# Set Theory is interpretable in Class Ordering Theory

Zuhair Al-Johar

(October 2023)

##  1 Abstract

Here it is shown that standard set theory can be interpreted in a theory about
order. The ordering here is about non-extensional flat classes, i.e. classes
that are not elements of classes. So, stipulating a nearly well order over all
those classes coupled together with indexing that order by elements of those
classes, thereby having those elements serve as ordinals; this together with
infinity and a replacement like axiom would be shown to interpret $\sf ZFC$.
Moreover, it is shown that a suitable version of this order theory is bi-
interpretable with Morse-Kelley set theory augmented with a well ordering on
classes.

##  2 Introduction

This theory align itself along attempts to formalize set theory in order
theory. In a series of articles Takeuti had formalized the theory of ordinal
numbers in $G^{\prime}LC$ and $FLC$ both of which are written in higher order
logic, and also in the ordinary first order logic with equality
[Tak53,54,57,65]. He proved that a model of $\sf ZFC$ can be constructed in
those theories. Koepke and Koerwien defined the theory of sets of ordinals
$\sf SO$ in bi-sorted first order logic with equality, and proved its bi-
interpretability with $\sf ZFC$ [KK05]. Here, a simpler axiomatic system, we’d
name as $\sf COT$, is presented about flat classes of ordinals equipped with a
near well-ordering on them together with an indexing function on that ordering
which recursively sends bounded classes to ordinal indices. It would be shown
here, that a version of such a theory is bi-interpretable with a more powerful
system than $\sf ZFC$, that of More-Kelley set theory equipped with a well
ordering on classes. The proof of bi-interpretability is vastly simpler than
the case of $\sf SO$ with $\sf ZFC$. On the other hand, there is a growing
interest in Class Choice principles extending $\sf MK$ set theory [G14], some
of which prove to be hard questions [GHK21]. The point is that those choice
principles are provable by adding a well-order on classes to $\sf MK$.

##  3 Class Ordering Theory “COT”

Language: Bi-sorted FOL with identity. First sort in lower case ranging over
ordinals. Second sort in upper case ranging over classes of ordinals.

Primitives: $\\{=,\varepsilon,\prec,\iota\\}$

Sorting: $=$ denoting equality holding only between same sort objects;
$\varepsilon$ standing for class element-hood relation holding only from first
to second sort objects; $\prec$ denoting prior relation holding only for
second sort objects; $\iota$ standing for the ordinal index of a class, a
partial unary function holding only from second to first sort objects.

Axioms: Bi-sorted ID axioms +:

$\displaystyle\textbf{1. Comprehension: }\exists X\forall x\,(x\ \varepsilon\
X\leftrightarrow\phi),\text{ if }X\text{ is not free in }\phi$
$\displaystyle\textbf{ Define: }X\equiv Y\iff\forall z(z\ \varepsilon\
X\leftrightarrow z\ \varepsilon\ Y)$ $\displaystyle\textbf{ Define:
}X\equiv\\{x\mid\phi\\}\iff\forall x\,(x\ \varepsilon\ X\leftrightarrow\phi)$
$\displaystyle\textbf{2. Well Ordering}\prec\text{is a co-well-ordering on
classes}$ $\displaystyle\textbf{ -Transitive:}X\prec Y\prec Z\to X\prec Z$
$\displaystyle\textbf{ -Co-Connected:}X\not\equiv Y\leftrightarrow[X\prec
Y\lor Y\prec X],$ $\displaystyle\textbf{ -Well-Founded:}\phi(X)\to\exists
M:\phi(M)\land\forall Y\,(\phi(Y)\to\neg Y\prec M);\text{ for every formula
}\phi$ $\displaystyle\textbf{ Define: }x<y\iff\\{x\\}\prec\\{y\\}$
$\displaystyle\textbf{ Define:}\lim X=\min l:\forall x\ \varepsilon\ X(x<l)$
$\displaystyle\textbf{3. Respective: }\lim X\leq y\ \varepsilon\ Y\to X\prec
Y$ $\displaystyle\textbf{ Define: }\operatorname{bnd}(X)\iff\exists
l:l=\lim(X)$ $\displaystyle\textbf{4. Indexing:
}\forall\operatorname{bnd}X\,\exists\iota(X):\iota(X)=\lim\\{\iota(Y)\mid
Y\prec X\\}$ $\displaystyle\textbf{5. Infinity: }\exists l:\exists
k<l\land\forall x<l\,\exists y\,(x<y<l)$ $\displaystyle\textbf{6. Boundedness:
}\operatorname{bnd}(X)\land X\operatorname{\sim}Y\to\operatorname{bnd}(Y)$

Where ”$\sim$” stands for equinumerousity defined in the usual way after
existence of a class bijection.

As a notation we use $\\{x\mid\phi\\}$ not to stand for a particular class up
to equality, but to stand for classes up to co-extensionality. So, it can be
treated as a variable ranging over all classes having the same membership. So,
when an expression $\psi(\\{x\mid\phi\\})$ is written, it denotes $\forall
X:[\forall x\,(x\ \varepsilon\ X\leftrightarrow\phi)]\to\psi(X)$

Lemma I: $<$ is a well-ordering on ordinals

Proof: The proofs of transitivity and well-foundedness for $<$ are
straightforward, what needs to be proved is connectedness of $<$, that is:

$x\neq y\iff(x<y\lor y<x)$

For left to right direction we have: $x\neq y\iff\\{x\\}\not\equiv\\{y\\}$
[Definitions of $\equiv$ and Singletons], thus by Co-connectedness, we have:
$\\{x\\}\prec\\{y\\}\lor\\{y\\}\prec\\{x\\}$, thus $x<y\lor y<x$ [Definition
of $<$]

For the opposite direction: $x<y\lor
y<x\to\\{x\\}\prec\\{y\\}\lor\\{y\\}\prec\\{x\\}$, thus by Co-connectedness,
we have: $\\{x\\}\not\equiv\\{y\\}$, thus $x\neq y$ [Definitions of $\equiv$
and Singletons]

Lemma II: Transfinite recursion over ordinals

$\phi(0)\land\forall x(\forall y(y<x\to\phi(y))\to\phi(x))\to\forall
x:\phi(x)$

Proof: as usual, suppose there is an $x$ such that $\neg\phi(x)$, by well-
foundedness of $<$, take the first such $x$, then it cannot be $0$, and so we
must have $\phi(x)$ (antecedent of the above rule), a contradiction.

Proposition I: every indexed class is bounded

$\iota(X)=x\to\operatorname{bnd}(X)$

Proof: Suppose there is an indexed unbounded class, let $X$ to be one of the
$\prec$-minimal indexed unbounded classes, then every subclass $Y$ of a class
$\\{x\mid x<\iota(X)\\}$ would be bounded and we’d have $\iota(Y)<\iota(X)$
[Respective and Indexing], take any class of all indices of those subclasses,
that are not elements of the classes they index, and any such class would be
bounded by $\iota(X)$ yet it won’t have an index, contradicting axiom of
Indexing.

Proposition II: There is no last ordinal

$\not\exists l:\forall x\,(x\leq l)$

Proof: if there is a last ordinal $l$, then there is a last bounded class that
$l$ indexes, i.e. $i^{-1}(l)$ is the last bounded class. Call a class _elite_
if its limit is $l$. Now, let $M$ be an $\prec$-minimal elite class; take any
class $K\equiv\\{x\mid x<\iota(M)\\}$, it is clear that any subclass of $K$
would be indexed by an element of $K$, since indexing ordinals $\geq\iota(M)$
are all reserved for indexing elite classes. Take a class of all elements of
$K$ that are not elements of the classes they index, this would be a diagonal
bounded class and it won’t have an index, contradicting axiom of Indexing.

Proposition III: every ordinal is an index

$\forall x\exists X:x=\iota(X)$

Proof: $0=\iota(\varnothing)$, let all ordinals $<l$ be indices, take any
class $X$ of all ordinals $<l$ . Now, it cannot be the case that all
subclasses of $X$ are indexed by ordinals prior to $l$, since if it is, then
we can simply run the diagonal argument over it. So, there are subclasses of
$X$ that are indexed with ordinals $\geq l$, take an $\prec$ least of those
subclasses, and that would have its index being $l$ [Indexing axiom]. Thus all
ordinals are indices [Lemma II].

Lemma III: every index is higher than any element in the class it indexes

$y\ \varepsilon\ X\to\iota(X)>y$

Proof: $0$ is greater than all elements of any null class. Now if all ordinals
$<l$ are greater than all elements in the classes they index, then $l$ must be
so! Remember from proposition III, $l$ is the index of a $\prec$-least
subclass of a class $\\{x\mid x<l\\}$ that has an index $\geq l$, and so all
elements of that subclass are $<l$. Thus establishing the result. [Lemma II]

##  4 COT interprets ZFC

Now, define $\in$ as:

$y\in x\iff y\ \varepsilon\ \iota^{-1}(x)\iff\exists X:y\ \varepsilon\ X\land
x=\iota(X)$

. It’ll be shown that $\sf ZFC$ would be interpreted over first sort world
using this membership relation.

Proof:

$\textbf{Extensionality: }\forall z(z\in x\leftrightarrow z\in y)\to x=y$

Proof: from the antecedent, we have $\iota^{-1}(x)\equiv\iota^{-1}(y)$. It can
be proved that for any two co-extensional classes $A,B$ if we have a class
$X\prec A$, then we must have $X\prec B$, the proof is easy, since if
otherwise we must have $B\prec X$, and therefore $B\prec A$ [Transitivity of
$\prec$], violating Co-connectedness of $\prec$. That said, then
$\iota^{-1}(x)$ and $\iota{-1}(y)$ would have the same preceding classes, thus
would be assigned the same limit by $\iota$, that is $x=y$. Since all ordinals
are indices of classes [Proposition III], then there are no Ur-elements

$\textbf{Regularity: }\exists x(x\in a)\to\exists y(y\in a\land\not\exists
c:c\in y\land c\in a)$

Proof: let $\phi(z)\iff z\ \varepsilon\ \iota^{-1}(a)$, then per well-
foundedness of $<$ there must exist an ordinal $m$ such that $\phi(m)$ and
every ordinal $<m$ doesn’t satisfy $\phi$, thus no $\varepsilon$-element of
$\iota^{-1}(m)$ can satisfy $\phi$ [Lemma III], since the elements of
$\iota^{-1}(a)$ are the ordinals that satisfy $\phi$, then we have $m\in a$
and $m$ is $\in$-disjoint of $a$ [Definition of $\in$]

$\textbf{Pairing: }\forall a\forall b\exists x\forall y(y\in x\leftrightarrow
y=a\lor y=b)$

Proof: every ordinals $a,b$, have a limit on the maximal one of them
[Proposition II], thus every class $\\{a,b\\}$ is bounded, thus has the index
$\iota(\\{a,b\\})$ which witness $x$ of the above formula.

$\textbf{Union: }\forall a\exists x\forall y(y\in x\leftrightarrow\exists z\in
a(y\in z))$

Proof: from [Lemma III] and the definition of $\in$, every $a$ has all its
$\in$-elements being prior (i.e. $<$) to it, that is $\forall y(y\in a\to
y<a)$, so all $\in$-members of $\in$-members of $a$ would also be $<a$, and
any class of all of those would be bounded by $a$ itself, and so must have an
ordinal index which would witness $x$ of the above formula.

$\textbf{Power: }\forall a\exists x\forall y(y\in x\leftrightarrow y\subseteq
a)$

Proof: any class indexed by $a$ is bounded by $a$ itself [Lemma III], so all
subclasses of an $\iota^{-1}(a)$ class are $\prec$ to every singleton class
$\\{a\\}$ [Respective], so the indices of all those subclasses are
$<\iota(\\{a\\})$ [Indexing], so any class of all those indices is bounded,
thus has an ordinal index and that index would witness $x$ of the above
formula.

$\textbf{Separation: }\forall a\exists x\forall y(y\in x\leftrightarrow y\in
a\land\phi)$; if $x$ doesn’t occur free in $\phi$.

Proof: Since sets here are ordinal indices of bounded classes, clearly any
sublcass of a bounded class is bounded, and so has an index that would witness
$x$ in the above formula.

Infinity:  The index of a class of all ordinals prior to the first ordinal
limit is an infinite set.

Choice:  This follows trivially actually the relation $<$ is a global well-
order over all ordinals, and so over all sets

Up till now, all axioms of $\sf ZC$ are interpreted.

Replacement:  This follows trivially from the axiom of Boundedness and the
definition of $\in$

##  5 Interpreting COT in MK

By MK we mean Morse-Kelley set theory, formalized as $\sf NBG$ but with
allowing class variables freely in comprehension [Men97]. However, we’ll adopt
what is basically the original Kelley’s system [Kel75] which is the well known
equivalent approach [Git14] of replacing the limitation of size axiom by two
axioms, one asserting that the range of every function from a set domain is a
set, the other is Global Choice.

Working in “$\sf ZFC\ +$$\ V=L\ +\text{ there is an inaccessible }$” take
$L_{\kappa}$ where $\kappa$ is the first inaccessible. Now
$\mathcal{P}(L_{\kappa})$ is a model of $\sf MK$ and moreover there is a well
ordering $R$ over $\mathcal{P}(L_{\kappa})$. Elements of $L_{\kappa}$ are the
$\operatorname{Sets}$ of $\sf MK$, while the rest of $\mathcal{P}(L_{\kappa})$
are the proper classes of $\sf MK$. Now we define a rank respective well
ordering $\prec$ over $\mathcal{P}(L_{\kappa})$, as:

$X\prec Y\iff\rho X<\rho Y\land(\rho X=\rho Y\to X\ R\ Y)$

; where $\rho$ is the usual rank function.

Define: ${\sf Ord}=\\{x\in L_{\kappa}\mid x\text{ is von Neumann ordinal}\\}$

Let $\Omega=\\{\\{0\\}\\}$

Define: $X$ is $class^{*}$ if and only if $\Omega\in X\land\forall Y\in
X(Y\neq\Omega\to Y\in{\sf ORD})$.

Define $\in^{*}$ as: $X\in^{*}Y\iff class^{*}(Y)\land X\in Y\land
X\neq\Omega$.

Define $\prec^{*}$ as:
$X\prec^{*}Y\iff\\{Z:Z\in^{*}X\\}\prec\\{Z:Z\in^{*}Y\\}$

Let, ${\sf Classes^{*}}=\\{K\mid class^{*}(K)\\}$

We’ll show that $({\sf Classes^{*}},{\sf Ord},=,\in^{*},\prec^{*},\iota^{*})$
is a model of $\sf COT$

Comprehension: $\exists X\in{\sf Classes^{*}}\forall x\in{\sf
Ord}:x\in^{*}X\leftrightarrow\phi(x)$

Proof: for any $\phi$ we have the class $A=\\{x\in{\sf Ord}\mid\phi(x)\\}$,

then easily get $X=A\bigcup\\{\Omega\\}$

Well-Ordering: $\prec^{*}$ is a well ordering on ${\sf Classes^{*}}$

Proof: Clearly the formula $X=\\{Y\mid Y\in^{*}X^{*}\\}$ describes a one-to-
one correspondence between $\mathcal{P}({\sf Ord})$ and $\sf Classes^{*}$, and
since $\prec$ is a well-ordering on the former, then from the definition of
$\prec^{*}$ it must be a well-ordering the later.

Define: $x<^{*}y\iff\\{x\\}\prec^{*}\\{y\\}$

Define: $\lim^{*}X=\min l:\forall x\in^{*}X(x<^{*}l)$

Respective: $X,Y\in\mathcal{\sf Classes^{*}}\land\lim^{*}X\leq y\in^{*}Y\to
X\prec^{*}Y$

Proof: No distinct ordinals have the same rank! Therefore, for any ordinals
$a,b$, the sets $\\{a\\};\\{b\\}$ are of distinct ranks. Moreover the rank
strict smaller than relation ”$<^{\rho}$” is a strict well-order on $\sf Ord$
and thus also on $\mathcal{P}_{1}({\sf Ord})$. That said, then for every $\sf
Class^{*}$ $X$ we have $\lim^{*}X$ having strictly greater rank than all
$\in^{*}$-elements of $X$, and so $\rho\\{x\mid
x\in^{*}X\\}<\rho\\{\lim^{*}X\\}$, accordingly if there is $\lim^{*}X\leq
y\in^{*}Y$, then $\rho\\{x\mid x\in^{*}X\\}<\rho\\{y\mid y\in^{*}Y\\}$ and so
$\\{x\mid x\in^{*}X\\}\prec\\{y\mid y\in^{*}Y\\}$ by definition, and so
$X\prec^{*}Y$ by definition

Define: $\operatorname{bnd}(X)\iff X\in\mathcal{P}({\sf Ord})\land\exists
l:l=\lim X$

Theorem 1: $\operatorname{bnd}(X)\iff X\in L_{\kappa}\cap\mathcal{P}({\sf
Ord})$

Proof: if $X$ is a set of ordinals in $L_{\kappa}$ that is bounded upwardly by
an ordinal in $L_{\kappa}$, then it has a rank in $L_{\kappa}$, and so it is
an element of $L_{\kappa}$. For the opposite direction if $X\in
L_{\kappa}\cap\mathcal{P}({\sf Ord}$), then it has a rank in $L_{\kappa}$ and
that rank is a limit on all elements of $X$ and so $\operatorname{bnd}(X)$

Define: $\operatorname{bnd^{*}}(X)\iff X\in{\sf Classes^{*}}\land\exists
l:l=\lim^{*}X$

Theorem 2:
$|\\{X:\operatorname{bnd^{*}}(X)\\}|=|L_{\kappa}\cap\mathcal{P}({\sf Ord})|$

Proof: Clearly the function $\eta(X)=\\{Y\mid Y\in^{*}X\\}$ is a bijection
from bounded∗ classes to bounded classes and the latter are the elements of
$L_{\kappa}\cap\mathcal{P}({\sf Ord})$ [Theorem 1]

Indexing: Define $\iota^{*}$ recursively on the bounded∗ classes using the
well ordering $\prec^{*}$ as:

$\iota^{*}(X)=\lim\\{\iota^{*}(Y)\mid Y\prec^{*}X\\}$

Infinity: There exists a limit ordinal.

Proof: $\omega$ is a limit ordinal with respect to $<$

Boundedness: $\operatorname{bnd^{*}}(X)\land Y\in{\sf
Classes^{*}}\land|\eta(X)|=|\eta(Y)|\to\operatorname{bnd^{*}}(Y)$

Proof: Since we have $\operatorname{bnd^{*}}(X)$ then $\eta(X)\in L_{\kappa}$,
so by Replacement we get $\eta(Y)\in L_{\kappa}$, and so
$\operatorname{bnd^{*}}(Y)$ by definition.

##  6 Bi-interpretability of COT and Set Theory

We set a claim that Extensional Tri-sorted $\sf COT$ is bi-interpretable with
$\sf MK$ extended with a well ordering $\prec$ on all classes, the latter
theory to be denoted by $\sf MKCWO$.

Extensional Tri-sorted $\sf COT$, hereafter symbolized simply as $\sf COT$,
adds on top of the above mentioned axioms of $\sf COT$ the axiom of
Extensionality over all classes, and also adds a third sort that ranges over
the first two sorts, we can use $\mathfrak{A,B,C,X,Y,Z,...}$ to denote the
third sort. So, we axiomatize: $\\\ \textbf{Sorting axioms: }\\\
\forall{\mathfrak{X}}:\exists x\,({\mathfrak{X}}=x)\lor\exists
X\,({\mathfrak{X}}=X)\\\ \forall x\exists{\mathfrak{X}}:x={\mathfrak{X}}\\\
\forall X\exists{\mathfrak{X}}:X={\mathfrak{X}}\\\ \forall x\forall X:x\neq X$

###  6.1 Definitions and Terminology:

The main definitions and terminology follows that of Visser in Categories of
Theories and Interpretations [Vis06]. However, for the purpose of self
inclusiveness they’d be concisely reintroduced here.

A translation $\tau$ of signature $\Sigma$ in signature $\Theta$, denoted
$\tau:\Sigma\longrightarrow\Theta$, is a pair $\langle\delta,F\rangle$, where
$\delta$ is a formula in signature $\Theta$ with exactly one free variable,
and $F$ is a function such that for each $n$-ary predicate (an $n$-ary
relation, or $n-1$-ary function) $P$ in $\Sigma$ we have $F(P)$ [also written
as $F_{\tau}(P)$] being an $n$-ary predicate defined by a formula in signature
$\Theta$ having exactly $n$ free variables.

Any formula $a_{i}$ written in $\Sigma$, its translation using $\tau$ is
denoted by $a_{i}^{\tau}$ and it is the sentence resulting from merely
replacing each element $P$ of the signature of $\Sigma$ occurring in $a_{i}$
by $F_{\tau}(P)$ and relativising all quantifiers in $a_{i}$ by
$\delta_{\tau}$. For connivance each translated predicate $F_{\tau}(P)$ is
written as $P_{\tau}$, also quantification is denoted by $\forall
x:\delta_{\tau};\exists x:\delta_{\tau}$, it is understood that $x$ is the
sole free variable of $\delta_{\tau}$.

Interpreting a multi-sorted theory follows Visser and Friedman [FV14], so when
using translation $\tau$, this is done by assigning a domain formula
$\delta^{i}_{\tau}$ for each sort $i$, then we require that for any $n$-ary
predicate $P$ in the signature of that theory that has $t_{1},..,t_{n}$ type
pattern, then the formula $A$ defining $P_{\tau}$ must assure that $\forall
x_{1},..,\forall
x_{n}(P_{\tau}(x_{1},..,x_{n})\to\delta_{\tau}^{t_{1}}(x_{1})\land...\land\delta_{\tau}^{t_{n}}(x_{n}))$

An identity translation on signature $\Theta$, denoted by $\sf id_{\Theta}$,
is $\sf id:\Theta\longrightarrow\Theta$ where $\delta_{\sf id}:=\Huge{\top}$,
and for every element $P$ of $\Theta$, $P_{\sf id}:=P$

Any two translations $\tau,\nu$, the composition $\tau\circ\nu$, abbreviated
as $\nu\tau$, is a translation with
$\delta_{\nu\tau}:=\delta_{\tau}\land(\delta_{\nu})^{\tau}$ and
$P_{\nu\tau}:=(P_{\nu})^{\tau}$

An interpretation $I_{\tau}$ of theory $U$ written in $\Sigma$ in theory $V$
written in $\Theta$, denoted $I_{\tau}:U\longrightarrow V$, is a triplet
$\langle U,\tau,V\rangle$ such that $\tau:\Sigma\longrightarrow\Theta$ and for
every axiom $a_{i}$ of $U$ we have $V\vdash a_{i}^{\tau}$.

We use similar terminology to that of translations, so for an interpretation
$K$ we have $\tau_{K}$ means the translation used in $K$, i.e.
$K=\langle.,\tau,.\rangle$, now $\delta_{K},E_{K},P_{K},A^{K}$ are actually
$\delta_{\tau_{K}};E_{\tau_{K}},P_{\tau_{K}},A^{\tau_{K}}$

Similarly we can have identity interpretations and compositions of
interpretations, so $I_{\sf id}:U\longrightarrow U$ is the triplet $\langle
U,{\sf id_{\Sigma}},U\rangle$, is the identity interpretation on theory $U$ in
signature $\Sigma$.

Also, for interpretations $I_{\tau}:U\longrightarrow
W;I_{\nu}:W\longrightarrow V$, the composition interpretation $I_{\nu}\circ
I_{\tau}:U\longrightarrow V$, is the triplet $\langle
U,\nu\circ\tau,V\rangle=\langle U,\tau\nu,V\rangle$

$F$ is an i-isomorphism from interpretation K to interpretation M, where
$K:U\longrightarrow V;M:U\longrightarrow V$, if and only if, $F$ is a formula
in the language of $V$ having exactly two free variables, that satisfy all the
following:

$\displaystyle V\vdash xFy\to(x:\delta_{K}\land y:\delta_{M})$ (1)
$\displaystyle V\vdash(x:\delta_{K}\land y:\delta_{M}\land
xE_{K}x^{\prime}Fy^{\prime}E_{M}y)\to xFy$ (2) $\displaystyle V\vdash\forall
x:\delta_{K}\exists y:\delta_{M}xFy$ (3) $\displaystyle V\vdash(xFy\land
xFy^{\prime})\to yE_{M}y^{\prime}$ (4) $\displaystyle V\vdash\forall
y:\delta_{M}\exists x:\delta_{K}xFy$ (5) $\displaystyle V\vdash(xFy\land
x^{\prime}Fy)\to xE_{K}x^{\prime}$ (6) $\displaystyle
V\vdash\vec{x}F\vec{y}\to(P_{K}\vec{x}\leftrightarrow P_{M}\vec{y})$ (7)

Here $\vec{x}F\vec{y}$ abbreviates $x_{0}Fy_{0}\land...\land x_{n-1}Fy_{n-1}$,
for appropriate $n$; and $E$ is the identity relation in the signature of $U$.

To establish the bi-interpretability claim, we need to prove that there are
interpretations

$\displaystyle\sf I_{\tau}:COT\longrightarrow MKCWO$ (1) $\displaystyle\sf
I_{\nu}:MKCWO\longrightarrow COT$ (2)

, such that:

$\displaystyle\sf I_{\nu}\circ I_{\tau}:COT\longrightarrow COT\text{ is
i-isomorphic to }\sf I_{id}:COT\longrightarrow COT$ (1) $\displaystyle\sf
I_{\tau}\circ I_{\nu}:MKCWO\longrightarrow MKCWO\text{ is i-isomorphic to }\sf
I_{id}:MKCWO\longrightarrow MKCWO$ (2)

###  6.2 Proof

####  6.2.1  First direction

Let
$\nu:\mathcal{L}(=,\in,\prec)\longrightarrow\mathcal{L}(=,\varepsilon,\prec,\iota)$

Define: $\delta_{\nu}:=\exists
x(\mathfrak{X}=x)\lor\neg\operatorname{bnd}(\mathfrak{X})$

Define: $\mathfrak{X}=_{\nu}\mathfrak{X}\iff\mathfrak{X}=\mathfrak{X}$

Define:

$\displaystyle{\mathfrak{A}}\in_{\nu}{\mathfrak{X}}\iff\exists
a:{\mathfrak{A}}=a\land[$ $\displaystyle(\exists x:{\mathfrak{X}}=x\land a\
\varepsilon\ \iota^{-1}(x))\lor$ $\displaystyle(\exists
X:{\mathfrak{X}}=X\land a\ \varepsilon\ X)]$

Define:

$\displaystyle\mathfrak{A}\prec_{\nu}\mathfrak{X}\iff$ $\displaystyle\exists
x\exists a:\mathfrak{X}=x\land\mathfrak{A}=a\land a<x\ \lor$
$\displaystyle\exists X\exists A:\mathfrak{X}=X\land\mathfrak{A}=A\land A\prec
X\ \lor$ $\displaystyle\exists X\exists a:\mathfrak{X}=X\land\mathfrak{A}=a$

This way we get to interpret $\sf MKCWO$, this is straightforward for the set
axioms of $\sf MK$, i.e. set Extensionality, Pairing, Union, Power,
Separation, Replacement, and Infinity, all follow by interpretability of $\sf
ZFC$. Now, foundation for classes clearly follows because all sets are well
founded so classes of them would be so. What remains is Class Extensionality,
Comprehension and Global Choice. For the first, it follows from
$\in_{\nu}$-co-Extensionality preserving the state of being bounded or not,
thus preserving set-hood and proper class-hood; then Class Extensionality
straightforwardly follow from Extensionality of classes in $\sf COT$.
Comprehension also follows easily from Comprehension in $\sf COT$. Global
Choice clearly follows from the well ordering on all classes. To be noted is
that quantification over classes is allowed both in Comprehension as well as
in Separation and Replacement.

####  6.2.2  Second direction

Let
$\tau:\mathcal{L}(=,\varepsilon,\prec,\iota)\longrightarrow\mathcal{L}(=,\in,\prec)$

Without loss of generality we take $\prec$ to be rank respective [section 4].

Fix a non ordinal set $\Omega$;

Define: $\\\ \delta^{1}_{\tau}:=x\in{\sf ORD}\\\ \delta^{2}_{\tau}:=\Omega\in
x\land\forall y\in x(y\neq\Omega\to y\in{\sf ORD})\\\
\delta^{3}_{\tau}:=\delta^{1}_{\tau}\lor\delta^{2}_{\tau}\\\ x=_{\tau}y\iff
x=y\\\ x\ \varepsilon_{\tau}\
y\iff\delta^{1}_{\tau}(x)\land\delta^{2}_{\tau}(y)\land x\in y\\\
\zeta(x)=i\iff\operatorname{Set}(x)\land
i=\lim\operatorname{ord}\\{\zeta(y)\mid y\prec x\\}\\\ x\prec_{*}y\iff
x\subseteq{\sf ORD}\land y\subseteq{\sf ORD}\land\\{\zeta^{-1}(m)\mid m\in
x\\}\prec\\{\zeta^{-1}(n)\mid n\in y\\}\\\
x\prec_{\tau}y\iff\delta^{2}_{\tau}(x)\land\delta_{\tau}^{2}(y)\land\eta(x)\prec_{*}\eta(y)\\\
\iota_{\tau}(x)=y\iff\delta^{2}_{\tau}(x)\land y=\zeta(\\{\zeta^{-1}(z)\mid z\
\varepsilon_{\tau}\ x\\})$

Interpreting $\sf COT$ follows exactly the same argument in section 4. Just
replace the domains appropriately, and switch $P^{*}$ to $P_{\tau}$. What
needs to be proved is:

$\iota_{\tau}(x)=\lim\operatorname{ord}\\{\iota_{\tau}(y)\mid
y\prec_{\tau}x\\}$

Proof: it is obvious that the well ordering relation $\prec$ on sets is copied
downwardly to $\prec_{*}$ on $\mathcal{P}({\sf ORD})$, which is in turn copied
downwardly to $\prec_{\tau}$ on $\sf Classes^{*}$ (i.e. $\delta^{2}_{\tau}$
objects). Then the $\zeta$ function which is an index of $\prec$ on sets,
would be copied downwardly to the $\iota_{\tau}$ function on $\sf
Classes^{*}$, and so $\iota_{\tau}$ would like-wisely index $\prec_{\tau}$
relation on $\sf Classes^{*}$. That said, it is easy to prove that whatever
ordinal $\alpha$ we have $\alpha<\iota_{\tau}(x)\iff\exists
y:\delta^{2}_{\tau}\ y\prec_{\tau}x\land\alpha=\iota_{\tau}(y)$. Thus
establishing the result.

####  6.2.3  Proving i-isomorphisms

First direction:

That $F$ is i-isomorphic from $K$ to $M$ where $K=\sf\langle MKCWO,{\sf
id},MKCWO\rangle$ and $M=\sf\langle MKCWO,\tau\circ\nu,MKCWO\rangle$ , is
proved as follows:

Working in $\sf MKCWO$, define:

$\displaystyle F(X)=\left\\{\begin{aligned} &\zeta(X)\quad&\text{if
}\operatorname{Set}(X)\\\ &\\{\zeta(Y)\mid Y\in
X\\}\cup\\{\Omega\\}\quad&\text{if }\operatorname{\neg
Set}(X)\end{aligned}\right.$

$\delta_{K}=\delta_{\sf id}:=x\in\mathcal{P}(L_{\kappa})$

$\delta_{M}=\delta_{\nu\tau}:=\delta^{3}_{\tau}\land(\delta_{\nu})^{\tau}$

$\delta_{M}:=y\in{\sf ORD}\cup\\{x\cup\\{\Omega\\}\mid x\subseteq{\sf
ORD}\land\neg\operatorname{Set}(x)\\}$

$xE_{K}x^{\prime}\iff x=x^{\prime}$

$yE_{M}y^{\prime}\iff y=y^{\prime}$

$x\in_{K}x^{\prime}\iff x\in x^{\prime}$

$y\in_{M}y^{\prime}\iff(y\in_{\nu}y^{\prime})^{\tau}$

$y\in_{M}y^{\prime}\iff(y^{\prime}\in{\sf ORD}\land
y\in\iota_{\tau}^{-1}(y^{\prime}))\lor(\delta^{2}_{\tau}(y^{\prime})\land\neg\operatorname{Set}(y^{\prime})\land
y\in y^{\prime})$

$x\prec_{K}x^{\prime}\iff x\prec x^{\prime}$

$y\prec_{M}y^{\prime}\iff(y\prec_{\nu}y^{\prime})^{\tau}$

$\displaystyle x\prec_{M}y\iff$ $\displaystyle x,y\in{\sf ORD}\land x<y\ \lor$
$\displaystyle\delta^{2}_{\tau}(x)\land\delta^{2}_{\tau}(y)\land\neg\operatorname{Set}(x)\land\neg\operatorname{Set}(y)\land
x\prec_{\tau}y\ \lor$ $\displaystyle x\in{\sf
ORD}\land\delta^{2}_{\tau}(y)\land\neg\operatorname{Set}(y)$

It’s an easy check to see that all 1-7 requirements of i-isomorphisms are met!

Second direction:

That $G$ is i-isomorphic from $K=\sf\langle COT,{\sf id},COT\rangle$ to
$M=\sf\langle COT,\nu\circ\tau,COT\rangle$ , is proved as follows:

Working in $\sf COT$, define:
$\zeta_{\nu}:=(\lim\operatorname{ord}\\{\zeta(y)\mid y\prec x\\})^{\nu}$

$\displaystyle\text{Define: }\ \ \ G(\mathfrak{X})=\left\\{\begin{aligned}
&\zeta_{\nu}(\mathfrak{X})\quad&\text{ if }\exists x:\mathfrak{X}=x\\\
&\iota(\\{\zeta_{\nu}(\mathfrak{A})\mid\mathfrak{A}\ \varepsilon\
\mathfrak{X}\\}\cup\\{\Omega\\})\quad&\text{ if }\exists
X:\mathfrak{X}=X\land\operatorname{bnd}(X)\\\
&\\{\zeta_{\nu}(\mathfrak{A})\mid\mathfrak{A}\ \varepsilon\
\mathfrak{X}\\}\cup\\{\Omega\\}\quad&\text{ if }\exists
X:\mathfrak{X}=X\land\neg\operatorname{bnd}(X)\end{aligned}\right.$

We’ll adopt the notation $x^{i}$ to stand for the $i^{th}$ sort, so
$\phi(x^{1});\phi(x^{2});\phi(x^{3})$ actually means
$\phi(x),\phi(X),\phi(\mathfrak{X})$

$\delta^{i}_{K}=\delta^{i}_{\sf id}:=x^{i}=x^{i}$

$\delta^{i}_{M}=\delta^{i}_{\tau\nu}:=\delta_{\nu}\land(\delta^{i}_{\tau})^{\nu}$

$x^{i}E_{K}y^{i}\iff x^{i}=y^{i}$

$z^{i}E_{M}u^{i}\iff z^{i}=u^{i}$

$x^{1,3}\ \varepsilon_{K}\ y^{2,3}\iff x^{1,3}\ \varepsilon\ y^{2,3}$

$z^{1,3}\ \varepsilon_{M}\ u^{2,3}\iff(z^{1,3}\ \varepsilon_{\tau}\
u^{2,3})^{\nu}$

$x^{2,3}\prec_{K}y^{2,3}\iff x^{2,3}\prec y^{2,3}$

$z^{2,3}\prec_{M}u^{2,3}\iff(z^{2,3}\prec_{\tau}u^{2,3})^{\nu}$

$\iota_{K}(x^{2,3})=y^{1}\iff\iota(x^{2,3})=y^{1}$

$\iota_{M}(z^{2,3})=u^{1}\iff(\iota_{\tau}(z^{2,3})=u^{1})^{\nu}$

It’s an easy check to see that all 1-7 requirements of i-isomorphism are met!

More demonstrative, this means that given whatever model $\mathcal{M}$ of $\sf
MKCWO$ we can use translation $\tau$ to build a model
$\mathcal{C}_{\tau}^{\mathcal{M}}$ of $\sf COT$ within it, and within
$\mathcal{C}_{\tau}^{\mathcal{M}}$, via translation $\nu$, we can build a
model $M_{\nu}^{\mathcal{C}_{\tau}^{\mathcal{M}}}$ of $\sf MKCWO$, and such
that we can define within $\mathcal{M}$ an isomorphism after formula $F$ from
$\mathcal{M}$ to $M_{\nu}^{\mathcal{C}_{\tau}^{\mathcal{M}}}$. And also given
whatever model $\mathcal{C}$ of $\sf COT$ we can use translation $\nu$ to
build a model $\mathcal{M}_{\nu}^{\mathcal{C}}$ of $\sf MKCWO$ within it, and
within $\mathcal{M}_{\nu}^{\mathcal{C}}$, via translation $\tau$, we can build
a model $C_{\tau}^{\mathcal{M}_{\nu}^{\mathcal{C}}}$ of $\sf COT$, and such
that we can define within $\mathcal{C}$ an isomorphism after formula $G$ from
$\mathcal{C}$ to
$C_{\tau}^{\mathcal{\mathcal{missing}}M_{\nu}^{\mathcal{C}}}$.

##  7 Conclusion

The powerful hierarchical class theory $\sf MKCWO$ is reducible in structure
to a theory about indexed ordering of bounded classes of ordinals. It needs to
be warned that this bi-interpretability is only with the extensional form of
class ordering theory. The non-extensional form of $\sf COT$ is most likely
not bi-interpretable with $\sf MKCWO$, but it might be bi-interpretable with a
non-extensional version of it. The point is that the massive hierarchical
structure of such a rich theory as $\sf MKCWO$ is in some sense redundant,
everything can be done in a single tier of membership, i.e. with flat classes,
and moreover the system is simple and the proofs of interpretation results in
it are also simple. This motivates this system as a strong and simple tool for
proving interpretability results.

##  8 References:

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