What are invisible sets? In order to illustrate what we mean here let me to explain some examples:
Consider you want to find the biggest treasure of the world and you have found a magic map of some hidden treasures. If you look at it in "sun light", you will see some objects, signs and sentences which guide you to a treasure. But if you look at it in "moon light", theses signs will vanish and you can see another objects, signs and sentences which guide you to a bigger treasure. But you know these treasures are not the "biggest" one which you want to discover. So what will you do? Probably you ask: "Is there any other light which can uncover guide signs of the biggest treasure of the world? What is the correct light? Candle or firefly?"
As a more real example note that a same situation takes place in astronomy. Many astronomical phenomenas are not detectable in the visible light spectrum. But if you look at them by other electromagnetic frequences such as infra – red, ultra – violet, X – ray or gamma spectrum you can see them but at the same time, most of visible objects will be vanished or deformed.
The above examples show that in many cases the objects which thought to have "no existence" are just "invisible" under our prevailing paradigm and we can uncover them easily by changing our glasses.
Now back to set theory. Redundancy of independence results in set theory says the current mathematics and its foundation ($ZFC^{-}$) are very weak. In the other words actually our mathematical game is too superficial and boring and we must richen it. But what is a "rich" mathematics? Intuitionally richness of a game such as mathematics has a direct relevance with the situations which can take place during its playing and this depends on the number of playable objects. So the richest mathematics must have the most number of mathematical objects (sets). As same as above examples, the world of "all" mathematical objects is such a magic map or cosmos and every axiomatization for set theory is a particular light. If you are looking on the map of mathematical objects using $ZFC$-light you can see $\emptyset$ , $\omega$, choice functions,… but the "non well founded sets" and "set of all sets" are invisible. Indeed under $NF$-light "set of all sets" is a visible object and with the light of $ZFC^{-}+AFA$ some non well founded sets will appear in the world. But beside the number of objects in theory, "consistency" is an important problem too. Although by an inconsistent light we can see whole of the secret objects and sentences of the magic map but this is completely useless and by these signs you can go nowhere because the treasure island is everywhere! Now the main question is:
Main Question: Which one of the axiomatizations of set theory are the best and give us the richest (relative consistent) mathematics? In other words: Which one of theses axiomatic systems have the most number of mathematical objects?
In order to clarify our main question we need to investigate the structure of producing objects in axiomatizations of set theory. In this direction we can consider any axiomatic foundation of mathematics as a "set producing factory". They have two main parts. First "atomic existencial axioms" which produce new sets with no need to any former existent object such as axioms of "empty set" and "infinity" in $ZFC$ which provide some "initial inputs" for this "factory". Second are "relative existential axioms" which produce new sets using former objects as same as a "machine" which takes some "inputs" and give us a "production". In $ZFC$, axioms of "union", "pairing", "separation", "replacement", "choice", … are relative existential. Now intuitionally we can consider a "rich" mathematics as a "factory" (theory) which has the most number of "initial inputs" and "machines" for a huge production of mathematical objects. Now we can explain an exact formalization for these notions:
Definition (1) : Let $\mathcal{L}=\lbrace \in \rbrace$ and $T$ be an $\mathcal{L}$ – theory. Define:
$O_{0}(T):=\lbrace\varphi(x)\in \mathcal{L}-Form |~T\vDash \exists ! x \varphi (x) \rbrace$
$\forall n>0~;~~O_{n}(T):=\lbrace\varphi(x,y_{1},…,y_{n})\in \mathcal{L}-Form |$
$T\vDash \forall y_{1},…,y_{n} \exists ! x\varphi (x, y_{1},…,y_{n}) \rbrace$
Informally any formula in $O_{0}(T)$ describes an "atomic set" and any formula in $O_{n}(T)$ (for some $n>0$) describes an $n$ - ary "set producing machine". For example consider:
$emp(x):~~\forall y~\neg (y\in x)$(which says $"x=\emptyset"$)
$uni(x,y):~~\forall z~(z\in x \longleftrightarrow \exists t~(z\in t \wedge t\in y))$ (which says $"x=\cup y"$)
$int(x,y,z):~~\forall t~(t\in x \longleftrightarrow t\in y \wedge t\in z)$ (which says $"x=y \cap z"$)
And we have:
$emp(x)\in O_{0}(ZFC)$
$uni(x,y)\in O_{1}(ZFC)$
$int(x,y,z)\in O_{2}(ZFC)$
Definition (2) : Let $\mathcal{L}=\lbrace \in \rbrace$, define a partial order (i.e. reflexive and transitive) on $\mathcal{L}$ -theories as follows:
$T\sqsubseteq T' \Longleftrightarrow \forall n\in \omega~~O_{n}(T)\subseteq O_{n}(T')$
Definition (3) : Let $\mathcal{L}=\lbrace \in \rbrace$, an $\mathcal{L}$ -theory $T$ called a "foundation of mathematics" iff $T\vDash ZFC^{-}$.
Definition (4) : Let $\mathcal{L}=\lbrace \in \rbrace$, an $\mathcal{L}$ -theory $T$ called "almost consistent" iff $Con(ZF+ "some~large~cardinal~axiom")\longrightarrow Con(T)$.
Definition (5) : Let $\mathcal{L}=\lbrace \in \rbrace$, then define $\mathcal{F}$ to be the set of all almost consistent foundations of mathematics.
Question (1) : Is there any maximal element in $\langle \mathcal{F},\sqsubseteq \rangle$?
Question (2) : Are there any better definitions for the "object oriented" notion of richness on $\mathcal{F}$?
Best Answer
I think the question loses some of its appeal once we note that the preorder $T\sqsubseteq T'$ defined by the OP coincides with the preorder defined by the inclusion $\mathtt{Th}(T)\subseteq\mathtt{Th}(T')$ when restricted to theories which are strong enough to prove say $\exists!x\forall y(y\not\in x)$ (a formula suggested by Francois' comment). So if we restrict our analysis to theories which are closed under deduction ($\{\phi:T\vdash\phi\}=\mathtt{Th}(T)=T$), this preorder is simply the usual inclusion and if we focus on consistent theories (as Noah suggests) we are simply asking if a consistent theory can be extended to a maximal consistent theory (yes it can and it will be complete hence not recursively axiomatizable). So in the light of this, I think Noah's answer could be made simpler (This should really be a comment of mine, but my rep doesn't allow me to use that option). To see that $T\sqsubseteq T'$ is equivalent to $\mathtt{Th}(T)\subseteq\mathtt{Th}(T')$ (for strong enough theories), the hardest part is to focus on $\Rightarrow$. So assuming that $T\sqsubseteq T'$ and $\phi$ is a sentence such that $T\vdash \phi$, we have $T\vdash\exists !x\phi\land\forall y(y\not\in x)$ from which we obtain $T'\vdash\exists !x\phi\land\forall y(y\not\in x)$ and finally $T'\vdash\phi$.