step isn’t a genuine interpretation. Interpretations always return models whose domain is a definable subset of the original model. Generic extensions always return models that are proper supersets of the original model. This can be addressed by using a Boolean valued ultrapower where we build a model using an ordinary ultrafilter over a complete Boolean algebra rather than a generic ultrafilter.Footnote ⁴⁰ This means that the Boolean ultrapower can be defined within the ground model, so we are able to define an internal model of $ZFC+V=L[c]$ within any model of $ZFC+V=L$ .Footnote ⁴¹ Exploiting the homogeneity of the forcing used to obtain Cohen reals, it can then be shown that regardless of which ultrafilter we use, the resultant model will have the same complete theory. Putting this together, we can show that the respective back and forth transitions between models of these theories always give us a model that is elementary equivalent to the one we started with.

This gives us our best prospect, so far, of an equivalence relation modeling disputants arguing past each other. It doesn’t appear to be practically trivial like mutual interpretability or its faithful improvement; and it doesn’t collapse into the triviality of identity like definitional equivalence and bi-interpretability. If you believed that every set is constructible and I believed that every set is constructible from a Cohen real over L, then we might use the interpretations above to translate this disagreement away by noting that the back and forth interpretations preserve the complete theory of the original models. I think many set theorists would agree that sententially equivalent theories like those above are close enough as foundations as to be practically the same. Anything you can do in one theory you can do in the other, via the translations. There is also a syntactic rendering of sentential equivalence.

Here we see that the translation functions return us sentence-by-sentence to exactly where we started according to the theory in question. We might see this as offering a more internalperspective on sentential equivalence. If you are an S user and I am a T user, then these translations reveal that I can translate any sentence of my theory into yours and then return to obtain a sentence that arguably has the same meaning.Footnote ⁴²

This looks like good news, however, we should also take into account what is lost in the move below iso-congruence and bi-intepretability. In general, the back and forth translations do not return us to a structure that is isomorphic to the one we started with: we only get elementary equivalence. As such, we might wonder whether this relationship is strong enough to satisfy the mathematical structuralist. I think the structuralist can reasonably complain that information has been lost in translation. For example, when forcing is involved, our use of Boolean valued ultrapowers means that the structure we end up with is typically ill-founded from the perspective of the original model. If we like transitive models, then the model we end up with seems obviously pathological. Nonetheless, if we consider the example above, I think it’s not so surprising that this kind of information loss occurs. In a nutshell, the information contained in ultra or generic filters is, so to speak, too random to be recovered by a definition or interpretation. Essentially, this is because generics and ultrafilters are only obtained by choice principles. The consequence of this is that when we use an interpretation that forgets a generic set—as in the example above—there is simply no way of defining it back. Given this, if we want forcing arguments to be in our toolkit for theory comparison, it seems that isocongruence is too much to demand and that sentential equivalence provides a reasonable way of modeling the idea that distinct theories are saying the same thing.

2.3.1 Pointwise equivalence

We’ve now finally found in sentential equivalence an equivalence relation that avoids the practical triviality of mutual interpretation and the collapse into identity of bi-interpretability. I think it is a good candidate for theoretical equivalence between strong set theories. Nonetheless, it also suffers a practical limitation in that examples of sententially equivalent theories extending $ZFC$ , tend to make use of very artificial theories. For example, while $ZFC+V=L$ is a natural albeit restrictive theory, the theory $ZFC+V=L[c]$ is not commonly used and certainly never proposed as a serious foundation for mathematics. Indeed, there is a sense in which this theory was cooked to provide a simple pair of sententially equivalent theories.Footnote ⁴³ It would be better if we had an equivalence relation between theories that worked between theories more commonly used by set theorists. More particularly, it would be good to identify a non-trivial equivalence relation that was able to enjoin theories of the form $ZFC+\varphi $ and $ZFC+\neg \varphi $ . In this little section, we’ll consider a significant weakening of sentential equivalence that will provide many more examples of equivalent theories. We’ll then discuss an equivalence relation from contemporary set theory that has a similar extension: the generic multiverse. As this is the first place in this article where forcing enters in a very general fashion, I should make a quick remark about nomenclature. Whenever I talk about forcing or generic extension, I shall mean that the underlying partial order is a set unless I state otherwise.

Our plan is to weaken sentential equivalence in two stages. For the first stage, we remove demand that we always have to use the same translations to get between a model of one theory and another. Rather, we may vary which translation we use in response to the model we are considering.

Definition 22. Let T and S be theories extending $ZFC$ . We say that T and S are pointwise sententially equivalent if:

• • For all $\mathcal {M}\models T,$ there is a model $\mathcal {N}$ of S and mod-functors $t,s$ such that

  $$\begin{align*}t(\mathcal{M})\equiv\mathcal{N}\ \&\ s(\mathcal{N})\equiv\mathcal{M}. \end{align*}$$

• • For all $\mathcal {N}\models S,$ there is a model $\mathcal {M}$ of T and mod-functors $s,t$ such that

  $$\begin{align*}s(\mathcal{N})\equiv\mathcal{M}\ \&\ t(\mathcal{M})\equiv\mathcal{N}. \end{align*}$$

The salient difference is that the translations are no longer deployed uniformlyacross the models of T and S. We can use different translations from model to model. This opens up the playing field substantially.Footnote ⁴⁴

To throw a little philosophical light on this equivalence relation, we return to our motivating example of a debate about competing foundational theories. Suppose that you are using theory S while I’m using theory T, where T and S are pointwise sententially equivalent theories extending $ZFC$ . We see that any model of my theory T can define a model of S and that model can in turn define a model of T that is elementary equivalent to the one we started with. This means that if you prove a theorem in S, there is a sense in which I can always access that by using the translation to move to a model of S where that theorem is true and further there is a sense in which this model of S is equivalent to the one I started with in the sense that it can define a model that is elementary equivalent to it. Similar remarks show that you, as an S user, can access theorems of T by moving to an equivalent model. Moreover, if some statement $\varphi $ is merely satisfiable according to S then I know there is a model of my theory T that can be defined in a model of $S,$ where $\varphi $ is true and that is equivalent in the sense that this model of T can define a model of S that has the same complete theory as the original model of S. Again, similar remarks apply when we start from a sentence $\psi $ that is satisfiable according to T.

I think these remarks speak of a tight connection between pointwise sententially equivalent theories, however, unlike our previous examples there does not appear to be a syntactic or internal counterpart to the pointwise equivalence relation. Since the pointwise equivalence allows us to vary the interpretations used as we move between models, the talk of models becomes an essential part of the definition and as such, its philosophical interpretation. Given that we noted that sentential equivalence may be more appealing to those with formalist tendencies, this talk of models makes it more challenging to provide a satisfying philosophical account of who should find pointwise sentential equivalence to be sufficient for us to say that we have equivalent ways of talking about the same subject. It seems to be that the easier we make it to find equivalences, the more difficult it is to explain the underlying philosophy.

The second stage of our weakening compounds this phenomenon. Given that the pointwise relation focuses on models, this opens up the possibility of using arbitrary parameters in the definitions of our interpretations. Then just as we allowed ourselves to vary the interpretation from model to model, we might also allow ourselves to vary the parameters used in those interpretations. To set this up, we first modify our interpretative mechanism to incorporate the use of parameters. Given that we are focused on models, we just give the generalized version of mod-functor.

Definition 23. Let us say that $t(\cdot )_{\cdot }:\sum _{\mathcal {M}\in mod(T)}M\to mod(\emptyset )$ is a parametric mod-functor determined by the formulae $\delta _{t}(x,z)$ and $\varepsilon _{t}(x,y,z)$ if all ordered pairs of a model $\mathcal {M}=\langle M,\in _{\mathcal {M}}\rangle $ of T and element $m\in M$ , we have $t(\mathcal {M})_{m}=\langle N,\in _{\mathcal {N}}\rangle ,$ where:Footnote ⁴⁵

1. (1) $N=\{x\in M\ |\ \mathcal {M}\models \delta _{t}(x,m)\}$ ; and

2. (2) $\in _{\mathcal {N}}=\{\langle x,y\rangle \in M^{2}\ |\ \mathcal {M}\models \varepsilon _{t}(x,y,m)$ .

Thus, the mod-functor definition remains the same except that we now allow the use of a parameter, $m\in M$ , to define the interpretations of domain and membership relation. With this in hand, we can provide our final weakening of sentential equivalence.

Definition 24. We say that S and T are pointwise, parametrically sententially equivalentif

• • For all $\mathcal {M}\models T,$ there exist $\mathcal {N}\models S$ , parametric mod-functors $t,s$ , and sets $m\in M$ and $n\in N$ such that

  $$\begin{align*}t(\mathcal{M})_{m}\equiv\mathcal{N}\ \&\ s(\mathcal{N})_{n}\equiv\mathcal{M}. \end{align*}$$

• • For all $\mathcal {N}\models S,$ there exist $\mathcal {M}\models T$ , parametric mod-functors $s,t$ , and sets $m\in M$ and $n\in N$ such that

  $$\begin{align*}s(\mathcal{N})_{n}\equiv\mathcal{M}\ \&\ t(\mathcal{M})_{m}\equiv\mathcal{N}. \end{align*}$$

Once again, we are allowed to vary the interpretation we use from model to model. However in addition to this, we are also allowed to pick parameters from the respective models to get the interpretations to do the required work. This opens up the playing field further and again complicates the philosophical story. While sentential equivalence seems more appealing to those with formalist tendencies, we’re now not only talking about models but making essential use of parameters within them. Nonetheless, it is arguably at this level that we see a more practical level of intervention by obtaining equivalences via forcing of distinct natural theories extending $ZFC$ . To demonstrate this, we first recall the following very useful theorem.

Fact 25 ([Reference Laver34]; Woodin).

If V is a generic extension of an inner model M, then M can be defined in V using a parameter.Footnote ⁴⁶

Informally speaking, this tells us that if we are in a generic extension, then we can define—using parameters—any of the ground models from which our universe eventuated. This fact, in combination with Boolean valued ultrapowers, allows us to show a striking equivalence.

Example 26. Consider the theories $ZFC+CH$ and $ZFC+\neg CH$ . We claim that they are parametrically, pointwise, sententially equivalent and provide a sketch of a proof. We start with a model $\mathcal {M}$ of $ZFC+CH$ . As is well known, in any such model, there is a complete Boolean algebra $\mathbb {B}$ such that

$$\begin{align*}(\Vdash_{\mathbb{B}}\neg CH)^{\mathcal{M}}. \end{align*}$$

Let U be an ultrafilter over $\mathbb {B}$ from $\mathcal {M}$ . Then the Boolean ultrapower $\mathcal {N}$ of $\mathcal {M}$ by U can be defined in $\mathcal {M}$ using the parameter U. Then it can be seen that $\mathcal {N}$ has an inner model $\mathcal {M}^{*}$ that is a generic refinement of $\mathcal {N}$ and is elementary equivalent to $\mathcal {M}$ . By Fact 25, $\mathcal {M}^{*}$ is definable using a parameter in $\mathcal {N}$ . Thus we have moved from a model of $ZFC+CH$ to a model of $ZFC+\neg CH$ and back to a model of $ZFC+CH$ that is elementary equivalent to the one we started with.

Next we start with a model $\mathcal {M}$ of $ZFC+\neg CH$ . Rather than going to $L^{\mathcal {M}}$ , we may force to collapse all the cardinals below $2^{\aleph _{0}}$ to obtain a model of $ZFC+CH$ . More precisely, there is a complete Boolean algebra $\mathbb {B}$ in any model $\mathcal {M}$ of $ZFC+\neg CH$ such that

$$\begin{align*}(\Vdash_{\mathbb{B}}CH)^{\mathcal{M}}. \end{align*}$$

The rest of this case is then analogous to the first case: we use an ultrafilter to define a Boolean ultrapower and then return to an elementary equivalent model using Fact 25.

It’s then easy to see that the argument above can be adapted to obtain the following result. Taking our lead from the example, let us say that a theory T interprets S via forcing if for all $\varphi \in S$ , T proves $\Vdash _{\mathbb {P}}\varphi $ , where $\mathbb {P}$ is uniformly defined over T by some formula.Footnote ⁴⁷

Corollary 27. Any pair of theories that can interpret each other via forcing are parametrically, pointwise sententially equivalent.Footnote ⁴⁸

This gives our best solution to the problem posed at the very beginning of this article: is there a way of deflating a debate between users of $ZFC$ who disagree about whether some sentence $\varphi $ is true or not? We wondered whether we could offer translations that would reveal that they were arguing past each. Could they be understood as using different languages to say the same thing? We then saw that: mutual interpretation was too easy to obtain; bi-interpretability collapsed into identity; and sentential equivalence only worked in special cases. But with the generalization to pointwise comparison with parameters, the generalized notion of sentential equivalence gives us an equivalence relation that allows us to deflate one of the fundamental problems of set theory: the continuum hypothesis. If we think that $CH$ holds, then we can travel to and return from an equivalent world where it is false. Similarly if $\neg CH$ is true.

2.3.2 The generic multiverse

We finally have an equivalence relation that is serviceable for the comparison of theories extending $ZFC$ . However, we’ve also noted some difficulties in arguing for the philosophical picture that motivates it. In an effort to alleviate this worry and to situate this relation in the context of contemporary set theory, we now highlight an intriguing relationship between pointwise parametric sentential equivalence and the generic multiverse. The latter notion has been investigated by Woodin and Steel in response to the incompleteness of $ZFC$ wrought by forcing [Reference Steel and Kennedy65, Reference Hugh Woodin77]. Very crudely, the generic multiverse responds to incompleteness by embracing it rather than attempting to remove it. Rather than attempting to solve problems shown to be be independent of $ZFC$ by forcing, we simply allow that there are set-concepts or worlds where such problems have affirmative solutions and worlds where they do not. The underlying idea is not dissimilar to the motivating problems of this article. Indeed, I think its relationship with pointwise parametric sentential equivalence gives a clear perspective of a role that the generic multiverse can play in set theory.

First we define the generic multiverse. Our definition is a little different from that offered in [Reference Hugh Woodin77] since we only demand that our models are countable but not transitive. Countability is required to ensure that generic filters exist, but we avoid transitivity since our approach to forcing uses Boolean valued ultrapowers which are generally ill-founded. This helps us to get the relationship with generalized sentential equivalence, however, a similar relation was also proposed by Woodin in [Reference Maddy and Meadows40]. It will be helpful to say that a generic refinement is obtained when we employ Fact 25 to define a ground model of the universe. Roughly speaking it is the inverse of the operation that gives us generic extensions.

Definition 28 (Woodin).

Let $\mathcal {M}$ be a countable model of $ZFC$ . Let the generic multiverse generated from $\mathcal {M}$ , denoted $\mathbb {V}_{\mathcal {M}}$ , be the set of models $\mathcal {N}$ such that there is a sequence $\langle \mathcal {M}_{0},...,\mathcal {M}_{n}\rangle ,$ where $\mathcal {M}=\mathcal {M}_{0}$ and $\mathcal {N}=\mathcal {M}_{n}$ such that for all $i<n$ , $\mathcal {M}_{i+1}$ is either a generic extension or a generic refinement of $\mathcal {M}_{i}$ .

Thus, the generic multiverse of some $\mathcal {M}$ consists of all those models that can be obtained by a sequence of generic extensions and refinements starting from $\mathcal {M}$ . We have a set of models closed under generic extension and its inverse. Using a result of [Reference Usuba68], it turns out that every such model can be obtained by a single generic refinement followed by a generic extension.Footnote ⁴⁹ Our next problem is to define a suitable equivalence relation using the generic multiverse.

Definition 29. Let us say that theories T and S extending $ZFC$ are generic multiverse equivalentif

$$\begin{align*}\{\mathbb{V}_{\mathcal{M}}\ |\ \mathcal{M}\models T\}=\{\mathbb{V}_{\mathcal{N}}\ |\ \mathcal{M}\models S\}, \end{align*}$$

where we restrict our attention to models that are hereditarily countable.

The essential idea here is to blur the models associated with a theory T by considering their generic multiverses. Note that whenever we have

$$\begin{align*}\{\mathcal{M}\ |\ \mathcal{M}\models T\}=\{\mathcal{N}\ |\ \mathcal{N}\models S\} \end{align*}$$

then S and T are the same theory. By using a collection of generically related models rather than a single model, we wipe away the generically superficial differences between those models. Thus, comparing these generic multiverses associated allows us to ignore the kinds of difference that are caused by forcing. If we don’t think these differences are significant, then generic multiverse equivalence is a good way to track equivalences that ignore these differences. As with pointwise parametric sentential equivalence, we see that $ZFC+CH$ and $ZFC+\neg CH$ are generic multiverse equivalent. Given a model $\mathcal {M}$ of $ZFC+CH$ , there is a model $\mathcal {M}[G]$ of $ZFC+\neg CH$ in $\mathbb {V}_{\mathcal {M}}$ ; and given a model $\mathcal {N}$ of $ZFC+\neg CH,$ there is a model $\mathcal {N}[H]$ of $ZFC+CH$ in $\mathbb {V}_{\mathcal {M}}$ .

Given the pointwise parametric sentential equivalence also obtains these kinds of equivalences, this raises questions about its relationship with generic multiverse equivalence. Our goal now is to show that—under certain restrictions—generic multiverse equivalence implies pointwise parametric sentential equivalence. The restrictions are necessary since generic multiverse equivalence uses genuine generic extension for forcing arguments while pointwise parametric sentential equivalence uses Boolean valued ultrapowers. It can be seen that there are complete theories given by generic filters that cannot be matched by ultrapower constructions. A relatively simple restriction is to restrict our attention to theories extending $ZFC$ by $\Sigma _{2}$ statements. Essentially, a $\Sigma _{2}$ statement says that something holds in an initial segment of the cumulative hierarchy.Footnote ⁵⁰ A typical example is the statement that a measurable cardinal exists. The following fact then underpins our claim.

Fact 30 (Woodin).

Suppose there are unboundedly many Woodin cardinals and that $\varphi $ is a $\Sigma _{2}$ sentence. Then whenever $\mathbb {P}\in V$ is such that $\Vdash _{\mathbb {P}}\varphi $ and $\mathbb {Q}$ is a poset, there is some $\dot {\mathbb {S}}\in V^{\mathbb {Q}}$ such that:

$$\begin{align*}\Vdash_{\mathbb{Q}*\dot{\mathbb{S}}}\varphi. \end{align*}$$

Informally speaking, this tells that whenever a $\Sigma _{2}$ sentence $\varphi $ can be forced to be true but is then forced to be false, it can still be forced to be true again. For this reason, it is often known as the $\Sigma _{2}$ -resurrection theorem. We can then state our claim and sketch its proof.

Theorem 31 [Reference Meadows48].

Suppose T and S are extensions of $ZFC$ plus there are unboundedly many Woodin cardinals by $\Sigma _{2}$ statements. Then if T and S are generic multiverse equivalent, then they are also parametrically, pointwise sententially equivalent.

Proof (Sketch)

Suppose T and S are as above and that they are generically multiverse equivalent. Let $ZFC^{*}$ denote $ZFC$ augmented with a proper class of Woodin cardinals. Then let T be $ZFC^{*}+\Phi ,$ where $\Phi $ is $\Sigma _{2}$ ; and S be $ZFC^{*}+\Psi ,$ where $\Psi $ is also $\Sigma _{2}$ . Now let $\mathcal {M}$ be a model of T. By symmetry of argument and results from above, it will suffice to show that there is a generic extension of $\mathcal {M}$ in which S holds. Since T and S are generic multiverse equivalent, there must be a generic refinement $\mathcal {M}^{*}$ of $\mathcal {M}$ with a generic extension $\mathcal {N}$ satisfying S. But then using Fact 30, we see that $\mathcal {M}$ must also have a generic extension satisfying S as required.

Thus, we see that parametric, pointwise sentential equivalence provides us with a non-trivial lower bound below generic multiverse equivalence. This speaks to both the naturalness of generic multiverse equivalence and the value of the framework we’ve been developing by being able to capture it. But is there some reason to prefer one of the other? In defense of parametric, pointwise sentential equivalence, we have in it an equivalence relation that emerges through a natural sequence of generalizations based on a linguistic translation: it’s just definability with parameters! By contrast, generic multiverse equivalence essentially relies on forcing and generic extension. These techniques are peculiar to set theory and as such, one might worry about that the resulting equivalence relation is too parochial to support much philosophical significance. Moreover, the paucity of generic sets means that we must restrict our attention to toy models that are countable. No such restriction is required with parametric, pointwise sentential equivalence. On the other hand, John Steel offers a defense of generic multiverse equivalence based on the quality of models it delivers in comparison to those offered by sentential equivalence relations [Reference Steel, Arbeiter and Kennedy62].Footnote ⁵¹ As we observed above, in order to simulate the effect of forcing arguments, we must use Boolean valued ultrapowers in parametric, pointwise sentential equivalence. In general, the effect of this is that the models produced by such ultrapowers will be ill-founded. Steel says that such interpretations fail to preserve meaning. I am unsure how to offer a complete theory of meaning here, but we can illustrate the effect by considering how the meaning of the term “ordinal” changes as we move from some ground model to an ill-founded Boolean ultrapower. While “ordinals” are well-founded in the ground model, in general, they are not in the ultrapower. We might say that the meaning of the term “ordinal” has not been preserved in this translation. By contrast, when we use generic extensions, we obtain a model with exactly the same ordinals as those of the ground model. Thus, the term “ordinal” and, indeed, much more is preserved in the move to a generic extension. For this reason, we may come to prefer generic multiverse equivalence. We leave any final adjudication on this matter to the reader, but note that the close relationship illustrated above is remarkable and arguably casts them both in a better light together.

This concludes our discussion of using “talking past each other” as a means of resolving debates about extensions of $ZFC$ . Our goal was to find a plausible way of modeling the idea that some mutually inconsistent set theories could be considered as providing merely different ways of saying essentially the same thing. The hope then was that the disagreement could be dissolved in translation. Having discarded a number of equivalence relations that were too weak or strong, we then slowly worked our way into a Goldilocks zone where equivalence wasn’t too easy to obtain and yet was applicable to contemporary set theory. In pointwise, parametric sentential agreement we found a plausible alignment with contemporary practice while—at the same time—making the philosophical story about the naturalness of this relation more challenging.

3.1 Partial equivalence

Let’s start with a plausible example of something like empirical equivalence in set theory. We’ll then use this as our guide to providing a general theory of partial equivalence using interpretability. After that, we’ll apply our framework to some other examples and then show that it has some disappointing limitations. Our prototypical example is well-known, however, it will be useful to sketch a quick proof of it since it provides the basis for our investigation. Let us say that T interprets S using an inner modelif the mod-functor $t:mod(T)\to mod(S)$ is such that for all models $\mathcal {M}$ of T, $t(\mathcal {M})$ is an inner modelof $\mathcal {M}$ ; i.e., $t(\mathcal {M})$ is a model of S with the same ordinals as $\mathcal {M}$ .

Proposition 33. Let T and S be theories extending $ZFC$ and suppose that T interprets S using an inner model while S interprets T via forcing. Then T and S agree on $\Pi _{2}^{1}$ sentences; i.e., for all $\Pi _{2}^{1}$ sentences $\varphi $ ,

$$\begin{align*}T\vdash\varphi\ \Leftrightarrow\ S\vdash\varphi. \end{align*}$$

The proof above is driven by the fact that we are using good interpretations in the context of theories that are strong enough to ensure that $\Pi _{2}^{1}$ sentences are preserved. The classic example here is $ZFC+CH$ and $ZFC+\neg CH$ . Traditionally, a model of $ZFC+CH$ can be obtained by using the interpretation that restricts us to L; and a model of $ZFC+\neg CH$ can be obtained by forcing to add $\aleph _{2}$ many reals. Informally speaking, this tells us that if we are happy—as the mathematical community generally is—to use $ZFC$ in our mathematical practice, then no matter what natural strengthening of this theory we take up, we cannot disagree about $\Pi _{2}^{1}$ -statements.Footnote ⁵⁴ Following the idea above, we might take the $\Pi _{2}^{1}$ -statements as the concrete ones that we really care about and the possible exotic extensions of $ZFC$ as mere instruments for learning more about what is concrete. This informal gloss only talks about natural strengthenings, so the scope of this claim is arguably limited. The idea that motivates this is that the ways of extending $ZFC$ that have turned out to have been mathematically fruitful are linked to the large cardinal hierarchy by inner model and forcing arguments.Footnote ⁵⁵

This example gives us a reasonable prototype for “empirical equivalence” in the foundations of mathematics. We now put our interpretative tools to work to abstract out what is arguably the underlying mechanism of the example. This will allow us to draw some links with the work of the previous sections and highlight the fact that “empirical equivalence” is fruitfully understood through the lens of relative interpretability. We start by describing the idea semi-formally and then give the proper definition. Suppose we have two theories T and S that purport to be foundations for mathematics. Then we might consider a particular topic of mathematics that is discussed in some language $\mathcal {L}^{*}$ and wonder whether they agree about it. For example, we might consider the ways in which T and S are applied to analysis. To do this, we consider the ways in which T and S interpret the language of analysis. Note, however, in the example above, we only obtain agreement over the $\Pi _{2}^{1}$ fragment of the language of analysis. In order to model this situation better we shall overload our notation and let $\mathcal {L}^{*}$ represent both a language and some fragment of the sentences associated with that language.Footnote ⁵⁶ We shall then speak of $\mathcal {L}^{*}$ -sentences as those sentences of the language associated with $\mathcal {L}^{*}$ that are members of the fragment of associated with $\mathcal {L}^{*}$ .

Definition 34. Let T and S be theories in languages $\mathcal {L}_{S}$ and $\mathcal {L}_{T,}$ respectively, and let $\mathcal {L}^{*}$ be some language associated with a fragment of its sentences. Then suppose we have translations p and q determining mod-functors such that:Footnote ⁵⁷

$$\begin{align*}p:mod(T)\to mod(\mathcal{L}^{*})\leftarrow mod(S):q. \end{align*}$$

We say that T and S are agree on $\mathcal {L}^{*}$ via p and q if for all $\mathcal {L}^{*}$ -sentences $\varphi $ Footnote ⁵⁸

$$\begin{align*}T\vdash p(\varphi)\ \Leftrightarrow\ S\vdash q(\varphi). \end{align*}$$

The idea here that we interpret $\mathcal {L}^{*}$ in T and S using p and q; and then we say that they agree about $\mathcal {L}^{*}$ if the respective translations give the same $\mathcal {L}^{*}$ -theory; i.e., the same theorems in the fragment associated with $\mathcal {L}^{*}$ . We might think of $\mathcal {L}^{*}$ as the “empirical” domain that we really care about. In contrast to most of the relationships discussed in Section 2, it appears to be very weak. For instance, there is no demand for any kind of back-and-forth agreement between the theories. Indeed, we haven’t even required that they be mutually interpretable. Rather, it seems most similar in structure to mutual faithful interpretability, as in Definition 10, and it is perhaps best understood as a partial version of that. Nonetheless, it does have a simple model-theoretic gloss.

Observe now that Proposition 33 can be understood within this framework. To see this, first recall that there are many natural translations of the language of analysis into the language of set theory. Let us fix one of them can call it the standard translation.

In Section 2.3, we noted above that forcing arguments using generic extension don’t strictly give us interpretations since they augment the domain. For this reason, we made use of Boolean valued ultrapowers to get around this. However, to make the passage of our current inquiry a little smoother, we’ll now proceed as if generic extensions do give us interpretations.Footnote ⁶⁰ It’s also worth noting that the standard translation works on every sentence in analysis, not just the $\Pi _{2}^{1}$ sentences. However, this has no effect on the results.

We now have a general notion of partial interpretation based on interpretability that fits our canonical example. It turns out that with the addition of large cardinal axioms, further examples of agreement can be gained that encompass the entire language of analysis and beyond. This phenomenon has been discussed extensively by John Steel in his work on eventual monotonicity and what he calls a theory of the concrete [Reference Feferman, Friedman, Maddy and Steel15, Reference Steel64, Reference Steel and Kennedy65]. For one example, we have the following.

Very roughly, we start by establishing the equiconsistency of these theories: a forcing argument takes us from $ZFC$ and the Woodin cardinals to a model of $ZFC+AD^{L(\mathbb {R})}$ ; and an inner model argument takes us from a model of $ZFC+AD^{L(\mathbb {R})}$ to a model of $ZFC$ with infinitely many Woodin cardinals.Footnote ⁶¹ The arguments then proceed by showing that sufficient structure is preserved in these interpretations to ensure that the theory of analysis is preserved.Footnote ⁶²

3.2 The limitations of empirical agreement

As we observed above, agreement seems very similar to mutual faithful interpretability. Moreover, we saw reasons in Section 2.1.2 to be pessimistic about the philosophical significance of faithful interpretability; in particular, it was too easy to obtain and provided an implausible analysis of our target idea of arguing past each other. Perhaps worryingly, with our definition of agreement above, we are only concerned with the translations of consequences of the theories. We are not offering a more natural, structural connection between the theories themselves. As such, the fact that T and S agree on $\Pi _{2}^{1}$ sentences doesn’t tell us much about T and S or their connection. Perhaps this is enough for an argument based on common ground, but we’ll see in this section that it leaves some unsatisfying equivalences on the table that are difficult to remove by formal means. To see this, recall that in the example from the previous section, we considered theories that were mutually related by forcing or inner model arguments. But this is not required to obtain agreement as we have defined it. Here is an arguably pathological illustration of this.

The connection between T and S from the proof above is clearly very weak as is evidenced by the fact that, in general, a model of S will only be able to define a model of T that is ill-founded. There is also something sufficiently odd about S to warrant consideration about whether such theories should be taken seriously. In favor of S, we might observe that it appears to provide a way for us to obtain more results in analysis without explicitly committing to the existence of large cardinals. This could be seen as an advantage by those who are shy about the apparent ontological excesses of contemporary set theory. However, the way in which this advantage is obtained is in some sense parasitic: we are enjoying the benefits of theft over honest toil by simply lifting the $\Pi _{2}^{1}$ theorems of T in order to even formulate S. John Steel is critical of this move and calls it an instrumental dodge [Reference Feferman, Friedman, Maddy and Steel15]. While it is true that S doesn’t commit itself to the existence of an inaccessible cardinal and this might seem like a benefit, we are left to wonder how one might extend or even just gain a proper understanding of S, without investigating and understanding T better. It seems that the proper way to investigate S must go through T and so the apparent alleviation of large cardinal burdens is more of a dodge than genuine ontological parsimony. More practically, the use of S invokes a kind of bad faith in that the only way to obtain the extra $\Pi _{2}^{1}$ theorems is to prove them using T and then sweep them up into S by fiat. One wonders whether the inaccessible cardinal of T has really gone away after all.

This kind of pathology motivates a brief (and somewhat disappointing) effort to obtain a deeper connection between theories and the languages they interpret. Our idea is to not merely demand that the $\mathcal {L}^{*}$ -consequence of two theories are the same, but the theories themselves are mutually interpretable in a way that preserves their interpretation into $\mathcal {L}^{*}$ . The following definition is intended to capture this idea.

Definition 39. Let T and S be theories in $\mathcal {L}_{S}$ and $\mathcal {L}_{T,}$ respectively, and let $\mathcal {L}^{*}$ be some language associated with a fragment of its sentences. Suppose we have $t:mod(T)\leftrightarrow mod(S)$ and

$$\begin{align*}p:mod(T)\to mod(\mathcal{L}^{*})\leftarrow mod(S):q. \end{align*}$$

We say that t (sententially) preserves $\mathcal {L}^{*}$ from T over p to S over q if for all $\mathcal {L}^{*}$ -sentences $\varphi $

$$\begin{align*}T\vdash p(\varphi)\leftrightarrow t\circ q(\varphi). \end{align*}$$

If in addition, there is some $s:mod(S)\to mod(T)$ such that s preserves $\mathcal {L}^{*}$ from S over q to T over p, we say that s and t mutually preserve $\mathcal {L}^{*}$ between S and T over q and p.

The key difference between preservation and agreement is that we also demand some connection between T and S in that they must be mutually interpretable, which then gives us a conduit through which we can track the information about $\mathcal {L}^{*}$ that is passed between these theories. Before we give an example, we note that a more convenient model-theoretic characterization is also available.

Suppose T and S are set theories and that $\mathcal {L}^{*}$ is the language of analysis. Then if the relation defined above holds and we are given some model $\mathcal {M}$ of T, $\mathcal {M}$ defines a model $p(\mathcal {M})$ of arithmetic that is elementary equivalent to the model of arithmetic $q\circ t(\mathcal {M})$ defined in the model $t(\mathcal {M})$ of S that is defined in $\mathcal {M}$ . It is then easy to see that mutual preservation between theories implies agreement.Footnote ⁶⁴ Moreover, we see that the scenario of Proposition 33 also gives us mutual preservation.

This suggests that our initial abstraction of Proposition 33 to obtain the definition of agreement missed an important feature of the relationship between such theories. So are there any theories that satisfy Definition 34 but not Definition 39? Here is an easy example where agreement holds but preservation fails.

This does give us a strict hierarchy between agreement and mutual preservation, but there is something cheap about the failure since we blocked it by selecting a pair of theories that are very close but are, somewhat annoyingly, not mutually interpretable. So can we can have mutually interpretable theories that agree on some language but do not mutually preserve it? An affirmative answer to this question might put us in a position to formally identify theories that exploit an instrumental dodge, like those from Proposition 38. Unfortunately and perhaps surprisingly, a generalization of Lindström’s argument for Theorem 11 shows that it many cases, agreement is sufficient for mutual preservation. To keep things short and simple, we provide a relatively specific example and then merely remark about how it can be generalized.

Proof Suppose $\mathcal {M}$ is a model of S. We aim to define a model $\mathcal {N}$ of T in $\mathcal {M}$ with the same theory of arithmetic. More precisely, we want to show that $\mathcal {M}$ can define a model of

$$\begin{align*}T^{*}=T\cup\{\varphi\in\mathcal{L}_{Ar}\ |\ \mathcal{M}\models\varphi\}. \end{align*}$$

Fix a formula defining $T^{*}$ and let $T^{*\mathcal {M}}$ be the denotation of that formula in $\mathcal {M}$ . Note that it can be arranged that the standard part of $T^{*\mathcal {M}}$ is $T^{*}$ . Now observe that for any finite subset $\Gamma _{0}$ of T and finite set $\Gamma _{1}$ of arithmetic sentences,

$$\begin{align*}T\vdash\bigwedge\Gamma_{1}\to Con(\Gamma_{0}\cup\Gamma_{1}). \end{align*}$$

This holds since T extends $ZFC$ and so it can prove that $\Gamma _{0}$ holds in some initial segment of the universe which, of course, contains its version of $\mathbb {N}$ . Now since this sentence is itself arithmetic and T and S agree on arithmetic, we see that S also proves this sentence.

We now claim that $\mathcal {M}\models Con(\Delta )$ for all finite subsets of $T^{*}$ . To see this, let $\Delta $ be such a set and let $\Delta =\Delta _{0}\cup \Delta _{1}$ , where $\Delta _{0}\subseteq T$ and $\Delta _{1}$ is a set of arithmetic sentences that are true in $\mathcal {M}$ . Then since $\Delta _{1}$ is satisfied in $\mathcal {M}$ and $S\vdash \bigwedge \Delta _{1}\to Con(\Delta _{0}\cup \Delta _{1})$ we see that $\mathcal {M}\models Con(\Delta _{0}\cup \Delta _{1})$ as required.

Now we can define the model of $T^{*}$ . We do this in two cases noting that either $\mathcal {M}\models Con(T^{*\mathcal {M}})$ or $\mathcal {M}\not \models Con(T^{*\mathcal {M}})$ . In the first case, we let $\mathcal {N}$ be what $\mathcal {M}$ thinks is the L-least model satisfying $T^{*\mathcal {M}}$ . In the second case, we let c be what $\mathcal {M}$ thinks is the largest natural number such that the set X of codes of formulae from $T^{*\mathcal {M}}$ up to and including c are consistent. Such a c must exist in $\mathcal {M}$ since it thinks that $T^{*\mathcal {M}}$ is inconsistent. Moreover, by a previous claim, we see that c must be in the nonstandard part of $\mathcal {M}$ ’s natural numbers. Thus, X includes $T^{*}$ and is consistent according to $\mathcal {M}$ . Thus, we let $\mathcal {N}$ be the L-least model of X.

Thus, we see that under a few conditions that are easily satisfied, mutual agreement between theories over some language implies that they also preserve it. Thus, the effort to demand more connection between theories puts us in no better position with regard to the problem of the instrumental dodge. Moreover, this result can easily be generalized as the following theorem demonstrates.

We omit the proof, but the key point is that since T and S both think that $0^{\#}$ exists, they can both define their theories of L.Footnote ⁶⁵ For a particularly pathological example, we could let T be $ZFC$ plus the existence of a measurable cardinal and S be $ZFC$ plus the statement that $0^{\#}$ exists and all of the L-consequences of T. Then T and S mutually preserve their theories of L. This is another obvious example of an instrumental dodge. So our efforts to avoid this pathology appear to have been in vain. For this reason, we’ll content ourselves with mere agreement for the rest of this article. Despite this, these results shed some interesting light on the relationship between some natural generalizations of mutual faithful interpretability and sentential equivalence to partial contexts and further work should be done here.

4.1 Too much to ask for

Now we aim to explore the limits of agreement using the framework proposed above. An obvious and natural first question leaps to mind: what is the strongest X-logic we could hope to incorporate into the common ground?Footnote ⁷⁰ The obvious and naive answer is: we’d like to preserve the logic that has just one model, the entire universe. But since the universe is not itself a set, there are tedious Tarskian issues around defining this as an X-logic. Moreover, by including the whole universe, we would have departed from our project to obtain merely partial agreement. So let’s look for what seems to be the next natural step down. Let $V_{\cdot }$ be the class consisting of all rank initial segments of the universe; i.e., $V_{\alpha }$ for all ordinals $\alpha $ . Then for $\Gamma \cup \{\varphi \}\subseteq \mathcal {L}_{\in }$ , we say that $\varphi $ is a $V_{\cdot }$ consequence of $\Gamma $ , abbreviated, $\Gamma \models _{V_{\cdot }}\varphi $ , if for all $\alpha $ , $V_{\alpha }\models \varphi $ whenever $V_{\alpha }\models \Gamma $ . Like $\beta $ -logic, $V_{\cdot }$ -logic has some very natural cousins.

Proof (1 $\leq _{T}$ 2)

Note that $\models _{V_{\cdot }}\varphi $ iff

(4.2) $$ \begin{align} \forall\alpha\forall M(M=V_{\alpha}\to M\models\varphi), \end{align} $$

where the right-hand side is $\Pi _{2}$ since $M=V_{\alpha }$ is $\Pi _{1}$ .Footnote ⁷² So the Turing machine taking $\varphi $ to the corresponding version of 4.2, we get a 1–1 reduction. (2 $\leq _{T}$ 1) Let $\varphi $ be $\Pi _{2}$ and $\psi $ be a sentence of $\mathcal {L}_{\in }$ saying that the universe is a $\beth $ -fixed point. Then it can be seen that $\varphi $ holds iff

$$\begin{align*}\models_{V_{\cdot}}\psi\to\varphi. \end{align*}$$

To see this, note that if $\alpha $ is a $\beth $ -fixed point, then $V_{\alpha }$ is a $\Sigma _{1}$ elementary submodel of V.Footnote ⁷³ Now suppose $\varphi $ is of the form $\forall x\chi (x)$ and suppose it is not true. Then we may fix some x such that $\neg \chi (x)$ . Let $\alpha $ be a $\beth $ -fixed point, where $x\in V_{\alpha }$ . Then $V_{\alpha }\models \psi \wedge \neg \chi (x)$ and so $\not \models _{V_{\cdot }}\psi \to \varphi $ . The proofs of (2 $\leq _{T}$ 3) and (3 $\leq _{T}$ 2) are very similar to (2 $\leq _{T}$ 1) and (1 $\leq _{T}$ 2), respectively. For (2 $\leq _{T}$ 3), we use a sentence of $SOL$ that says its universe is a $\beth $ -fixed point. For (3 $\leq _{T}$ 2), we exploit the fact that being a full model of $SOL$ is $\Pi _{1}$ to see that $\models _{SOL}$ is $\Pi _{2}$ .

I think it’s fair to say that these equivalences reveal that $V_{\cdot }$ -logic is a very natural X-logic. We might even say that $V_{\cdot }$ -logic gives us the strongest, natural notion of satisfiability we could hope to make sense of. When $\varphi $ is $V_{\cdot }$ -satisfiable it is true in an initial segment of the universe that is maximal in the sense that nothing can be added to it without increasing its rank. But is it a plausible place to find agreement between strong set theories? Can it fit in the common ground? Given this relationship with second-order logic, we might wonder if a triviality result like Zermelo’s Theorem 1 or Enayat’s Corollary 19 is on the cards. Perhaps the only way for theories to agree about $V_{\cdot }$ -logic is for those theories to be identical. This is not quite the case.

Thus, we see that agreement between theories over $V_{\cdot }$ -logic doesn’t collapse to identity. But unlike the example from Proposition 33 and its refinement in Theorem 48, it’s not clear that some kind of structural feature is being preserved in the interpretations used above. The forcing arguments that link these theories are, so to speak, tinkering with the heavens rather than preserving something feature that occurs a little closer to the ground. Worse still, agreement over $V_{\cdot }$ -logic also distinguishes theories that seem to be very close to each other. Here is a simple example.

To see this, let $\varphi $ be a sentence of $\mathcal {L}_{\in }$ that says there is a largest cardinal and it is $\aleph _{17}$ . Then $T_{0}$ proves that $\models _{V_{\cdot }}\varphi \to 2^{\aleph _{16}}=\aleph _{17}$ while $T_{1}$ does not. From a practical perspective of applicable set theory, there is not much difference between these theories. But $V_{\cdot }$ -logic is designed to take such differences seriously. It is not difficult to see that this phenomena can be generalized to larger definable cardinals. For these reasons, we now leave $V_{\cdot }$ -logic behind, although we’ll return to a generalization of this X-logic when we come to the end of this investigation.

4.2 The long road

Our search for the common ground now has two examples of X-logics at opposite ends of the common ground spectrum: $\beta $ -logic and $V_{\cdot }$ -logic. While $\beta $ -logic delivers a modest but well-understood amount of common ground, we argued that $V_{\cdot }$ -logic was beyond the pale. In this section, we shall explore some of the vast region between them, with a view to opening up an even wider body of common ground. Our plan is to lift the ideas underlying $\beta $ -logic agreement to deliver a sequence of stronger and stronger X-logics: our long road. While we cannot hope to provide a comprehensive survey here, we will aim to generalize our analysis in such a way that we end up with a template for logical agreement. Our main tools for this journey will be from inner model theory and large cardinals. This section is somewhat technical in comparison to those preceding it and there are fewer philosophical remarks to adorn it. However in essence, our goal is to simply enumerate a number of natural strengthenings of $\beta $ -logic and illustrate the tools that can be used to achieve this effect. I believe a clear and patient illustration of these techniques should provide the first steps toward a more edifying foundational understanding of them in this context. The reader less willing to get bogged in details may simply focus on the defining conditions of these X-logics and the results delimiting the common ground that they obtain.

4.2.1 $\gamma $ -logic

We start by recalling that $\beta $ -consequences form a $\Pi _{2}^{1}$ set and set our initial sights on defining a natural X-logic whose consequence relation is $\Pi _{3}^{1}$ . To do this, we must place a stronger restriction on our target models than mere well-foundedness and for this purpose, we appeal to the theory of sharps.Footnote ⁷⁶ Given a set x we say that $x^{\#}$ exists if there is a non-trivial elementary embedding $j:L(x)\to L(x),$ where $L(x)$ the smallest inner model of $ZF$ which contains x as an element.Footnote ⁷⁷ This fact can be used to obtain a transitive model $M_{0}^{\#}(x)$ of the form $\langle L_{\lambda }(x),\in ,U\rangle $ satisfying certain conditions,Footnote ⁷⁸ the most salient of which are that:

• • $M_{0}^{\#}(x)$ thinks U is a $\kappa $ -complete, normal ultrafilter on some $\kappa <\lambda $ ; and

• • $M_{0}^{\#}(x)$ is linearly iterable by U and its images; i.e., for all $\alpha \in Ord$ , $Ult_{\alpha }(M_{0}^{\#}(x),U)$ is well-founded.Footnote ⁷⁹

Such models are known as mice. The final condition above has a useful implication: by iterating $M_{0}^{\#}(x)$ more than $\omega _{1}$ -times, we obtain an elementary embedding $j:M_{0}^{\#}(x)\to M^{*}$ , where $M^{*}$ is a transitive model with $\omega _{1}\subseteq M^{*}$ . The Lévy–Shoenfield theorem then tells us that $M^{*}$ is correct about $\Pi _{2}^{1}$ statements and thus, so is $M_{0}^{\#}(x)$ . So mice of the form $M_{0}^{\#}(x)$ are, in effect, perfect calculators of $\Pi _{2}^{1}$ truth, much in the same way that countable $\beta $ -models are perfect calculators of $\Pi _{1}^{1}$ truth. We put them to work in our project by using them to define a new X-logic.

Definition 52. Let $\gamma $ be the X-logic consisting of the set of countable transitive models closed under the operation $x\mapsto M_{0}^{\#}(x)$ taking a set to the associated mouse containing it.

We now record a few useful properties of $\gamma $ -logic.Footnote ⁸⁰ Recall that a model N is -correct if for all statements $\varphi (x)$ about the reals and any real x, we have $\varphi (x)^{N}\leftrightarrow \varphi (x)^{V}$ .

Proposition 53. (1) Being an element of $\gamma $ is a $\Pi _{2}^{1}$ property.

(2) $\models _{\gamma }$ is $\Pi _{3}^{1}$ .

(3) If N is in $\gamma $ and $N\models ZFC^{-}$ , then $N[g]$ is correct whenever g is (set) generic over N.

Proof (1) Noting that iterability through just the countable ordinals is sufficient for iterability through all the ordinals, it can be seen that the property of being $M_{0}^{\#}(x)$ is a $\Pi _{2}^{1}$ -property.Footnote ⁸¹ Then we observe that $N\in \gamma $ iff

$$\begin{align*}\underbrace{\forall x\in N\exists y\in N\ \underbrace{(y=M_{0}^{\#}(x))}_{\Pi_{2}^{1}}.}_{\Pi_{2}^{1}} \end{align*}$$

Note that the first two quantifiers don’t increase the complexity since they quantify over elements of N and are thus, in effect, arithmetic. (2) Observe that $\models _{\gamma }\varphi $ iff

$$\begin{align*}\underbrace{\forall\mathcal{M}\in\mathbb{R}(\underbrace{\mathcal{M}\in\gamma}_{\Pi_{2}^{1}}\to\underbrace{\mathcal{M}\models\varphi}_{\Delta_{1}^{1}})}_{\Pi_{3}^{1}}. \end{align*}$$

(3) Let $N\in \gamma $ and let g be N-generic. Then it can be seen that $N[g]$ is still closed under sharps. Now suppose $\varphi (y)$ is $\Pi _{2}^{1}$ , where $y\in \mathbb {R}^{N[g]}$ . We claim that

$$\begin{align*}\varphi(y)^{V}\Leftrightarrow\varphi(y)^{N[g]}. \end{align*}$$

The ( $\Rightarrow $ ) direction follows by downward absoluteness from V to $N[g]$ . For the ( $\Leftarrow $ ) direction, we see that $M_{0}^{\#}(y)$ exists and is a member of $N[g]$ . Suppose that $\varphi (y)^{V}$ . Since $M_{0}^{\#}(y)$ is linearly iterable, there is an elementary embedding $j:M_{0}^{\#}(y)\to M^{*}$ , where $\omega _{1}^{V}\subseteq M^{*}$ . It then follows by the Lévy–Shoenfield theorem that $M^{*}\models \varphi (y)$ and so by the elementarity of j, $M_{0}^{\#}(y)\models \varphi (y)$ .Footnote ⁸² By downward absoluteness again, we see that $N[g]$ thinks that $M_{0}^{\#}(y)$ possesses the $\Pi _{2}^{1}$ property of being linearly iterable, and so we may work in $N[g]$ to obtain $i\in N[g]$ such that $i:M_{0}^{\#}(y)\to M^{\dagger }$ is an elementary embedding, where $\omega _{1}^{N[g]}\subseteq M^{\dagger }$ . Then we see that $M^{\dagger }\models \varphi (y)$ by the elementarity of i, and $N[g]\models \varphi (y)$ by the Lévy–Shoenfield theorem as required.

We, thus, have some understanding of the complexity of this X-logic and a demonstration that it is a natural generalization of $\beta $ -logic. Note that in the argument for (3) as in the argument for Theorem 48, we make heavy use of the Lévy–Shoenfield theorem. In the case of Theorem 48, we did this with the inner model L, whereas here we use the merely countable $M_{0}^{\#}$ in (3) to get the same effect. We might say that $M_{0}^{\#}$ localizes the inner model L. Indeed we can obtain L from $M_{0}^{\#}$ by iterating $M_{0}^{\#}$ along the entirety of the ordinals.Footnote ⁸³ These observations are useful for our current problem: generalizing Theorem 48 to obtain agreement on $\gamma $ -logic. While we demanded that the L was preserved in Theorem 48, we are now going to need to preserve a richer structure. For this purpose, we introduce the inner model $M_{1}$ . In essence, $M_{1}$ is the canonical inner model containing one Woodin cardinal. A little more specifically, $M_{1}$ can be obtained from its sharp $M_{1}^{\#}$ by iterating its active measure out of the universe. $M_{1}^{\#}$ is then a stronger version of a mouse than $M_{0}^{\#}$ that contains a Woodin cardinal and satisfies a minimality condition.Footnote ⁸⁴ Moreover, the iterability required of $M_{1}^{\#}$ makes use of a non-linear iteration described through the device of an iteration tree. We shall be particularly interested in what is known as $\omega _{1}$ -iterability.Footnote ⁸⁵ The following remarkable fact is the driving force behind our current interest in $M_{1}$ and other mice with Woodin cardinals.

Fact 54 [Reference Neeman52]Footnote ⁸⁶ .

Let $x\in \mathbb {R}$ and let $M $ be an $\omega _{1}$ -iterable premouse satisfying $ZF^{-}$ plus the existence of a Woodin cardinal $\delta ,$ where $\delta $ is countable in V. Then there is an elementary embedding $i:M\to M^{*}$ fixing $\delta $ such that there is a $Col(\omega ,\delta )$ -generic g over $M^{*}$ , where $x\in M^{*}[g]$ .

Informally speaking, this tells us that every real number can be captured in a generic extension of an iteration of such a mouse. Or as a slogan: every real is a generic somewhere! This process is known as genericity iteration. This fact will allow us to gain a kind of access to witnesses of existential statements involving the reals. With this in hand, we may obtain our agreement result.

Theorem 55. Suppose T and S are theories that extending $ZFC$ that imply that $M_{1}$ -exists, it is $\omega _{1}$ -iterable, and that its Woodin cardinal is countable. Suppose also that S and T are mutually interpretable by interpretations that preserve $M_{1}$ and its iterability. Then T and S completely agree on $\gamma $ -logic.

Proof Since $\models _{\gamma }$ is $\Pi _{3}^{1}$ , it will suffice to show that T and S agree on $\Pi _{3}^{1}$ statements. Let $\forall x\psi (x)$ be $\Pi _{3}^{1}$ , where $\psi (x)$ is $\Sigma _{2}^{1}$ . Suppose that $T\nvdash \forall x\psi (x)$ and fix a model $\mathcal {M}$ of T such that $\mathcal {M}\models \exists x\neg \psi (x)$ and fix x from $\mathcal {M}$ witnessing this. We aim to show that there is a model of S where this statement is also true. To this end, let $t(\mathcal {M})$ be the model of S given by the $M_{1}$ -preserving interpretation, so we have $M_{1}^{\mathcal {M}}=M_{1}^{t(\mathcal {M})}$ . Working in $\mathcal {M}$ , we then use Fact 54 to obtain and elementary embedding $i:M_{1}\to M^{*}$ such that there is an $M^{*}$ -generic g for $Col(\omega ,\delta ),$ where $x\in M^{*}[g]$ . Since $M^{*}$ is an inner model and $\neg \psi (x)$ is $\Pi _{2}^{1}$ , we see that $\psi (x)^{M^{*}[g]}$ and so we have

$$ \begin{align*} & (\Vdash_{Col(\omega,\delta)}\exists x\neg\psi(x))^{M^{*}}\\ \Leftrightarrow & (\Vdash_{Col(\omega,\delta)}\exists x\neg\psi(x))^{M_{1}} \end{align*} $$

by the elementarity of i. Now working in $t(\mathcal {M})$ we may let h be $Col(\omega ,\delta )$ -generic over $M_{1}$ and so we see $(\exists x\neg \psi (x))^{M[h]}$ . Then since $M_{1}[h]$ is -correct in $t(\mathcal {M})$ and $\exists x\neg \psi (x)$ is $\Sigma _{3}^{1}$ , we see that $t(\mathcal {M})\models \exists x\neg \psi (x)$ and so, $S\nvdash \forall x\psi (x)$ as required. The rest of the cases are similar.

Informally speaking, we start with a model $\mathcal {M}$ of T satisfying $\exists x\neg \psi (x)$ . We use a genericity iteration to capture a witness of this. Elementarity then ensures that this existential statement is also forced in $M_{1}$ . Then since $M_{1}$ is shared by $\mathcal {M}$ and $t(\mathcal {M})$ we may work in $t(\mathcal {M})$ to obtain a generic extension of $M_{1}$ , where the existential statement holds. Absoluteness between $M_{1}$ and $t(\mathcal {M})$ then finishes the proof. The key element, however, is the use of the genericity iteration to capture the required witness. This gives us a nice analogy and generalization of Theorem 48. Instead of merely preserving $L,$ we preserve $M_{1}$ and instead of merely agreeing on $\beta $ -logic, we agree on $\gamma $ -logic. Here is a helpful illustration of its application.

Example 56. Consider the following theories:Footnote ⁸⁷

1. (1) $ZFC$ plus there is a Woodin cardinal;

2. (2) $ZFC$ plus $\Pi _{2}^{1}$ -determinacy;

3. (3) $ZFC$ plus there is an $\omega _{2}$ -saturated ideal on $\omega _{1}$ .

Woodin showed that these theories are equiconsistent. Moreover, the interpretations witnessing this can be seen to be $M_{1}$ -preserving and so by Theorem 55, we see that they all agree on $\gamma $ -logic.

Thus, we have a number of natural extensions of $ZFC$ that all agree on $\gamma $ -logic and indeed on any $\Pi _{3}^{1}$ statement. This cements our claim to having generalized $\beta $ -logic.

4.2.2 $\gamma _{n}$ -logic

Our next goal is to show how this technique can be taken further up through the projective hierarchy by preserving more structure in the form of inner models with the pleasing side effect of also obtaining agreement on stronger X-logics. First we describe our next X-logic. For $n\geq 1$ and $x\in \mathbb {R}$ , $M_{n}^{\#}(x)$ is the canonical countable mouse with n Woodin cardinals. Its canonicity is witnessed by a minimality condition.Footnote ⁸⁸

Definition 57. For $n\geq 1$ , let $\gamma _{n}$ be the set of countable transitive models N closed under $x\mapsto M_{n}^{\#}(x)$ for for $x\in \mathbb {R,}$ where $M_{n}^{\#}(x)$ is $\omega _{1}$ -iterable in N and V.

For the inner models that our interpretations will preserve, we note that $M_{n}(x)$ is the inner model obtained from $M_{n}^{\#}(x)$ by iterating its topmost and active extender out of the universe.Footnote ⁸⁹ It is the canonical inner model containing n many Woodin cardinals and $x\in \mathbb {R}$ .Footnote ⁹⁰ The $M_{n}$ s will play a similar role to L and $M_{1}$ in our previous agreement results. Toward our agreement claim, we then observe that the models in $\gamma _{n}$ are increasingly correct; and that this correctness corresponds with preservation of inner models of the form $M_{n}$ . We note that for some technical issues, these results are sensitive to whether we use odd or even $n\in \omega $ .Footnote ⁹¹ The correctness result rests on the following fact.

Fact 58. Suppose $n\geq 0$ is even and $M_{n}^{\#}(x)$ exists and is $\omega _{1}$ -iterable for all $x\in \mathbb {R}$ . Then $M_{n}^{\#}(x)$ is -correct.Footnote ⁹²

Lemma 59. (1) If n is odd and $N\in \gamma _{n}$ , then N is -correct.Footnote ⁹³

(2) If V and W are models of $ZFC,$ where $M_{n}^{V}=M_{n}^{W}$ for odd n, then for all $\varphi \in \Pi _{n+2}^{1}$

$$\begin{align*}\varphi^{V}\Leftrightarrow\varphi^{W}. \end{align*}$$

Proof (1) We proceed by induction supposing that we have established the claim for $m<n$ . Fix N as described above and let $\exists x\psi (x,y)$ be $\Sigma _{n+2}^{1}$ with $y\in \mathbb {R}^{N}$ and $\psi (x,y)\in \Pi _{n+1}^{1}$ .Footnote ⁹⁴ Now fix $x\in V$ witnessing this. We claim that

$$\begin{align*}\exists x\psi(x,y)^{V}\leftrightarrow N\models\exists x\psi(x,y). \end{align*}$$

( $\Rightarrow $ ) Since $N\in \gamma _{n}$ , we know that $M_{n}^{\#}(y)\in N$ and is $\omega _{1}$ -iterable in N and V. Let $\delta _{0}$ be the least Woodin cardinal according to $M^{\dagger }=M_{n}^{\#}(y)$ . Using Fact 54, we may fix $i:M^{\dagger }\to M^{*}$ and g that is $Col(\omega ,\delta _{0})$ -generic over $M^{*}$ such that $x\in M^{*}[g]$ . Now it can be seen that $M^{*}[g]$ is $M_{n-1}^{\#}(x\oplus y)$ and so by Fact 58, it is -correct.Footnote ⁹⁵ This means that $M^{*}[g]\models \psi (x,y)$ and so we have by elementarity thatFootnote ⁹⁶

$$ \begin{align*} & (\Vdash_{Col(\omega,\delta_{0})}\exists x\psi(x,y))^{M^{*}}\\ \Leftrightarrow & (\Vdash_{Col(\omega,\delta_{0})}\exists x\psi(x,y))^{M^{\dagger}}. \end{align*} $$

Now working in N we may fix $h\in N$ that is $Col(\omega ,\delta )$ -generic over $M^{\dagger }$ . So we have $M^{\dagger }[h]\models \exists x\psi (x,y)$ . Fixing some $x\in \mathbb {R}^{M^{\dagger }[h]}$ witnessing this, it can be seen that $M^{\dagger }[h]$ is $M_{n-1}^{\#}(x\oplus y)$ and thus -correct by Fact 58.Footnote ⁹⁷ Finally, by upward absoluteness, we see that $\exists x\psi (x,y)$ holds in V.

( $\Leftarrow $ ) Suppose $\psi (x,y)$ is of the form $\forall z\chi (x,y,z),$ where $\chi (x,y,z)$ is $\Sigma _{n}^{1}$ . Now fix $x\in N$ such that $N\models \forall z\chi (x,y,z)$ and suppose toward a contradiction that

$$\begin{align*}\exists z\neg\chi(x,y,z)^{V}. \end{align*}$$

Noting that $\exists z\neg \chi (x,y,z)$ is $\Pi _{n+1}^{1}$ and thus $\Sigma _{n+1}^{1}$ we may reuse the ( $\Rightarrow $ ) part of proof to obtain the result. The proof of (2) is very similar.

Using this lemma, we can now obtain agreement over $\gamma _{n}$ -logics using strong theories and interpretations that preserve the right inner models.

Corollary 60. Let $n\in \omega $ be even. Suppose T and S are theories extending $ZFC$ that imply that $M_{n+1}$ -exists, it is $\omega _{1}$ -iterable, and that its Woodin cardinal is countable. Suppose also that S and T are mutually interpretable by interpretations that preserve $M_{n+1}$ and its iterability. Then T and S completely agree on $\gamma _{n}$ -logic.

Proof To see this first note that $M_{n}^{\#}(x)$ is a $\Pi _{n+2}^{1}$ -singleton.Footnote ⁹⁸ Thus, by a similar calculation to that in Proposition 53 (1) and (2), we see that the consequence relation of $\gamma _{n}$ is $\Pi _{n+3}^{1}$ . Then Lemma 59 can be used to close out the proof.

Through the use of Woodin cardinals and preservation of their canonical models, we now have a way of obtaining agreement for X-logics representing any odd level of the projective hierarchy. Thus, we’ve generalized the ideas of $\gamma $ -logic to through every rung of the theory of analysis.