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5. Universes are Models, asserting that each member of the multiverse consists of an ordered pair $(V,\in_V)$ where $V$ is a class, called a universe$^*$, and $\in_V$ is a relation on $V$ called membership in $V$. $$\forall X\Big(X\in\mathcal{M}\implies\exists(V,\in_V)\big(X=(V,\in_V)\wedge\in_V\subseteq V\times V\big)\Big).$$ We will abuse notation and denote a member $(V,\in_V)\in\mathcal{M}$ of the multiverse by its first coordinate $V$ ($^*$bringing us into alignment with the terminology and notation defined in the heading). Call a universe standard iff membership in that universe is actual membership. $$V\ \text{is standard}\iff\forall x,y\in V(x\in_Vy\iff x\in y).$$ Call a universe transitive iff it is transitive in the usual sense, and complete iff it is transitive and contains all subsets of its members as members. $$V\ \text{is transitive}\iff\forall X\forall Y(X\in Y\in V\implies X\in V),$$ $$V\ \text{is complete}\iff\Big(V\ \text{is transitive}\wedge\forall X\forall Y(X\subseteq Y\in V\implies X\in V)\Big).$$ We use the word 'models' in the usual sense; a universe $V$ models a predicate $\phi$, written $V\models\phi$, iff relativizing $\phi$ to $V$ and replacing every instance of $\in$ in $\phi$ with $\in_V$ yields a true sentence (so if a standard universe models a sentence it is 'externally' true about members of that universe as well as 'true in that universe').

6. Internal Empty Set, asserting that every universe thinks it has an empty set. $$\forall V\in\mathcal{M}\Big(V\models\big(\exists z\forall x(x\notin z)\big)\Big).$$

7. Standard Transitive Universe, asserting that a standard transitive universe exists. $$\exists V\in\mathcal{M}(V\ \text{is standard and transitive}).$$ Note that $(6.)$ implies that all standard transitive universes actually have the empty class as a set.

At least one issue is that we may want to assert that we have standard/transitive/complete universes for each predicate safe above and independent of a given universe; I suspect that we at least want to assert a standard one, but completeness/transitivity seem like they may conflict with some of the predicates we might want to 'add' to the new universe.

9. Standard Theories. For any well-established set theory $T$ (for example all variants of $ZFC$ with/without large cardinals, NF, KP, etc.) let $T_\wedge$ denote the conjunction of all axioms of $T$; then there exists a complete universe $V_T$ such that $$V_T\models T_\wedge.$$

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

5. Universes are Models, asserting that each member of the multiverse consists of an ordered pair $(V,\in_V)$ where $V$ is a class, called a universe$^*$, and $\in_V$ is a relation on $V$ called membership in $V$. $$\forall X\Big(X\in\mathcal{M}\implies\exists(V,\in_V)\big(X=(V,\in_V)\wedge\in_V\subseteq V\times V\big)\Big).$$ We will abuse notation and denote a member $(V,\in_V)\in\mathcal{M}$ of the multiverse by its first coordinate $V$ ($^*$bringing us into alignment with the terminology and notation defined in the heading). Call a universe standard iff membership in that universe is actual membership. $$V\ \text{is standard}\iff\forall x,y\in V(x\in_Vy\iff x\in y).$$ Call a universe transitive iff it is transitive in the usual sense, and supertransitive iff it is transitive and contains all subsets of its members as members. $$V\ \text{is transitive}\iff\forall X\forall Y(X\in Y\in V\implies X\in V),$$ $$V\ \text{is supertransitive}\iff\Big(V\ \text{is transitive}\wedge\forall X\forall Y(X\subseteq Y\in V\implies X\in V)\Big).$$ We use the word 'models' in the usual sense; a universe $V$ models a predicate $\phi$, written $V\models\phi$, iff relativizing $\phi$ to $V$ and replacing every instance of $\in$ in $\phi$ with $\in_V$ yields a true sentence (so if a standard universe models a sentence it is 'externally' true about members of that universe as well as 'true in that universe').

6. Internal Empty Set, asserting that every universe thinks it has an empty set. $$\forall V\in\mathcal{M}\Big(V\models\big(\exists z\forall x(x\notin z)\big)\Big).$$

7. Standard Transitive Universe, asserting that a standard transitive universe exists. $$\exists V\in\mathcal{M}(V\ \text{is standard and transitive}).$$ Note that $(6.)$ implies that all standard transitive universes actually have the empty class as a set.

At least one issue is that we may want to assert that we have standard/transitive/supertransitive universes for each predicate safe above and independent of a given universe; I suspect that we at least want to assert a standard one, but (super)transitivity seems like it may conflict with some of the predicates we might want to 'add' to the new universe.

9. Standard Theories. For any well-established set theory $T$ (for example all variants of $ZFC$ with/without large cardinals, NF, KP, etc.) let $T_\wedge$ denote the conjunction of all axioms of $T$; then there exists a supertransitive universe $V_T$ such that $$V_T\models T_\wedge.$$

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

In response to Mikhail's comment below, in non-technical terms what I'd like to accomplish is a 'maximally powerful theory of the multiverse' in the consistency strength sense of power, which is also 'maximally permissive' in terms of what kinds of universes it allows for, and which is 'ontologically clean' in the sense that it's power is derived from axioms which come as close as possible to directly capturing intuition for what a multiverse 'should' contain, with minimal extra technical jargon introduced to capture this intuition. (I make no claim to have done a good job at accomplishing any of these goals.)

In response to James E Hanson's two comments, I intended for the 'safe above' clause to prevent us from referring to a whole universe in the predicate we use to apply truth closure -- both of your examples explicitly use a symbol to refer to the whole universe. I think saying that the universe is standard accomplished your 'is the real' part of the clause, so it may be best to omit the requirement that the new universes instantiated by truth closure be standard.

In response to Mikhail's comment below, in non-technical terms what I'd like to accomplish is a 'maximally powerful theory of the multiverse' in the consistency strength sense of power, which is also 'maximally permissive' in terms of what kinds of universes it allows for, and which is 'ontologically clean' in the sense that its power is derived from axioms which come as close as possible to directly capturing intuition for what a multiverse 'should' contain, with minimal extra technical jargon introduced to capture this intuition. (I make no claim to have done a good job at accomplishing any of these goals.)

In response to James E Hanson's two comments, I intended for the 'safe above' clause to prevent us from referring to a whole universe in the predicate we use to apply truth closure -- both of your examples explicitly use a symbol to refer to the whole universe. I think saying that the universe is standard accomplishes your 'is the real' part of the predicate, so it may be best to omit the requirement that the new universes instantiated by truth closure be standard.

Edit:

In response to Mikhail's comment below, in non-technical terms what I'd like to accomplish is a 'maximally powerful theory of the multiverse' in the consistency strength sense of power, which is also 'maximally permissive' in terms of what kinds of universes it allows for, and which is 'ontologically clean' in the sense that it's power is derived from axioms which come as close as possible to directly capturing intuition for what a multiverse 'should' contain, with minimal extra technical jargon introduced to capture this intuition. (I make no claim to have done a good job at accomplishing any of these goals.)

In response to James E Hanson's two comments, I intended for the 'safe above' clause to prevent us from referring to a whole universe in the predicate we use to apply truth closure -- both of your examples explicitly use a symbol to refer to the whole universe. I think saying that the universe is standard accomplished your 'is the real' part of the clause, so it may be best to omit the requirement that the new universes instantiated by truth closure be standard.