Proof. Since the rank-initial segments $V_\beta $ are linearly ordered, it follows by Theorem 2 that every assertion in S4.3 is valid for rank potentialism, with respect to any language, verifying the first inclusion. The central inclusion $\mathord {\textrm {Val}}_{\mathcal {V}}(V_\beta ,\mathcal {L}^\Diamond _{\in ,V_\beta })\subseteq \mathord {\textrm {Val}}_{\mathcal {V}}(V_\beta ,\mathcal {L}_\in )$ follows from the fact that it is a less severe requirement for a propositional modal assertion to be valid with respect to less expressive language. Finally, to show the last inclusion $\mathord {\textrm {Val}}_{\mathcal {V}}(V_\beta ,\mathcal {L}_\in )\subseteq S5$ , it suffices by Theorems 3 and 4 to show that we have an infinite dial that works in all these worlds. For $j<\omega $ , let $d_j$ be the assertion that the height of the ordinals is $\lambda +j$ , where $\lambda $ is a limit ordinal or zero. So $d_j$ is true in $V_\beta $ if and only if $\beta =\lambda +j$ for some limit ordinal $\lambda $ or zero. These statements are expressible in the language of set theory $\mathcal {L}_\in $ without parameters or modal vocabulary. And they form a dial, because every ordinal $\beta $ is uniquely expressed as $\lambda +j$ for some limit ordinal $\lambda $ or zero and some finite $j<\omega $ , and from any $V_\beta $ we can extend to a larger $V_{\lambda +j}$ so as to realize any desired j.

Proof. Let $r_k$ be the assertion that the ordinal $\omega \cdot k$ exists. These statements are expressible in the language of set theory without parameters, since $r_k$ is equivalent in any $V_\beta $ to the assertion that there are at least k distinct limit ordinals. The statements constitute a ratchet for the system of rank potentialism, since once the ordinal $\omega \cdot k$ exists in $V_\beta $ , then all smaller $\omega \cdot r$ also exist; they continue to exist in all larger $V_\delta $ ; and if $\omega \cdot k$ does not yet exist in $V_\beta $ , then we can extend $V_\beta $ to $V_{\omega \cdot k+1}$ , where $\omega \cdot k$ exists but $\omega \cdot (k+1)$ does not yet exist. For any fixed $n<\omega $ and any $i<n$ , let $d_i^*$ assert that the dial value $d_j$ of the previous theorem has $i=j\mod n$ . That is, we use the dial values of the previous theorem modulo n. The point is that we can realize any dial value for $d_i^*$ without raising the ratchet volume, since we simply move to the next block of size n, and so we have independent ratchet and dials of any desired size, and these are expressible in the language of set theory without parameters or modal operators. It follows by Theorem 6 that the validities of $V_\beta $ for the language of set theory are precisely S4.3, for sufficiently small $\beta $ , as desired.

Proof. Theorems 8 and 9 show that S4.3 is valid in all rank-potentialist worlds $V_\beta $ , and there is some world realizing only the validities in S4.3. So it is precisely S4.3 that is valid in all worlds, and consequently $\mathord {\textrm {Val}}_{\mathcal {V}}(\mathcal {L}_\in ) = S4.3$ . It follows by Theorem 2 that $\mathord {\textrm {Val}}_{\mathcal {V}}(\mathcal {L}_\in ^+) = S4.3$ for any extension of the language, since the $V_\beta $ are linearly ordered.

Proof. ( $2\mathrel {\rightarrow } 1$ ) Assume that $\delta $ is $\Sigma _3$ -correct. Since we already know that S4.3 is valid at $V_\delta $ , it suffices to verify the validity of Axiom 5. So suppose that for some assertion $\varphi $ in the language of set theory with parameters $\vec a\in V_\delta $ , using the rank-potentialist semantics. This means that there is some $\lambda \geq \delta $ , such that for all $\theta \geq \lambda $ we have $V_\theta \models \varphi (\vec a)$ . The assertion $\exists \lambda \,\forall \theta \geq \lambda \,V_\theta \models \varphi (\vec a)$ has complexity $\Sigma _3(\vec a)$ in the language of set theory,Footnote ⁵ and so this assertion must be true in $V_\delta $ , thereby verifying this instance of S5. And so the potentialist maximality principle is true at $V_\delta $ for assertions in the language of set theory with parameters from $V_\delta $ .

( $1\mathrel {\rightarrow } 2$ ) Conversely, assume that S5 is valid in $V_\delta $ for assertions in the language of set theory with parameters in $V_\delta $ . This implies that $\delta $ is a $\beth $ -fixed point, since for any ordinal $\alpha $ it is possibly necessary that $\beth _\alpha $ exists (one need only grow the universe sufficiently tall), and therefore, for any $\alpha <\delta $ we know that $\beth _\alpha $ exists already in $V_\delta $ , and so $\delta =\beth _\delta $ . This implies $V_\delta =H_\delta $ and so $V_\delta \prec _{\Sigma _1} V$ by the Lévy reflection theorem; so $\delta $ is $\Sigma _1$ -correct. A similar argument works more generally to establish $\Sigma _3$ -correctness. Specifically, suppose that a $\Sigma _3$ assertion holds in V, using some parameters $\vec a\in V_\delta $ . Every $\Sigma _3$ assertion is equivalent to an assertion of the form $\exists x\forall \beta \,V_\beta \models \psi (x,\vec a)$ . Thus, , since one need only grow the universe sufficiently tall until the witness x is found. By S5, we may conclude that $V_\delta \models \exists x\forall \beta \, V_\beta \models \psi (x,\vec a)$ , showing that the $\Sigma _3$ assertion holds in $V_\delta $ . So $\delta $ is $\Sigma _3$ -correct, as desired.

Proof. Theorem 12 shows that this is true for any $\Sigma _3$ -correct cardinal $\delta $ , and there is a proper-class club of such cardinals.

Proof. ( $2\mathrel {\rightarrow } 1$ ) Assume that $\delta $ is correct, so that we have the scheme $V_\delta \prec V$ , and suppose that is true at $V_\delta $ , where $\varphi $ is an assertion in the potentialist language $\mathcal {L}_\in ^\Diamond $ and $\vec a\in V_\delta $ . Thus, $\exists \lambda \forall \theta \geq \lambda \, V_\theta \models \varphi (\vec a)$ is true in V. Since the rank-potentialist modalities are expressible in the language of set theory, it follows from the $V_\delta \prec V$ scheme that $V_\delta \models \exists \lambda \forall \theta \geq \lambda \, V_\theta \models \varphi (\vec a)$ . Note that since $V_\delta \prec V$ , it follows that $V_\delta $ and V agree on whether some particular $V_\theta $ satisfies a given rank-potentialist assertion. Since $V_\delta \prec V$ , it follows that the witness $\lambda <\delta $ also works in V, and so $V_\delta \models \varphi (\vec a)$ , verifying this instance of S5. So $V_\delta $ satisfies the desired potentialist maximality principle.

( $1\mathrel {\rightarrow } 2$ ) Conversely, suppose that S5 holds at $V_\delta $ for assertions in the potentialist language $\mathcal {L}_\in ^\Diamond $ with parameters in $V_\delta $ . We know from Statement 1 that $\delta $ is $\Sigma _3$ correct. Suppose that $\delta $ is $\Sigma _n$ -correct, and that a $\Sigma _{n+1}$ assertion is true in V. So $V\models \exists x\,\psi (x,\vec a)$ , where $\psi $ has complexity $\Pi _n$ . By Theorem 1, the set-theoretic assertion $\psi (x,\vec a)$ is true in V if and only if $\psi ^\Diamond (x,\vec a)$ is true at any $V_\beta $ containing x and $\vec a$ . Thus, , since once one goes high enough, the witness x from V will exists. By S5, therefore, there is $x\in V_\delta $ with $V_\delta \models \psi ^\Diamond (x,\vec a)$ , which means that $V\models \psi (x,\vec a)$ , and so we have found the desired witness x inside $V_\delta $ . So $\delta $ is $\Sigma _{n+1}$ -correct, as desired.

Proof. Theorem 14 shows that this is true exactly at the (fully) correct cardinals $\delta $ . The existence of a correct cardinal is independent of but equiconsistent with ZFC.

Proof. Since the worlds $V_\beta $ in $\mathcal {V}_A$ are linearly ordered, it follows that every assertion of S4.3 will be valid for the potentialist semantics with respect to any of the languages. To show that some worlds validate only S4.3, it suffices by Theorem 7 to find uniform long ratchets. Consider first the case where the definition of A is absolute to $V_\beta $ for every $\beta \in A$ . For example, this would be true if A was the class of strong limit cardinals, or the class of $\Sigma _n$ -correct cardinals, or if A was the class of $\beta $ for which $V_\beta $ satisfied a certain specific c.e. theory T. For any ordinal $\alpha $ , let $r_\alpha $ be the assertion that there are at least $\alpha $ many ordinals in A. These statements are expressible in the language of set theory and they form a long ratchet, since once they are true in some $V_\beta $ , they remain true in all taller models $V_\delta $ , by the absoluteness assumption on A; they are all false in the smallest model $V_\beta $ where $\beta $ is the least element of A; they have the necessary downward implications; and for any $\alpha $ , we can let $\beta $ be the $(\alpha +1)^{\textrm {st}}$ element of A and observe that $V_\beta $ thinks that there are precisely $\alpha $ many elements of A (again using the absoluteness of A); it follows that $r_\alpha $ is true in $V_\beta $ but not $r_{\alpha +1}$ . Because we have therefore produced a uniform long ratchet, expressible in the language of set theory, it follows from Theorem 7 that the propositional modal validities of any sufficiently small world $V_\beta $ of $\mathcal {V}_A$ , that is, where the ratchets have not yet been cranked, will be exactly

$$ \begin{align*}S4.3=\mathord{\mathrm{Val}}_{\mathcal{V}_A}(V_\beta,\mathcal{L}_{\in,V_\beta}^\Diamond)=\mathord{\mathrm{Val}}_{\mathcal{V}_A}(V_\beta,\mathcal{L}_\in),\end{align*} $$

realizing the lower bound in a strong way. Furthermore, Theorem 7 also shows that every world of $\mathcal {V}_A$ will have its validities contained in S5, establishing the upper bound.

Consider now the case where A is definable, but not necessarily absolute to all these $V_\beta $ . In this case, the statements $r_\alpha $ of the previous paragraph might not be a ratchet, since perhaps some $V_\beta $ thinks wrongly that there are $\alpha $ many elements of A, but a larger $V_\delta $ recognizes that many of those ordinals are not actually in A. Nevertheless, we can fix this problem simply by making our ratchet assertions in the potentialist modal language $\mathcal {L}_\in ^\Diamond $ , rather than in the language of set theory. Specifically, for our ratchet statements, we use instead the assertion $\bar r_\alpha $ that asserts that there are $\alpha $ many ordinals $\xi $ for which $(\xi \in A)^\Diamond $ , using the potentialist translation of the definition of A in V. By Theorem 1, the assertion $(\xi \in A)^\Diamond $ in some $V_\beta $ is equivalent to $\xi \in A$ in V, and so $\bar r_\alpha $ is true in some $V_\beta $ just in case there are at least $\alpha $ many elements of A below $\beta $ . (This is a sneaky trick, and the cost is that our analysis on the bounds will apply only for $\mathcal {L}^\Diamond $ -substitution instances; it could be that additional validities hold when one considers only substitution instances in the language of set theory $\mathcal {L}_\in $ alone.) These revised statements in effect are able to refer via the potentialist translation to the actual class A, and they form a long ratchet for the same reasons that $r_\alpha $ did in the previous paragraph. Thus, in any case, the modal validities of the smallest world of $\mathcal {V}_A$ is exactly S4.3 for assertions in the potentialist language, again realizing the lower bound, and every world has its validities for that language contained in S5, establishing the upper bound.

Proof. This is simply an adaptation of Theorem 16. Since we have added A explicitly to the structures, it is in effect absolute between these structures, and so we get the stronger conclusions of that theorem, but only in the language in which A is a predicate.

Proof. The proof is essentially identical to the proofs of Theorems 8 and 9. Namely, the S4.3 assertions are valid since the $H_\kappa $ are linearly ordered. Conversely, one can define ratchets for the $H_\kappa $ by letting $r_\alpha $ assert that there are $\alpha $ many regular cardinals $\kappa $ for which $H_\kappa $ is allowed as a universe fragment, using the potentialist translation as in the proof of Theorem 16. So there are worlds $H_\kappa $ in which only S4.3 is valid, and meanwhile, every world has its validities within S5.

Proof. This is an immediate consequence of Theorem 16. Note that because the class of inaccessible cardinals is absolute to $V_\kappa $ , we obtain the stronger conclusions of that theorem.

Proof. ( $2\to 1$ ) Assume that $\kappa $ is a $\Sigma _3$ -reflecting cardinal. So $\kappa $ is inaccessible and $V_\kappa $ is a Grothendieck–Zermelo universe. Since $V_\kappa \prec _{\Sigma _3}V$ and V has an inaccessible cardinal, it follows that $V_\kappa $ must have an inaccessible cardinal. Similarly, for any $\beta <\kappa $ we know that V has an inaccessible cardinal above $\beta $ , and so $V_\kappa $ must agree. So the inaccessible cardinals are unbounded in $\kappa $ . Thus, $V_\kappa $ satisfies the Grothendieck universe axiom. This axiom is itself a $\Pi _3$ assertion, and so it is true in V. So the Grothendieck universe axiom holds. Now suppose that holds at $V_\kappa $ in the GZ-potentialist semantics, where $\varphi $ is an assertion in the language of set theory and $\vec a\in V_\kappa $ . Thus, there is some inaccessible cardinal $\lambda $ such that for all inaccessible cardinals $\theta \geq \lambda $ we have $V_\theta \models \varphi $ . As before, this is a $\Sigma _3$ assertion, and so it is true in $V_\kappa $ . So there is a $\lambda <\kappa $ , such that every inaccessible cardinal $\theta \geq \lambda $ in $V_\kappa $ has $V_\theta \models \varphi (\vec a)$ . The assertion that $\lambda $ has this property has complexity $\Pi _2$ , and so this $\lambda $ also works in V. Therefore, $V_\kappa \models \varphi (\vec a)$ , verifying this instance of the maximality principle, as desired.

( $1\to 2$ ) Assume that the Grothendieck universe axiom holds and that the GZ-potentialist maximality principle is true at GZ-universe $V_\kappa $ . In particular, $\kappa $ is an inaccessible cardinal. So it is a $\beth $ -fixed point and therefore $\Sigma _1$ -correct. If a $\Sigma _2$ assertion is true in V, then this is witnessed in all large enough $V_\theta $ , including when $\theta $ is inaccessible, and so by the maximality principle, it is witnessed in $V_\kappa $ , so $\kappa $ is $\Sigma _2$ -correct. Finally, suppose that some $\Sigma _3$ -assertion is true. All such statements have the form $\exists x\forall y\, \psi (x,y,\vec a)$ , where $\psi $ has complexity $\Sigma _1$ . This implies at $V_\kappa $ , since once $\lambda $ is large enough, then x will exist in $V_\theta $ for any larger $\theta $ . So by the maximality principle, we conclude that $V_\kappa \models \exists x\forall y\,\psi (x,y,\vec a)$ , verifying this instance of $\Sigma _3$ -correctness. Since $\kappa $ is inaccessible, we have proved that $\kappa $ is $\Sigma _3$ -reflecting.

Proof. Simply adapt the proof of Theorem 14 in the same way that Theorem 22 adapts the proof of Theorem 12.

Proof. Since the union of two transitive sets is transitive, it follows that the potentialist system of all transitive sets is directed and therefore by Theorem 2 every assertion of S4.2 is valid. To achieve S5 as an upper bound, it suffices by Theorems 3 and 4 to show that this system admits an infinite dial. For this, we can use the same dial $d_j$ as in Theorem 8, namely, the assertion $d_j$ that the ordinals have the form $\lambda +j$ where $\lambda $ is either a limit ordinal or zero. This is expressible by a sentence in the language of set theory, correctly interpreted inside any transitive set, and since any transitive set can be extended to any desired larger ordinal height, it forms a dial.

To show that the lower bound is sharp, it suffices by Theorem 5 to find an independent family of buttons, independent of this dial. For this, let $\left \langle B_k\mid k\in \omega \right \rangle $ be an infinite list of distinct arithmetically definable infinite, co-infinite sets of natural numbers. Let $b_k$ be the assertion, “ $V_\omega $ exists and also $B_k$ exists.” Note that this assertion is upward absolute for transitive sets, since if $V_\omega $ and $B_k$ exist in some transitive set, then this will continue to be true in any larger transitive set. So each $b_k$ is a button. Furthermore, these assertions form an independent family of buttons, since if M is any transitive set, then $M\cup V_\omega \cup \left \{{B_k}\right \}$ is the union of transitive sets and hence also transitive, and we have added $B_k$ without adding any other $B_j$ for $j\neq k$ . Since these buttons and dial values can be controlled independently of each other without interference, it follows by Theorem 5 that the modal validities in any world where infinitely many of the buttons are not yet pushed will be contained within and hence equal to S4.2, as desired.

Proof. ( $2\mathrel {\rightarrow } 1$ ) Consider $M=V_\delta $ , where $\delta $ is $\Sigma _2$ -correct, and assume holds in M. So there is a transitive set N such that in all larger transitive sets $U\supseteq N$ , we have $U\models \varphi (\vec a)$ . This is a $\Sigma _2$ assertion about $\vec a$ , and so it must already be true in M. So there is transitive set $N\in M$ such that, inside M, every transitive $U\supseteq N$ satisfies $\varphi (\vec a)$ . This is a $\Pi _1$ -property about $\vec a$ and this particular N, which therefore holds in V. Since M itself is such a transitive set, we conclude $M\models \varphi $ , as desired.

( $1\mathrel {\rightarrow } 2$ ) Assume that the potentialist maximality principle holds in M for assertions in the language of set theory with parameters from M. First, we claim that M is correct about power sets. If a is any set in M, then we claim that . In  other words, the existence of the power set of a is possibly necessary. This is simply because once you move to a transitive set that has the actual power set of a, then it necessarily is the power set of a in all larger transitive sets. So M thinks $P(a)$ exists (that is, the power set relativized to M). If M does not have the actual power set of a, then there is some $u\subseteq a$ with $u\notin M$ , in which case $M\cup \left \{{u}\right \}$ is a transitive set containing M, which does not think that the power set of a exists, contrary to this being necessary over M. So M computes the power sets correctly. A similar argument shows that M computes $V_\alpha $ correctly for any ordinal $\alpha \in M$ , since the existence of $V_\alpha $ is possibly necessary, and M cannot have a fake version of some $V_\alpha $ , since in that case it would have to be wrong about some power sets. Also, for any set a it is possible necessary that $a\in V_\alpha $ for some ordinal $\alpha $ , and so this is already true in M. Thus, $M=V_\delta $ for some ordinal $\delta $ . To see that $\delta $ must be $\Sigma _2$ -correct, suppose that a $\Sigma _2$ assertion $\varphi (\vec a)$ is true in V. This is witnessed by the existence of some ordinal $\beta $ for which $V_\beta \models \psi (\vec a)$ for some assertion $\psi $ . So it is possibly necessary that, “there is an ordinal $\beta $ for which $V_\beta $ exists and satisfies $\psi (\vec a)$ .” Thus, this must already be true in M, and so $\varphi (\vec a)$ is true already in M, as desired. So $\delta $ is $\Sigma _2$ -correct.

Proof. This is like the proof of Theorem 14 combined with the ideas of Theorem 25. If the maximality principle holds for world M in the potentialist language, then we know by Theorem 25 that $M=V_\delta $ for a $\Sigma _2$ -correct cardinal $\delta $ . But since the transitive sets form a potentialist account of V in the manner of Theorem 1, we can as in Theorem 14 show that $\delta $ is $\Sigma _n$ -correct for every n.

Conversely, assume that $M=V_\delta \prec V$ and holds in M, where $\varphi $ is a $\mathcal {L}_\in ^\Diamond $ assertion. So there is a transitive set N, such that in all larger transitive sets $U\supseteq N$ we have $U\models \varphi (\vec a)$ . Since the existence of a such a set N and the potentialist semantics are expressible in the language of set theory, it follows from $M\prec V$ that there is such a set already in M, and consequently $M\models \varphi (\vec a)$ , as desired.

Proof. The collection of models $\mathcal {T}_T$ provides a potentialist account of V, since if $W\in \mathcal {T}_T$ , then $\left \langle W,\in \right \rangle $ is a substructure of $\left \langle V,\in \right \rangle $ and if x is any set in V, then by the assumption on T we have a transitive model containing $\{W,x\}$ as an element, which must therefore have $W\subseteq U$ . So the weak directedness property is fulfilled. But actually, a similar argument shows that $\mathcal {T}_T$ is fully directed: if $W_0,W_1\in \mathcal {T}_T$ , then the assumption on T ensures that there is $U\in \mathcal {T}_T$ with $\{W_0,W_1\}\in U$ , which ensures $W_0,W_1\subseteq U$ , as desired. By Theorem 2, therefore, S4.2 is valid at every world for the potentialist account provided by $\mathcal {T}_T$ , verifying Statement 1.

Meanwhile, observe that $\mathcal {T}_T$ is well-founded with respect to $\in $ , and so for any ordinal $\alpha $ , we may let $r_\alpha $ be the assertion, “the $\in $ -relation on the transitive models of T has rank at least $\alpha $ .” Our hypothesis in the theorem that T is sufficient is meant to ensure that models of T are able to refer to T by means of its arithmetic definition and are also able to calculate the rank of a well-founded relation (so even some very weak set theories suffice). Since the rank can never go down as the transitive sets get larger, it follows that each $r_\alpha $ will necessarily imply all earlier $r_\beta $ for $\beta <\alpha $ . And if $r_\alpha $ is not true in some transitive $W\models T$ , then we may simply extend W to a transitive set $U\models T$ of rank $\alpha +1$ , so that $W\subseteq U$ and $U\models r_\alpha \wedge \neg r_{\alpha +1}$ . So this is a uniform long ratchet, and therefore by Theorem 7 any world in which this ratchet has not yet begun will have its validities contained within S4.3. And furthermore, the same theorem provides S5 as an upper bound for all the worlds.

Let us now give some examples showing that various modal theories can arise, depending on the theory and on the set-theoretic background. For the easiest case, if we take the empty theory, then $\mathcal {T}_T$ is the same as the class $\mathcal {T}$ of all transitive sets, whose modal validities we have already established as S4.2 in Theorem 24. Next, assume $V=L$ in the background theory, and consider the theory T consisting of the assertion “V=L,” so that the transitive models of this theory are exactly the models $L_\alpha $ appearing in the constructibility hierarchy. These are linearly ordered, and so S4.3 is valid for this potentialist system, and the validities are contained within S4.3 by the previous paragraph, yielding exactly S4.3 as the common modal theory.

Finally, assume $V=L$ in the background, and let’s describe a theory T for which the corresponding potentialist system $\mathcal {T}_T$ exhibits modal validities strictly in between S4.2 and S4.3. Let ${\textrm {ZFC}}^*$ be a fixed finite fragment of ${\textrm {ZFC}}$ , which can prove the existence of the reals and which furthermore is invariant by Cohen-real forcing and by random-real forcing. Let T be the theory asserting that either (i) ${\textrm {ZFC}}^*+V=L$ ; or (ii) ${\textrm {ZFC}}^*+V=L[c]$ for an L-generic Cohen real c, but there is no transitive set model of ${\textrm {ZFC}}^*$ ; or (iii) ${\textrm {ZFC}}^*+V=L[r]$ for an L-generic random real, but there is no transitive set model of ${\textrm {ZFC}}^*$ . For example, since ${\textrm {ZFC}}^*$ is true in L, it follows by the finiteness of ${\textrm {ZFC}}^*$ that it is true in some large $L_\gamma $ , and by condensation the least $\gamma $ for which $L_\gamma \models {\textrm {ZFC}}^*$ is countable. Thus, we can easily construct $L_\gamma $ -generic reals c or r and form the extensions $L_\gamma [c]$ and $L_\gamma [r]$ . By using the least possible $\gamma $ , these models will have no set models of ${\textrm {ZFC}}^*$ , and so we can see that both statements (ii) and (iii) are possible from some $L_\gamma $ . Note that if one goes to a higher model, however, then (ii) and (iii) become impossible, because of the requirement that there is no transitive set model of ${\textrm {ZFC}}^*$ . In particular, both (ii) and (iii) are possible from the smallest $L_\gamma $ satisfying ${\textrm {ZFC}}^*$ , but once you have one of them, the other becomes impossible, since the forcing notions do not add generic reals of the other type. So this is exactly a violation of axiom (.3) in $L_\gamma $ , and therefore the validities of this potentialist system are not S4.3. Meanwhile, we claim that the validities exceed S4.2. To see this, note that because of the homogeneity of Cohen forcing and random-real forcing, all the models of statement (ii) have the same theory, and similarly for statement (iii). Thus, every transitive model of T is either a model of statement (i), and these are all linearly ordered, or else is a model of the theory of statement (ii) or of statement (iii). So it is impossible that a model of T has three independent unpushed buttons. In particular, the assertion “p, q and r are not three independent unpushed buttons,” which is expressible in propositional modal logic, is valid for the potentialist system determined by the transitive models of T, but this assertion is not an S4.2 theorem, since some models of S4.2 do have three independent unpushed buttons. So the modal validities of $\mathcal {T}_T$ for this theory are strictly intermediate between S4.2 and S4.3, as claimed.

Proof. We’ve already mentioned that S4.2 is always valid for the potentialist semantics of the generic multiverse, which is the same as the modal logic of forcing for those models, and exactly S4.2 is realized for every model when parameters are allowed. Meanwhile, we also know that some models of set theory can exhibit S5 as their forcing validities, as before. In order to prove the theorem, therefore, what we shall show is that every model of set theory contains a ground whose validities are exactly S4.2, without parameters. For this, it will suffice to show that every model of set theory M has a ground model W by set-forcing such that W has arbitrarily large finite families of independent buttons and switches, expressible without parameters. In order to prove this fact, we shall rely on the recent breakthrough result of Usuba [Reference Usuba27], establishing the strong downward-directed grounds hypothesis DDG, which asserts that the ground models of the set-theoretic universe V are downward set-directed: for any set-indexed family of grounds $\{{W_i\mid i\in I}\}$ in the uniform ground-model enumeration, there is a ground model W with $W\subseteq W_i$ for all $i\in I$ . It follows that for every model of set theory M and for every ordinal $\gamma $ in M, there is a ground model $W\subseteq M$ such that $V_\gamma ^W$ is the same as the $V_\gamma $ of the mantle of M, the intersection of all the grounds of M. In other words, $V_\gamma ^W$ is least among the $V_\gamma $ ’s of the grounds of M. In particular, the cardinal and GCH structure of $V_\gamma ^W$ on the cardinals up to, say, $\aleph _{\omega +\omega }$ is absolutely definable in every model of the generic multiverse of M, since the mantle itself is definable in those models and W agrees with the mantle in that realm.

We claim that this ground W has independent buttons and switches, and therefore its valid principles of forcing are exactly S4.2. For this, we cannot seem to use the stationary-set buttons that Hamkins and Löwe described in [Reference Hamkins, Leibman and Löwe14, sec. 4], since those make reference to an absolutely definable partition using the L-order. Nevertheless, we may modify the buttons identified by Rittberg, or alternatively modify the buttons of Friedman, Fuchino and Sakai, but using in each case the cardinals as defined in the mantle rather than L. Since as we mentioned the mantle is forcing-invariant and therefore an absolutely definable class common to all the models in the generic multiverse, the referent of such expressions as $\aleph _2^{\textrm {Mantle}}$ , for example, is invariant in this generic multiverse. For this reason, the buttons and switches relativized to the mantle rather than L will have the absoluteness properties that made them function as independent buttons and switches. Thus, the modal logic of forcing over W is exactly $\mathord {\textrm {Val}}(W,\mathcal {L}_\in )=S4.2$ , as desired.

Proof. Since the inclusion relation is both transitive and reflexive on $\mathcal {M}$ , it follows that S4 is valid at every world, for any language interpreted in those worlds. For the validity of .2, we cannot appeal to directedness, since as we noted earlier this collection of models has non-amalgamation and therefore is not directed; nevertheless, .2 is valid as in the modal logic of forcing because of product forcing. Namely, if holds at world $V_\beta ^W$ , where $W=W_r[G]$ and $r,G,{\mathbb Q}\in V_\beta ^W$ , then it means there is some larger $V_\gamma ^{U}$ , with $U=U_s[H]$ and forcing $s,H,{\mathbb P}\in V_\gamma ^U$ , such that any still larger $V_\delta ^{U'}$ satisfies $\varphi (\vec a)$ . Thus, this situation is forced by some condition in P over $U_s$ . So for any $V_\theta ^{W'}$ extending $V_\beta ^W$ , we can still force over $W'$ with that forcing and find a common extension $V_\lambda ^{W''}$ satisfying $\varphi (\vec a)$ . So S4.2 is valid.

Meanwhile, it is easy to see that the validities of any particular world are contained within S5, since one can use the GCH pattern as a family of independent switches. One can always force over any given model so as to achieve any desired finite pattern for the GCH at the cardinals $\aleph _n$ .

Finally, consider the generic-multiverse rank-potentialist system $\mathcal {M}$ arising from a model of ${\textrm {ZFC}}+V=L$ . We may use the same independent buttons and switches of Theorem 28, the point being that one doesn’t need the entire universe to verify the status of the buttons and switches that were used there. So it is enough to have $V_\beta ^{L[G]}$ for large enough $\beta $ in order to operate the buttons and switches.

Proof. The potentialist accessibility relation is inclusion, which is transitive and reflexive. Thus, S4 is valid at every world, even with respect to expansions of the language interpreted in every world of $\mathcal {C}$ . Meanwhile, if W is a world in $\mathcal {C}$ , then W admits arbitrarily large independent families of switches, such as the assertions expressing the GCH patterns on the $\aleph _n$ ’s. The truth of these assertions can be controlled independently by forcing over any countable transitive model of set theory. Thus, by Theorem 3, the modal validities of W are contained within S5.

Proof. Certainly every countable transitive set is a substructure of $\left \langle H_{\omega _1},\in \right \rangle $ , and every element of $H_{\omega _1}$ can be coded by a real and therefore placed into a countable transitive model of ZFC, under our assumption. It follows that $\mathcal {C}$ is directed (and even $\sigma $ -directed), and so $\mathcal {C}$ provides a potentialist account of $H_{\omega _1}$ . Since this system is directed, it follows that S4.2 is valid at every world, regardless of what other structure is imposed or which language or parameters are used.

Let $L_\alpha $ be the smallest transitive model of ZFC. This world has independent families of buttons and switches, using a modified version of those in the modal logic of forcing. Namely, the button $b_n$ asserts that $n^{\textrm {th}}$ piece of the L-least partition of $\omega _1^{L_\alpha }$ is no longer stationary (note that $L_\alpha $ is definable as a class in any transitive model of ZFC), and the switch $s_m$ asserts that the GCH holds at (the current) $\aleph _m$ , for $m\geq 1$ . These buttons and switches can be controlled independently by forcing, although the buttons are indeed buttons with respect to the potentialist semantics. In particular, moving to a taller model will push all the buttons, since $L_\alpha $ will be realized as countable there. Since we have sufficient independent buttons and switches, it follows by Theorem 5 that the validities of this world, with respect to substitution instances in the language of set theory, are exactly $\mathord {\textrm {Val}}(L_\alpha ,\mathcal {L}_\in )=S4.2$ . The same idea works in any model, if parameters are allowed, showing $S4.2=\mathord {\textrm {Val}}(W,\mathcal {L}_{\in ,W})$ for every world. Meanwhile, the GCH-pattern switches are operable over any model of set theory, and so the validities of any particular world are contained within S5.

The upper bound is provided by Theorem 33.

Proof. Assume that every real is an element of a countable transitive model of ZFC. It follows as we noted earlier that any countably many worlds of $\mathcal {C}$ have a common upper bound. Consider any particular world $U\in \mathcal {C}$ and any countable language $\mathcal {L}^+$ extending the language of set theory. In particular, the new language might have constants for every element of U. Enumerate the assertions of $\mathcal {L}^+$ as $\sigma _0$ , $\sigma _1$ , and so on. Let $W_0=U$ be the initial world. Given $W_n$ , ask whether in this potentialist system. If so, then there is some ; otherwise, choose $W_{n+1}$ extending $W_n$ as you like. Having selected $W_n$ for all $n<\omega $ , let W be any world with $W_n\subseteq W$ for all n. If , then since $W_n$ can access W, it follows that , in which case , and so $W\models \sigma _n$ . Thus, W satisfies every instance of Axiom 5 in this language, and since it also satisfies S4.2, it follows that S5 is valid in W. And the same conclusion holds with respect to any larger world than W, since S5 is persistent.

Proof. The main theorem of [Reference Hamkins9] shows that the countable models of ZFC are linearly pre-ordered by embeddability: for any two models, one of them is isomorphic to a substructure of the other, and furthermore, one model embeds into another just in case the ordinals of the first order embed into the ordinals of the second. It follows that the nonstandard models of ZFC are universal under embeddability.

Because the models are linearly pre-ordered by embeddability, it follows that S4.3 is valid in every world. Because these are ZFC models, the GCH-pattern statements are independent switches, and so the validities of any world are contained in S5. Meanwhile, for any world W, let $a=\emptyset ^W$ , and let $r_n$ be the statement “a has at least n distinct elements.” These statements are all false in W, of course, since a has no elements in W. But the particular embeddings constructed in [Reference Hamkins9] and variations of them show that there are larger models U into which W embeds, in which as many new elements of a as desired are introduced. Thus, these statements form a ratchet, and the GCH-pattern statements are a mutually independent family of switches. So the modal validities of W for such statements are contained within S4.3, and so the first chain of equations and inclusions is proved.

Finally, for the final statement of the theorem, it is a consequence of the fact that the nonstandard models all access isomorphic copies of each other—the main theorem of [Reference Hamkins9] shows that they are all bi-embeddable—that S5 is valid in each of them for sentences of $\mathcal {L}_\in ^\Diamond $ .