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Define $\emptyset=\{V\in\mathcal{M}:V\neq V\}$ by class separation.

For the final axiom, we refer to the class theory given by the primitives above and axioms $(1.)-(4.)$ as $T_\emptyset$. For a fixed universe $V$ say that a predicate $\psi$ is safe above $V$ iff the multiverse doesn't occur in it and no universes containing $V$ as a subclass occur in it, including $V$, and let $\Phi_V$ denote the class of all predicates safe above $V$. Then we have

This should give us universes with any axioms we could possibly want by starting with our very weak empty set universes and observing that the first four axioms don't prove that they model anything, but I can't tell if this is even a legitimately phrased axiom (or if it is wether it blows the consistency strength all the way up to inconsistent). There is obviously a question of what is meant by 'the axioms of a universe' in this setting; it might mean $(1.)^V$, $(2.)$, $(3.)^V$, parameter-free $(4.)$ and whatever other axioms we've already recursively added, but if this is the correct interpretation and how to phrase it precisely isn't clear.

Define $\emptyset=\{V\in\mathcal{M}:V\neq V\}$.

For the final axiom, we refer to the class theory given by the primitives above and axioms $(1.)-(4.)$ as $T_\emptyset$. For a fixed universe $V$ say that a predicate $\psi$ is safe above $V$ iff the multiverse doesn't occur in it and no universes containing $V$ as a subclass occur in it (including $V$), and let $\Phi_V$ denote the class of all predicates safe above $V$. Then we have

This should give us universes with any axioms we could possibly want by starting with our very weak empty set universes and observing that the first four axioms don't prove that they model anything, but I can't tell if this is even a legitimately phrased axiom (or if it is wether it blows the consistency strength all the way up to inconsistent).

1. Extensionality, in the usual sense applied to classes.

2. Completeness for Universes, asserting that all members of $\mathcal{M}$ are closed under their member’s members and member’s subsets. $$\forall V\in\mathcal{M}\big((Y\in X\in V\implies Y\in V)\wedge(Z\subseteq X\in V\implies Z\in V)\big).$$

3. Class Separation, asserting that for any predicate $\phi(\cdot,Y)$ where $Y$ stands for any finite number of class parameters, and for every class $Z$, there exists a class $A$ whose members are exactly those members of $Z$ satisfying $\phi(\cdot,Y)$. We use this implicitly in the usual way, writing $$\{X\in Z:\phi(X,Y)\}$$ for the class $A$ guaranteed by the instance of this axiom scheme for a predicate $\phi(\cdot,Y)$ and class $Z$ as above.

Let $\mathscr{L}_\alpha$ denote the $\alpha^{th}$ order language of class theory with identity on the first sort. For each universe $V\in M$ say that a predicate $\psi$ is $\alpha^{th}$-order $V$-safe iff $\psi$ is expressed in $\mathscr{L}_\alpha$ and only contains $V$-set parameters and no universes occur in $\psi$, and also $\mathcal{M}$ does not occur in it. Further, use the standard definitions of when a predicate is $\Sigma_n$ or $\Pi_n$. Say that a predicate $\psi$ is $V$-safe iff it is $\alpha^{th}$-order $V$ safe for some $\alpha$. Then we have

4. Varied Set Existence, asserting that for any universe $V\in\mathcal{M}$ and any $1^{st}$-order $V$-safe parameter free predicate $\phi$, if all the classes for which $\phi$ holds are $V$-sets then the class of these sets is a $V$-set. Further, for each ordinal $\alpha$ and pair of ordinals $n,m\leq\omega$ there exists a universe $V_{\alpha,n,m}$ such that for any $\alpha^{th}$-order $V_{\alpha,n,m}$-safe predicate $\phi(\cdot,y)\in\Sigma_n\cup\Pi_m$, where $y$ stands for any finite number of $V_{\alpha,n,m}$-set parameters, if all the classes for which $\phi(\cdot,y)$ holds are $V_{\alpha,n,m}$-sets then the class of these sets is a $V_{\alpha,n,m}$-set.

5. Optional Regularity, asserting that if we have a universe $V$ where regularity is independent of the axioms present relativized to that universe then there exists a universe $V^{Reg}$ with the same axioms as $V$ plus regularity and there exists a universe $V^{\neg Reg}$ with the same axioms as $V$ plus anti-regularity.

6. Optional Choice, asserting that if we have a universe $V$ where choice is independent of the axioms present relativized to $V$ then there exists a universe $V^{Ch}$ with the same axioms as $V$ plus choice and a universe $V^{\neg Ch}$ with the same axioms as $V$ plus the negation of choice.

The motivation for these axioms is populating the multiverse with some of the canonical universes we care about; $ZFC$ will be $V_{1,\omega,\omega}$, $V_{n,\omega,\omega}$ for $1<n<\omega$ will have indescribable cardinals as described here (where it is also mentioned by Joel that $V_{\alpha,\omega,\omega}$ for $\omega\leq\alpha$ will have strongly unfoldable cardinals). Further, the $V_{1,n,m}$'s for $n,m<\omega$ should be weak set theories but I'm not familiar enough with set theories weaker than $ZF$ to comment on what interesting consequences we might have. We are being loose mixing theories and models of those theories; by saying that $V_{1,\omega,\omega}$ 'is $ZFC$', we mean that the theory obtained by relativizing all axioms here to $V_{1,\omega,\omega}$ 'is' ZFC (in the same sense as in Lévy and Vaught's paper on partial reflection in $ZF$ and $A$ linked below).

The phrasing of the last two axioms, and the idea that 'all possible truths should be realized in the multiverse', suggests the following different approach. Use $(1.)-(3.)$ above, and the parameter free version of $(4.)$ to get a universe with the empty set. Then add

Forcing Closure. For every universe $V$ and every predicate $\phi$ such that $\phi^V$ is independent of the other axioms present, there exists a universe $V+\phi^V$ with the same axioms as $V$ plus $\phi^V$, and a universe $V+\neg\phi^V$ with the same axioms as $V$ plus $\neg\phi^V$.

This should give us universes with any axioms we could possibly want by starting with our very weak parameter-free set existence axiom universes, but I can't tell if this is even a legitimately phrased axiom (or if it is wether it blows the consistency strength all the way up to inconsistent). There is obviously a question of what is meant by 'the axioms of a universe' in this setting; it might mean $(1.)^V$, $(2.)$, $(3.)^V$, parameter-free $(4.)$ and whatever other axioms we've already recursively added, but if this is the correct interpretation and how to phrase it precisely isn't clear.

Question 1. Is this a legitimately phrased axiom, and if so is it obviously inconsistent?

The goal is to obtain one class theory that allows for a simultaneous consideration of all 'set theories' and the relationships between them, in a 'multiverse' sense as proposed by Hamkins. In particular, things like

and the like should be possible in this theory (other relationships like being an elementary submodel etc. should also be formalizable, perhaps providing vertical/horizontal arrows or arrows between arrows in ${\bf Uni}$).

This came up while reading through A. Lévy, R. Vaught, Principles of partial reflection in the set theories of Zermelo and Ackermann -- Lévy and Vaught essentially view Ackermann's class theory concentrated on his individual constant $V$ to be 'set theory', and show that any theorem about sets in standard set theory also holds relativized to $V$ in Ackermann's theory.$^1$

$^1$Although Reinhardt famously showed that Ackermann's original theory is equiconsistent with $ZF$, a modification laid out by F. A. Muller very close in spirit to the above axiom list is claimed to be stronger in consistency strength and thereby avoids the philosophical position that 'all classes in Ackermann's theory are really just sets'. Of course these classes can be modeled as sets by adding a large cardinal axiom $\phi$ to $ZFC$, but if we can also add $\phi^V$ to Ackermann's modified theory and obtain something stronger in consistency strength than $ZFC+\phi$ we will always have new classes that 'aren't sets' even when we try to use large cardinals to 'make everything a set'.

1. Extensionality, in the usual sense applied to classes.

2. Class Separation, asserting that for any predicate $\phi(\cdot,Y)$ where $Y$ stands for any finite number of class parameters, and for every class $Z$, there exists a class $A$ whose members are exactly those members of $Z$ satisfying $\phi(\cdot,Y)$. We use this implicitly in the usual way, writing $$\{X\in Z:\phi(X,Y)\}$$ for the class $A$ guaranteed by the instance of this axiom scheme for a predicate $\phi(\cdot,Y)$ and class $Z$ as above.

Define $\emptyset=\{V\in\mathcal{M}:V\neq V\}$ by class separation.

3. Populated Multiverse, asserting that the multiverse isn't empty. $$\mathcal{M}\neq\emptyset.$$
4. Empty set, asserting that the empty class is a set in every universe. $$\forall V\in\mathcal{M}\big(\emptyset\in V\big).$$

For the final axiom, we refer to the class theory given by the primitives above and axioms $(1.)-(4.)$ as $T_\emptyset$. For a fixed universe $V$ say that a predicate $\psi$ is safe above $V$ iff the multiverse doesn't occur in it and no universes containing $V$ as a subclass occur in it, including $V$, and let $\Phi_V$ denote the class of all predicates safe above $V$. Then we have

5. Truth Closure. For every universe $V$ and every predicate $\phi$ which is safe above $V$, if $V$ modeling $\phi$ is independent of $T_\emptyset$ then there exists a universe $V+\phi$ modeling $\phi$ such that $V$ is an elementary submodel of $V+\phi$. $$\forall V\in\mathcal{M}\forall\phi\in\Phi_V\Big(T_\emptyset\nvdash(V\models\phi)\wedge T_\emptyset\nvdash(V\models\phi)$$ $$\implies\exists V+\phi\in\mathcal{M}(V+\phi\models\phi\wedge V\prec V+\phi)\Big).$$

This should give us universes with any axioms we could possibly want by starting with our very weak empty set universes and observing that the first four axioms don't prove that they model anything, but I can't tell if this is even a legitimately phrased axiom (or if it is wether it blows the consistency strength all the way up to inconsistent). There is obviously a question of what is meant by 'the axioms of a universe' in this setting; it might mean $(1.)^V$, $(2.)$, $(3.)^V$, parameter-free $(4.)$ and whatever other axioms we've already recursively added, but if this is the correct interpretation and how to phrase it precisely isn't clear.

Question 1. Is this a legitimately phrased axiom, and if so is it obviously inconsistent?

If this last axiom doesn't work, we could use the following axioms instead -- they seems to give a more limited multiverse 'centered around $ZFC$', but I'm more confident that the following axioms are legitimate and not obviously inconsistent.

Let $\mathscr{L}_\alpha$ denote the $\alpha^{th}$ order language of class theory with identity on the first sort. For each universe $V\in M$ say that a predicate $\psi$ is $\alpha^{th}$-order $V$-safe iff $\psi$ is expressed in $\mathscr{L}_\alpha$ and only contains $V$-set parameters and no universes occur in $\psi$, and also $\mathcal{M}$ does not occur in it. Further, use the standard definitions of when a predicate is $\Sigma_n$ or $\Pi_n$. Say that a predicate $\psi$ is $V$-safe iff it is $\alpha^{th}$-order $V$ safe for some $\alpha$. Then the axioms are extensionality and class separation as above, together with

3*. Completeness for Universes, asserting that all members of $\mathcal{M}$ are closed under their member’s members and member’s subsets. $$\forall V\in\mathcal{M}\big((Y\in X\in V\implies Y\in V)\wedge(Z\subseteq X\in V\implies Z\in V)\big).$$

4*. Varied Set Existence, asserting that for any universe $V\in\mathcal{M}$ and any $1^{st}$-order $V$-safe parameter free predicate $\phi$, if all the classes for which $\phi$ holds are $V$-sets then the class of these sets is a $V$-set. Further, for each ordinal $\alpha$ and pair of ordinals $n,m\leq\omega$ there exists a universe $V_{\alpha,n,m}$ such that for any $\alpha^{th}$-order $V_{\alpha,n,m}$-safe predicate $\phi(\cdot,y)\in\Sigma_n\cup\Pi_m$, where $y$ stands for any finite number of $V_{\alpha,n,m}$-set parameters, if all the classes for which $\phi(\cdot,y)$ holds are $V_{\alpha,n,m}$-sets then the class of these sets is a $V_{\alpha,n,m}$-set.

5*. Optional Regularity, asserting that if we have a universe $V$ where regularity is independent of the axioms present relativized to that universe then there exists a universe $V^{Reg}$ with the same axioms as $V$ plus regularity and there exists a universe $V^{\neg Reg}$ with the same axioms as $V$ plus anti-regularity.

6*. Optional Choice, asserting that if we have a universe $V$ where choice is independent of the axioms present relativized to $V$ then there exists a universe $V^{Ch}$ with the same axioms as $V$ plus choice and a universe $V^{\neg Ch}$ with the same axioms as $V$ plus the negation of choice.

The motivation for these axioms is populating the multiverse with some of the canonical universes we care about; $ZFC$ will be $V_{1,\omega,\omega}$, $V_{n,\omega,\omega}$ for $1<n<\omega$ will have indescribable cardinals as described here (where it is also mentioned by Joel that $V_{\alpha,\omega,\omega}$ for $\omega\leq\alpha$ will have strongly unfoldable cardinals). Further, the $V_{1,n,m}$'s for $n,m<\omega$ should be weak set theories but I'm not familiar enough with set theories weaker than $ZF$ to comment on what interesting consequences we might have. We are being loose mixing theories and models of those theories; by saying that $V_{1,\omega,\omega}$ 'is $ZFC$', we mean that the theory obtained by relativizing all axioms here to $V_{1,\omega,\omega}$ 'is' ZFC (in the same sense as in A. Lévy, R. Vaught, Principles of partial reflection in the set theories of Zermelo and Ackermann).

The goal is to obtain one class theory that allows for a simultaneous consideration of all 'set theories' and the relationships between them, in a 'multiverse' sense as proposed in Hamkins, The set-theoretic multiverse. In particular, things like

and the like should be possible in this theory (other relationships like being an elementary submodel, a ground, a mantle, etc. should also be formalizable, perhaps providing vertical/horizontal arrows or arrows between arrows in ${\bf Uni}$).

This came up while reading through Lévy and Vaught's paper linked above -- they essentially view Ackermann's class theory concentrated on his individual constant $V$ to be 'set theory', and show that any theorem about sets in standard set theory also holds relativized to $V$ in Ackermann's theory.$^1$

$^1$Although Reinhardt famously showed that Ackermann's original theory is equiconsistent with $ZF$, a modification laid out by F. A. Muller very close in spirit to the second axiom list above is claimed to be stronger in consistency strength and thereby avoids the philosophical position that 'all classes in Ackermann's theory are really just sets'. Of course these classes can be modeled as sets by adding a large cardinal axiom $\phi$ to $ZFC$, but if we can also add $\phi^V$ to Ackermann's modified theory and obtain something stronger in consistency strength than $ZFC+\phi$ we will always have new classes that 'aren't sets' even when we try to use large cardinals to 'make everything a set'.

Let $\mathscr{L}_\alpha$ denote the $\alpha^{th}$ order language of class theory with identity on the first sort. For each universe $V\in M$ say that a predicate $\psi$ is $\alpha^{th}$-order $V$-safe iff $\psi$ is expressed in $\mathscr{L}_\alpha$ and only contains $V$-set parameters and no universes occur in $\psi$, and also $\mathcal{M}$ does not occur in it. Further, use the standard definitions of when a predicate is $\Sigma_n$, $\Pi_n$ or $\Delta_n=\Sigma_n\cap\Pi_n$. Say that a predicate $\psi$ is $V$-safe iff it is $\alpha^{th}$-order $V$ safe for some $\alpha$. Then we have

4. Varied Set Existence, asserting that for any universe $V\in\mathcal{M}$ and any $1^{st}$-order $V$-safe parameter free predicate $\phi$, if all the classes for which $\phi$ holds are $V$-sets then the class of these sets is a $V$-set. Further, for each ordinal $\alpha$ and pair of ordinals $n,m\leq\omega$ there exists a universe $V_{\alpha,n,m}$ such that for any $\alpha^{th}$-order $V_{\alpha,n,m}$-safe predicate $\phi(\cdot,y)\in\Sigma_n\cap\Pi_m$, where $y$ stands for any finite number of $V_{\alpha,n,m}$-set parameters, if all the classes for which $\phi(\cdot,y)$ holds are $V_{\alpha,n,m}$-sets then the class of these sets is a $V_{\alpha,n,m}$-set.

5. Optional Regularity, asserting that if we have a universe $V$ where regularity is independent of the axioms present relativized to that universe then there exists a universe $V^{Reg}$ with the same axioms as $V$ plus regularity and there exists a universe $V^{\neg Reg}$ with the same axioms as $V$ plus anti-regularity.

6. Optional Choice, asserting that if we have a universe $V$ where choice is independent of the axioms present relativized to $V$ then there exists a universe $V^{Ch}$ with the same axioms as $V$ plus choice and a universe $V^{\neg Ch}$ with the same axioms as $V$ plus the negation of choice.

Let $\mathscr{L}_\alpha$ denote the $\alpha^{th}$ order language of class theory with identity on the first sort. For each universe $V\in M$ say that a predicate $\psi$ is $\alpha^{th}$-order $V$-safe iff $\psi$ is expressed in $\mathscr{L}_\alpha$ and only contains $V$-set parameters and no universes occur in $\psi$, and also $\mathcal{M}$ does not occur in it. Further, use the standard definitions of when a predicate is $\Sigma_n$ or $\Pi_n$. Say that a predicate $\psi$ is $V$-safe iff it is $\alpha^{th}$-order $V$ safe for some $\alpha$. Then we have

4. Varied Set Existence, asserting that for any universe $V\in\mathcal{M}$ and any $1^{st}$-order $V$-safe parameter free predicate $\phi$, if all the classes for which $\phi$ holds are $V$-sets then the class of these sets is a $V$-set. Further, for each ordinal $\alpha$ and pair of ordinals $n,m\leq\omega$ there exists a universe $V_{\alpha,n,m}$ such that for any $\alpha^{th}$-order $V_{\alpha,n,m}$-safe predicate $\phi(\cdot,y)\in\Sigma_n\cup\Pi_m$, where $y$ stands for any finite number of $V_{\alpha,n,m}$-set parameters, if all the classes for which $\phi(\cdot,y)$ holds are $V_{\alpha,n,m}$-sets then the class of these sets is a $V_{\alpha,n,m}$-set.

5. Optional Regularity, asserting that if we have a universe $V$ where regularity is independent of the axioms present relativized to that universe then there exists a universe $V^{Reg}$ with the same axioms as $V$ plus regularity and there exists a universe $V^{\neg Reg}$ with the same axioms as $V$ plus anti-regularity.

6. Optional Choice, asserting that if we have a universe $V$ where choice is independent of the axioms present relativized to $V$ then there exists a universe $V^{Ch}$ with the same axioms as $V$ plus choice and a universe $V^{\neg Ch}$ with the same axioms as $V$ plus the negation of choice.