Where is the end of universe?

Question

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is endless from the top. Even in a straight forward way one can find an equiconsistent theory (respect to $ZFC$) which its world is endless from the bottom and bounded from the top by the set of all sets. It is sufficient to consider the theory $ZFC^{-1}$ ($ZFC$ inverse) which is obtained from $ZFC$ by replacing each phrase $x\in y$ in the axioms of $ZFC$ by the phrase $\neg (x\in y)$. This operation for example transforms the axiom of empty set of $ZFC$ to an statement which asserts "the set of all sets exists".

$[\exists x \forall y~~\neg(y\in x)]\mapsto [\exists x \forall y~~\neg \neg(y\in x)] $

Even the axiom of extensionality remains unchanged because we have:

$[\forall x\forall y~~(x=y\longleftrightarrow \forall z~~(z\in x\longleftrightarrow z\in y))]\mapsto [\forall x\forall y~~(x=y\longleftrightarrow \forall z~~(\neg (z\in x)\longleftrightarrow \neg (z\in y)))]$

So the "set of all sets" is unique in this theory. Even the equiconsistency simply follows from the fact that for all set (or proper class) $M$ and for all binary relation $E$ on it we have:

$\langle~M~,~E~\rangle \models ZFC \Longleftrightarrow \langle~M~,~M\times M\setminus E~\rangle \models ZFC^{-1}$

So it is trivial that $ZFC^{-1}\models \neg (\exists x \forall y~~\neg(y\in x))$ in the same way which one can prove $ZFC\models \neg (\exists x \forall y~~y\in x)$ by the Russell's paradox. But the situation seems rather strange when one wants to find an equiconsistent theory with $ZFC$ which has end points in both up and down direction because the existence of two contradictory objects like $\emptyset$ and $G$ seems ontologically incompatible in a particular "$ZFC$-like" world. So the question is:

Question (1): Is there an $\mathcal{L}=\lbrace \in\rbrace$-theory $T$ such that the following conditions hold:

$(1)~Con(ZFC)\Longleftrightarrow Con(T)$

$(2)~T\models \exists !x~\forall y~~(y\in x)$

$(3)~T\models \exists !x~\forall y~~\neg (y\in x)$

Remark (1): Quine's new foundation axiomatic system ($NF$) is not an answer because its equiconsistency with $ZFC$ is still unknown.

Even one can define two dual sets from empty and global sets. The set which does not belong to any other set ($\emptyset^{\star}$) and the set which belongs to any set ($G^{\star}$).Now one can restate the question (1) as follows:

Question (2): Is there an $\mathcal{L}=\lbrace \in\rbrace$-theory $T$ such that the following conditions hold:

$(1)~Con(ZFC)\Longleftrightarrow Con(T)$

$(2)~T\models \exists !x~\forall y~~(x\in y)$

$(3)~T\models \exists !x~\forall y~~\neg (x\in y)$

Even it is interesting to have an equiconsistent theory which has no end points in both up and down directions.So:

Question (3): Is there an $\mathcal{L}=\lbrace \in\rbrace$-theory $T$ such that the following conditions hold:

$(1)~Con(ZFC)\Longleftrightarrow Con(T)$

$(2)~T\models \neg (\exists x~\forall y~~(y\in x))$

$(3)~T\models \neg (\exists x~\forall y~~\neg (y\in x))$

Answer 1

Here's an answer which is a bit better than the one I suggested in the comment, but unfortunately still quite unnatural.

Question 1

Define $U(x) := \forall y\;y\in x$. Then adjust the axioms of $\mathsf{ZF}$ (other than extensionality) so that they are "bounded over sets that aren't universal." That is, replace every universal quantifier $\forall x \;\ldots$ with $\forall x\;\neg U(x) \rightarrow \ldots$ and every existential quantifier $\exists x \; \ldots$ with $\exists x \; \neg U(x) \wedge \ldots$. Then keep the extensionality axiom the same and add the axiom $\exists x\;U(x)$. Call this theory $\mathsf{T}$. Note that we can easily produce models of $\mathsf{T}$ from models of $\mathsf{ZF}$ by adding one extra element for the universal set. We can produce models of $\mathsf{ZF}$ from models of $\mathsf{T}$ by removing the universal set.

Question 2

This follows from the answer to question 1 by replacing $x \in y$ by $y \in x$ in every axiom of $\mathsf{T}$.

Question 3

Define $Q(x) := \forall y\;y \in x \leftrightarrow y = x$. That is, $Q(x)$ says that $x$ is a Quine atom. Let $\mathsf{T}$ be the theory with the axiom $\exists ! x \;Q(x)$ and with the axioms of $\mathsf{ZF}$ adjusted as follows. Like in question 1, quantifiers should be bounded to non-Quine atoms. This time we also require that whenever $\mathsf{ZF}$ would assert the existence of a set, $\mathsf{T}$ asserts the existence of the same set but with a Quine atom added. So for example, empty set becomes $$ \exists x \; \neg Q(x) \wedge (\exists z \; Q(z) \wedge z \in x) \wedge (\forall y \; \neg Q(y) \rightarrow \neg y \in x) $$ Separation would become $$ \forall x\,\neg Q(x) \rightarrow (\exists y\, \neg Q(y) \wedge (\exists z\; Q(z) \wedge z \in y) \wedge \forall z \; (\neg Q(z) \rightarrow (z \in y \leftrightarrow (\,z \in x \wedge \phi(z)\,)))) $$ Similarly to question 1, we can convert between models of $\mathsf{ZF}$ and models of $\mathsf{T}$ by just adding or removing a single element (this time corresponding to the Quine atom).

Answer 2

Have a look at Church’s (first) Set Theory with a Universal Set, which is equiconsistent with ZFGC, though Church didn’t publish a full proof, and I think he abandoned the proof in his archives at Princeton. (My forthcoming doctoral thesis completes the proof for a variant with the singleton function as a set.) His archives also contain lecture notes on some more complicated theories with approximately the same name, but I believe he gave up on those theories, not just the details of the consistency proofs.

• Alonzo Church (1974). “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297–308.

• T. E. Forster (1995). Set Theory with a Universal Set: Exploring an Untyped Universe (Oxford Logic Guides 31). Oxford University Press. ISBN 0-19-851477-8.

• T. E. Forster (2001). “Church’s Set Theory with a Universal Set.”

See also the Wikipedia article (disclaimer: I started it in its current form.)

Answer 3

Please forgive the fact that what follows these remarks is in ASCII format. I am an amateur; my work shall never be published; and, there is simply no personal benefit to be gained by converting the text into anything else.

To the best that I can ascertain, one cannot even approach question 1 from the standpoint of the first-order paradigm. Whatever the set universe might be, its formulation must satisfy the first-order presupposition of identity criteria without assuming that criteria. This is the case because it must be sufficient to serve as the interpretation of the universal quantifier.

Harry Deutsch discusses the problem of using the first-order paradigm for such purposes in the link:

http://plato.stanford.edu/entries/identity-relative/#1

Later, when critiquing Geach's arguments, he states exactly the problem that a formulation of the set universe presents. The notion of indiscernibility for the theory must be made to coincide with the notion of identity for the paradigm of first-order logic. According to Deutsch, arguments for language relativity improperly assume that first-order identity defines indiscernibility for a theory.

This analysis is in the section:

http://plato.stanford.edu/entries/identity-relative/#5

To understand the difference, one must discriminate between the referential purport of singular terms and the fact that a system of objects is simultaneously a system of relations between objects. Relative to the purport of singular terms, there are four necessary relations: the full relation, the empty relation, the identity relation, and the diversity relation.

But, again, this is precisely what cannot be assumed when formulating a theory that includes reference to the set universe. Such reference must be formulated with respect to some non-reflexive relations.

The sentences which follow attempt such a formulation. I could try to explain them in a variety of ways, but it is unlikely that my explanations would carry the day. By any account they are non-standard.

I have discussed them once before at the link,

Why hasn't mereology succeeded as an alternative to set theory?

That may help somewhat.

The only additional remark I could add is that the primitive relations of this theory are introduced using circular syntax. In "Model Theory", Hodges makes the observation that such definitions are recursive definitions to which model theory has no application. So, with that in mind, interpret them as you see fit.

For what this is worth, established mathematics has no interest in these sentences. And, I am not competent enough to analyze them properly. What I do know, however, is that introducing a term for the set universe is not a simple matter.

---

We take the consequences of the following as the basic theory.

It's signature is given by

( (M, |M|), (c, 2), (e, 2) )

with models interpreted coherently according to

M=V()

in the extended signature

( (M, |M|), (c, 2), (e, 2), (EQ, 2), (=, 2), (V, 0), (null, 0), (set, 1), (S, 1), (P, 1) )

Definition: AxAy(xcy <-> (Az(ycz -> xcz) /\ Ez(xcz /\ -ycz)))

Provable: AxAy(xcy -> -ycx)

Provable: AxAyAz((xcy /\ ycz) -> xcz)

Provable: Ax(-xcx)

Definition: AxAy(xey <-> (Az(ycz -> xez) /\ Ez(xez /\ -ycz)))

Definition: AxAy(xEQy <-> (Az(xcz <-> ycz) /\ Az(zcx <-> zcy) /\ Az(xez <-> yez) /\ Az(zex <-> zey)))

Definition: AxAy(x=y <-> Az(xez <-> yez))

Assumption: AxAy(Az(xcz <-> ycz) -> Az(xez <-> yez))

Assumption: AxAy(Az(zex -> zey) -> Az(ycz -> xcz))

Assumption: AxEyAz(zey <-> zcx)

Provable: AxAy(xcy -> Az(zex -> zey))

Provable: AxAy(Az(ycz -> xcz) -> Az(zex -> zey))

Provable: AxAy(xcy <-> (Az(zex -> zey) /\ Ez(zey /\ -zex)))

Provable: AxAy(Az(zex -> zey) -> Az(zcx -> zcy))

Provable: AxAy(Az(xez -> yez) -> Az(xcz -> ycz))

Provable: AxAy(xEQy <-> Az(xcz <-> ycz)

Provable: AxAy(xEQy <-> Az(zex <-> zey))

Provable: AxAy(xEQy <-> Az(xez <-> yez))

Provable: AxAy(xEQy <-> x=y)

Assumption: AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) -> Az((xez /\ -ycz) -> (Ew(xew /\ wcy) / Aw(zcw -> ycw))))

Assumption: AxAy((Ez(xcz) /\ Ez(ycz)) -> EwAz(zew -> (z=x / z=y)))

Definition: Ax(x=V() <-> Ay(-(ycx <-> y=x)))

Assumption: ExAy(-(ycx <-> y=x))

Assumption: Ax(Ey(xcy) -> Ey(xey))

Definition: Ax(set(x) <-> Ey(xcy))

Definition: Ax(x=null() <-> Ay(-(xcy <-> x=y)))

Assumption: ExAy(-(xcy <-> x=y))

Assumption: Ax(Ey(ycx) -> Ey(yex /\ -Ez(zex /\ zey)))

Assumption: AxEy(Az(zey <-> Ew(wex /\ zew)) /\ (Ez(xcz) -> Ez(ycz)))

Assumption: AxEy(Az(zey <-> Aw(wex -> zew)) /\ (Ez(zcx) -> Ez(ycz)))

Definition: AxAy(x=P(y) <-> (Ez(ycz) /\ Az(zex <-> (zcy / z=y))))

Assumption: Ax(Ey(xcy) -> Ey(Az(zey <-> (zcx / z=x)) /\ Ez(ycz)))

Definition: AxAy(x=S(y) <-> (Ez(ycz) /\ Az(zex <-> (zey / z=y))))

Assumption: Ax(Ey(xcy) -> Ey(Az(zey <-> (zex / z=x)) /\ Ez(ycz)))

Assumption: Ex(Ey(xcy) /\ null()cx /\ Ay(ycx -> Ez(zcx /\ ycz)))

Let the restricted quantifier

Ap[pEQp]

be interpreted as

Ap[pEQp] (phi(p)) <-> Ap(pEQp -> phi(p))

Then for each n and each well-formed formula phi(y, p_0, ..., p_n),

assume

=================

Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0] AxAy( Ew(ycw) -> (Ez((Ew(zcw) /\ (yez <-> (yex /\ phi(y, p_0, ..., p_n))))) <-> Ew(xcw)) )

=================

and assume

=================

Ap_n[p_nEQp_n]...Ap_0[p_0EQp_0] ( AxAyAz( ( ((Ew(xcw) /\ Ew(ycw)) /\ (phi(x,y, p_0, ..., p_n)) /\ ((Ew(xcw) /\ Ew(zcw)) /\ (phi(x,z, p_0, ..., p_n)) ) -> (y=z) ) -> AxAy( Ew(ycw) -> (Ez((Ew(zcw) /\ (yez <-> Ew(wex /\ phi(z,w, p_0, ..., p_n))))) <-> Ew(xcw)) ) )

=================

Answer 4

Work within $\sf NBG$, define ultraset as:

$ultraset(x) \iff x \in V \lor x=V$

Define a new membership relation $\in^*$ over ultrasets as:

$y \in^* x \iff ultraset(y) \land (x \neq V \to y \in x) $

Now it can be proven in $\sf NBG$ that the world of ultrasets would satisfy the existence of a universal set, that it: $$ \exists x \forall y (y \in^* x)$$ And that all axioms of ZF with all of its quantifiers bounded by non universal sets [i.e. written as $\forall x (\neg \forall y (y \in^* x) \to...); \exists x (\neg \forall y (y \in^* x) \land ...)$], and each symbol $\in$ replaced by $\in^*$, would hold over the world of ultrasets!

Now take the axioms of your theory to be the universal set axiom and the re-written axioms of $\sf ZFC$ as above, and I think this would be equi-consistent with $\sf ZFC$, since $\sf NBG$ itself is conservative over $\sf ZFC$.

There are other tricky ways of doing that also. For example also work in $\sf NBG$, define a function $j$ that sends the empty set to $V$, sends each nonempty Zermelo natural to its predecessor, and fix all sets otherwise.

Now define a membership relation $\in'$, over domain $V$, as: $$y \in' x \iff y \in j(x)$$, so the $\in$ empty set would turn into the universal $\in'$ set.

We can define a predicate "$Original$" to be the subclass of $V$ that is the hierarchy raised over $1$, i.e. it starts with set $1=\{\emptyset\}$, (i.e. the $\in$ singleton set of the $\in$ empty set $\emptyset$), and each successive stage replaces $1$ instead of $\emptyset$; that is, we have: $V^*_{\alpha+1}= (\mathcal P(V^*_\alpha) \setminus \{\emptyset\}) \cup \{1\}$.

Now we return to our world which has $V$ as its domain and $\in'$ membership relation, now clearly the $Original$ sets all belong to that world, and it is straighforward to see that $\sf ZFC$ axioms would hold over the world $Original$ with respect to $\in'$, i.e. if we re-write all axioms of $\sf ZFC$ restricted to $Original$ sets, and with $\in'$ replacing $\in$, then all of those would hold.

So, this theory would also interpet $\sf ZFC$ over the $Original$ sector of its world. And I also think it would be equi-consistent with $\sf ZFC$ since it is in whole definable in $\sf NBG$ which is a conservative extension over $\sf ZFC$