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 ‎$‎‎‎\mathcal{F}‎$‎‎?‎ ‎

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    $\begingroup$ Interestingly enough, $\sf ZF$ and $\sf ZF+DC+AD$ are not foundations of mathematics... :-) $\endgroup$ – Asaf Karagila Aug 7 '13 at 8:33
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    $\begingroup$ Q1: No. Given a sentence $\theta$, take $\phi_\theta(x)$ to be $(\forall z)(z \notin x) \lor \theta$. Then $\phi_\theta \in O_0(T)$ iff $T$ proves $\lnot\theta$. If $\theta$ is independent of $ZFC^-$ then no consistent "foundations of mathematics" $T$ is such that $\phi_\theta \in O_0(T)$ and $\phi_{\lnot\theta} \in O_0(T)$. $\endgroup$ – François G. Dorais Aug 7 '13 at 8:48
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    $\begingroup$ @ Asaf: Dear Asaf the theory $ZFC^{-}$ is not essential. Everybody can choose his foundation. This is very dependent to one's personal ideology. $\endgroup$ – user36136 Aug 7 '13 at 9:40
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    $\begingroup$ Oh, of course. I'm just making a trivial an unimportant observation. $\endgroup$ – Asaf Karagila Aug 7 '13 at 10:45
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    $\begingroup$ "Almost" consistent? Really? "Almost" would mean it is not consistent. $\endgroup$ – Andrés E. Caicedo Aug 7 '13 at 15:22

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 $T'\vdash\phi$.

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    $\begingroup$ This is much simpler than my answer - somehow, I didn't notice that these two preorders are the same. Thanks! $\endgroup$ – Noah Schweber Aug 8 '13 at 23:15
  • $\begingroup$ It took me some time before I realized it myself... $\endgroup$ – Noel Vaillant Aug 8 '13 at 23:19

If you restrict attention to recursively axiomatized theories, then the answer to your Question 1 is "no." (I'm replacing "consistent relative to a large cardinal axiom" with "consistent," here, for simplicity; it doesn't make much difference.)

EDIT: This argument is much more involved than it needs to be; see Noel's answer.

Suppose $T$ were a maximal element of $\mathcal{F}$. Then we must have that for every formula $\varphi(x; \overline{y}),$ $$ \text{Either }T\vdash\forall\overline{y}\exists!x\varphi(x; \overline{y})\text{ or $T\vdash\neg(\forall\overline{y}\exists!x\varphi(x; \overline{y})).$}$$ Otherwise, we could adjoin $\forall \overline{y}\exists!x\varphi(x;\overline{y})$ to $T$ to get a still consistent theory $> T$ in $\mathcal{F}$. So $T$ has a weak form of completeness. I'll show that - under the assumption that $T$ contains $ZF^-$ (actually, much less is needed) - this contradicts Goedel's Theorem. (Actually, the $\overline{y}$s are completely unnecessary here - this argument still applies if we just pay attention to how big $O_0(T)$ is.)

The key is our assumption that $T$ is recursively axiomatizable. This, together with the assumption that $T$ is a "foundation of mathematics," lets us perform Goedel's arguments formalizing $T$-provability inside $T$. Now consider the formula $$ \beta(x)\equiv\text{$x=0$ OR (For every $T$-proof of $\beta(x)$ there is a shorter $T$-proof of $\neg\beta(x)$)}.$$ Let $(*)$ be the statement "$\exists!x\beta(x)$." By our assumption on $T$, we have $$ T\vdash (*)\quad OR \quad T\vdash\neg(*).$$ But for the former to be true, $T$ would have to disprove its own Rosser sentence; and for the latter to be true, $T$ would have to prove its own Rosser sentence. Either way, this contradicts the assumption that $T$ is consistent. $\Box$

If we drop the restriction that $T$ be recursively axiomatizable, this of course fails completely; but then again, the theories also become much less interesting (at least from a foundations point of view). In this case, the answer to your Question 1 becomes "yes," for trivial reasons:

Let $\mathbb{P}$ be the poset of all sequences of sets of formulas $(F_i)_{i\in\omega}$ such that for some consistent foundation of mathematics $T$, we have $O_i(T)=F_i$; with elements of $\mathbb{P}$ ordered by inclusion. Then since first-order logic is compact, chains in $\mathbb{P}$ have upper bounds; we could use Zorn's Lemma here for overkill, but actually we don't need any choice in this context, and can now directly prove the existence of a maximal element, since we can well-order the set of formulas. $\Box$

As for your Question 2, one partial ordering that is frequently studied is the interpretability ordering on theories (not just "foundations of mathematics," or even recursively axiomatized theories) in a finite signature (interpretability gets weird if the signature is infinite). One theory $T$ interprets a theory $T'$ if there is a (finite collection of) formulas $\Phi$ which, in any model of $T$, define a model of $T'$. So, for example, the theory $ZFC^-$ interprets $PA$, by identifying hereditarily finite sets with natural numbers in definable fashion.

Motivated by this partial ordering, set theories which "interpret more" are more desirable; and axioms such as $V=L$, for example, are undesirable since they limit the achievable interpretive strength of the theory. Some philosophers of mathematics have argued - most notably, Penelope Maddy - that this "maximality" principle should be taken as a key criterion for evaluating the philosophical suitability of set theories (see http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.lnl/1235415905). However, there are intuitively "bad" theories which have extremely high interpretability strength (I believe this is detailed in Benedikt Loewe's "A second glance at non-restrictiveness," but it's behind a paywall so I can't check my memory), and so obviously this picture needs more details.

CAVEAT: this is all half-remembered, so I might have made some very silly mistakes in the previous couple paragraphs.

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    $\begingroup$ Regarding the point in your penultimate paragraph, there is also a view based on the upwardly extensible concept of ordinal, a kind of upward maximality in place of Maddy's outward maximality, on which $V=L$ is seen as much more appealing and natural. I discuss this in my paper jdh.hamkins.org/multiverse-perspective-on-constructibility, which expands on Benedikt's idea and engages Maddy's argument. $\endgroup$ – Joel David Hamkins Aug 8 '13 at 23:03

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