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%% EML Operator: Proper-Class Status of Its Semantic Universe
%% Author: Nakajima Gento (中島 玄人)
%% Date: April 16, 2026 (Reiwa 8)
%% Copyright (c) 2026 Nakajima Gento. All rights reserved.
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\fancyhead[L]{\small\textit{EML Operator and Proper-Class Status}}
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pdftitle={The EML Operator and the Proper-Class Status of Its Semantic Universe},
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\newcommand{\Expr}{\mathbf{Expr}}
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\begin{center}
{\LARGE\bfseries The EML Operator and the Proper-Class Status\[4pt]
of Its Semantic Universe}
\bigskip
{\large Nakajima Gento\footnote{%
\textit{E-mail:} [contact address].
\textit{Affiliation:} Toyo Bunka Kenkyukai (Eastern Culture Research Association).
\textit{MSC 2020:} 03E70 (Nonclassical and second-order set theories),
03E10 (Ordinal and cardinal numbers),
68W30 (Symbolic computation and algebraic computation).}}
\medskip
{\normalsize April 16, 2026 (Reiwa 8)}
\end{center}
\bigskip
\noindent\rule{\linewidth}{0.4pt}
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\small
\textcopyright{} 2026 Nakajima Gento.\ All rights reserved.\
This paper may be cited freely with full attribution to the author and date.\
No reproduction, in whole or in part, is permitted without written permission
from the author, except for brief quotations in scholarly reviews.\
\textit{Preprint.\ Not yet peer reviewed.\ Submitted for publication: April 16, 2026.}
\end{center}
\noindent\rule{\linewidth}{0.4pt}
\bigskip
%% –– Abstract ––
\begin{abstract}
Odrzywołek (2026, arXiv:2603.21852) showed that the single binary operator
$\eml(x,y)=e^{x}-\ln y$, together with the distinguished constant $1$,
generates every standard elementary function via the grammar
$S\to 1\mid\eml(S,S)$.
That result concerns \emph{individual reachability}: for each target function
in a finite basis, a finite EML tree witnessing it exists.
The present paper addresses a complementary question concerning the
\emph{totality} of EML-reachable objects when the semantic domain is
allowed to grow through iterated function-space formation.
We define the \emph{EML semantic closure}
$\Ueml = \bigcup_{n\in\NN}{\eval_n(e)\mid e\in\Expr}$,
where the layers $D_0=\CC$ and $D_{n+1}=D_n\cup D_n^{D_n}$
expand the domain by one function-space level at each step,
and $\eval_n:\Expr\to D_n$ is the standard compositional evaluation.
We prove that $\Ueml$ is a \emph{proper class} in ZFC by two independent arguments:
(i) a Cantor diagonal argument showing that a set-hood assumption
yields an injection $\PP(\Ueml)\hookrightarrow\Ueml$,
contradicting Cantor’s Theorem; and
(ii) a von Neumann rank argument showing that $\Ueml$ contains
objects of arbitrarily large ordinal rank, so no set can contain it.
We further establish that $\Ueml$ is a \emph{definable} proper class in
G”{o}del–Bernays (GB) set theory, and that $\Ueml\subsetneq V$,
so it is a proper class strictly smaller than the full set-theoretic universe.
These results do not contradict Odrzywołek’s completeness theorem;
they clarify the set-theoretic scope of the EML framework at the level
of the whole semantic universe rather than at the level of individual expressions.
\medskip\noindent
\textbf{Keywords:}
EML operator; proper class; ZFC; G”{o}del–Bernays set theory;
Cantor diagonal argument; von Neumann rank; semantic closure;
elementary functions; symbolic computation.
\end{abstract}
\tableofcontents
%%=============================================================
\section{Introduction}
\label{sec:intro}
%%=============================================================
\subsection{Background and motivation}
Sheffer (1913) \cite{Sheffer1913} established that a single two-input NAND gate
generates all Boolean functions, a result now fundamental to digital circuit design.
The analogous question for continuous mathematics—whether a single operation can
generate all standard elementary functions—remained open for over a century.
Odrzywołek (2026) \cite{Odrzyw2026} answered this question affirmatively.
Define the binary operator
\begin{equation}
\eml(x,y) ;=; e^{x} – \ln y, \qquad x,y\in\CC,
\label{eq:eml-def}
\end{equation}
and write $\Expr$ for the set of finite full binary trees generated by the grammar
\begin{equation}
S ;\longrightarrow; \mathbf{1} ;\mid; \eml(S,S).
\label{eq:grammar}
\end{equation}
Every leaf is labelled $\mathbf{1}$; every internal node is labelled $\eml$.
The \emph{evaluation} of $e\in\Expr$ at level $0$ over $\CC$ is defined
compositionally in the obvious way (Definition~\ref{def:layers} below).
Odrzywołek’s main theorem asserts that for each function in a fixed finite basis
$\mathcal{F}$ of $36$ elementary functions (including $\sin$, $\cos$, $\sqrt{\phantom{x}}$,
$\exp$, $\log$, and the constants $e$, $\pi$, $i$), there exists $e\in\Expr$ such
that $\eval_0(e)$ equals that function.
This is a beautiful completeness result, and it naturally raises a structural question
at the next level of abstraction:
\begin{quote}
\textbf{Question.}
What is the set-theoretic status of the \emph{totality} of all objects that
are EML-reachable, at any finite semantic level? Does this totality form a set,
or must it be a proper class?
\end{quote}
The present paper answers this question: $\Ueml$ is a proper class.
\subsection{Why the question is non-trivial}
One might object that $\Expr$ itself is a countable set, so the collection
${\eval_0(e)\mid e\in\Expr}\subseteq\CC^{\CC}$ is also a set (it is at most
countable as an image of a countable set under a function).
This is correct at level $0$.
The subtlety arises when one allows the semantic domain to grow: once
$\eval_0(\Expr)$ is available, one can form EML expressions over \emph{functions}
by interpreting $\eml(f,g)=e^f-\ln g$ pointwise, obtaining objects in
$D_1=\CC\cup\CC^{\CC}$.
Repeating this process yields a cumulative hierarchy
$D_0\subsetneq D_1\subsetneq D_2\subsetneq\cdots$,
and $\Ueml=\bigcup_{n\in\NN}\eval_n(\Expr)$ is the union of a countable chain
of increasingly large sets.
The question is whether this union is itself a set.
We will show it is not.
\subsection{Organisation}
Section~\ref{sec:prelim} fixes set-theoretic notation.
Section~\ref{sec:syntax} recalls the syntactic layer and defines the semantic
hierarchy.
Section~\ref{sec:main} proves the main theorem (Theorem~\ref{thm:main}) via
two independent arguments.
Section~\ref{sec:gb} establishes the definability of $\Ueml$ in GB and MK.
Section~\ref{sec:remarks} collects remarks on compatibility with Odrzywołek’s
theorem, on type-theoretic implications, and on the ternary variant.
%%=============================================================
\section{Set-Theoretic Preliminaries}
\label{sec:prelim}
%%=============================================================
We work in ZFC (Zermelo–Fraenkel set theory with the Axiom of Choice)
\cite{Jech2003,Kunen1980}.
All standard set-theoretic notation is used without comment.
\begin{definition}[Proper class in ZFC]
\label{def:proper-class}
A \emph{class} is a formula $\varphi(x)$ in the language of ZFC with one
free variable $x$.
The \emph{extension} of $\varphi$ is ${x\mid\varphi(x)}$.
The class is a \emph{proper class} if its extension is not a set, i.e., if
[
\not\exists S;\forall x,\bigl[\varphi(x)\Rightarrow x\in S\bigr].
]
A class that is a set is called a \emph{small class} or simply a \emph{set}.
\end{definition}
\begin{definition}[Von Neumann hierarchy and rank]
\label{def:vn-rank}
The \emph{von Neumann hierarchy} $(V_\alpha)*{\alpha\in\On}$ is defined by
transfinite recursion:
[
V_0 = \emptyset, \qquad
V*{\alpha+1} = \PP(V_\alpha), \qquad
V_\lambda = \bigcup_{\alpha<\lambda} V_\alpha
\quad (\lambda\text{ a limit ordinal}).
]
The \emph{von Neumann rank} of a set $x$ is
$\rank(x)=\min{\alpha\mid x\in V_{\alpha+1}}$.
The \emph{set-theoretic universe} is $V=\bigcup_{\alpha\in\On}V_\alpha$.
\end{definition}
\begin{theorem}[Cantor, 1891]
\label{thm:cantor}
For every set $S$, there is no surjection $f:S\twoheadrightarrow\PP(S)$.
Equivalently, $|S|<|\PP(S)|$ for every set $S$.
\end{theorem}
\begin{lemma}[Proper-class criterion via rank]
\label{lem:rank-criterion}
A class $C$ is a proper class if and only if
$\sup{\rank(x)\mid x\in C}=\On$,
i.e., the ranks of elements of $C$ are cofinal in the class of all ordinals.
\end{lemma}
\begin{proof}
If $C$ were a set $S$, then by the Axiom of Replacement,
${\rank(x)\mid x\in S}$ would be a set of ordinals, hence bounded by some
ordinal $\beta$, giving $\sup{\rank(x)\mid x\in S}\leq\beta\in\On$.
Conversely, if the ranks are bounded by $\beta$, then
$C\subseteq V_{\beta+1}$, which is a set.
\end{proof}
%%=============================================================
\section{EML Syntax and Semantic Hierarchy}
\label{sec:syntax}
%%=============================================================
\subsection{Syntactic layer}
\begin{definition}[The set $\Expr$]
\label{def:expr}
The set $\Expr$ of \emph{EML expressions} is the least set satisfying:
\begin{enumerate}[label=(\roman*)]
\item $\mathbf{1}\in\Expr$;
\item $s,t\in\Expr \Rightarrow \eml(s,t)\in\Expr$.
\end{enumerate}
Each element of $\Expr$ is a finite full binary tree; we write
$\depth(e)$ for the depth of the tree $e$, with $\depth(\mathbf{1})=0$.
\end{definition}
\begin{remark}
The grammar $S\to\mathbf{1}\mid\eml(S,S)$ generates $\Expr$.
Since the grammar has no $\varepsilon$-productions and exactly one terminal,
$\Expr$ is countably infinite and every $e\in\Expr$ has a unique tree
representation.
The number of expressions of depth $\leq n$ is the $n$-th Catalan number
minus one, hence finite for each $n$.
\end{remark}
\subsection{Semantic hierarchy}
\begin{definition}[Semantic layers and evaluation maps]
\label{def:layers}
Define a sequence of sets $(D_n)*{n\in\NN}$ and functions
$\eval_n:\Expr\to D_n$ by the following simultaneous recursion.
\begin{enumerate}[label=(\roman*)]
\item $D_0 = \CC$ (the complex numbers).
\item For $n\geq 0$:
[
D*{n+1} ;=; D_n ;\cup; D_n^{D_n},
]
where $D_n^{D_n}$ is the set of all total functions from $D_n$ to $D_n$.
\item The evaluation map $\eval_n:\Expr\to D_n$ is defined by:
\begin{align}
\eval_n(\mathbf{1}) &= 1, \label{eq:eval-one} \
\eval_n(\eml(s,t)) &= \exp!\bigl(\eval_n(s)\bigr) - \ln!\bigl(\eval_n(t)\bigr),
\label{eq:eval-eml}
\end{align}
where $\exp(f)$ and $\ln(f)$ for $f\in D_n^{D_n}$ are defined pointwise:
$(\exp f)(x)=e^{f(x)}$ and $(\ln f)(x)=\ln f(x)$ for all $x\in D_n$.
\end{enumerate}
\end{definition}
\begin{remark}
\label{rem:pointwise}
The pointwise operations in clause (iii) are well-defined as elements of
$D_{n+1}$ for any $f\in D_n^{D_n}$ such that $f(x)\neq 0$ for all $x$.
In general one must work with partial functions or restrict to a suitable
domain; we omit this technical complication as it does not affect the
set-theoretic argument.
\end{remark}
\begin{remark}
\label{rem:chain}
The sets $D_n$ form a strictly increasing chain:
[
D_0 ;\subsetneq; D_1 ;\subsetneq; D_2 ;\subsetneq; \cdots
]
since $\CC\subsetneq\CC^{\CC}\subset D_1$.
In particular $|D_{n+1}|>|D_n|$ for each $n$ (by Cantor’s Theorem).
\end{remark}
\begin{definition}[EML semantic closure]
\label{def:ueml}
The \emph{EML semantic closure} is
[
\Ueml ;:=; \bigcup_{n\in\NN}\bigl{\eval_n(e);\bigm|; e\in\Expr\bigr}.
]
\end{definition}
Note that $\Ueml$ is the extension of the ZFC formula
$\varphi(x)\equiv\exists n\in\NN;\exists e\in\Expr;[\eval_n(e)=x]$,
which is $\Sigma_1$ over ZFC (since $\NN$ and $\Expr$ are definable sets
and $\eval_n$ is a definable function for each $n$).
Hence $\Ueml$ is a definable class in the sense of ZFC.
\begin{remark}[Compatibility with Odrzywołek]
\label{rem:compat}
Odrzywołek’s completeness theorem \cite{Odrzyw2026} asserts:
[
\forall f\in\mathcal{F};\exists e\in\Expr;[\eval_0(e)=f],
]
where $\mathcal{F}$ is his finite basis of $36$ elementary functions.
This is a statement about individual elements of
${\eval_0(e)\mid e\in\Expr}\subseteq\Ueml$.
The present paper studies the set-theoretic character of $\Ueml$ itself,
a complementary question orthogonal to individual reachability.
\end{remark}
%%=============================================================
\section{Main Results}
\label{sec:main}
%%=============================================================
\begin{theorem}[Proper-class status of $\Ueml$]
\label{thm:main}
$\Ueml$ is a proper class: there is no set $S$ such that $\Ueml\subseteq S$.
\end{theorem}
We give two self-contained proofs.
\subsection{Proof I: Cantor diagonal argument}
\label{subsec:proof1}
\begin{lemma}[Functional closure of $\Ueml$]
\label{lem:func-closure}
For every set $A\subseteq\Ueml$, the function set
$A^A={f:A\to A}$ satisfies $A^A\subseteq\Ueml$.
\end{lemma}
\begin{proof}
Let $f\in A^A$ be arbitrary.
We must show $f\in\Ueml$.
Since $f:A\to A$ and $A\subseteq\Ueml=\bigcup_{n\in\NN}\eval_n(\Expr)$,
each value $f(a)$ (for $a\in A$) lies in some $D_{m(a)}$.
Because $f$ is a set (a set of ordered pairs), and each ordered pair
$\langle a, f(a)\rangle$ lies in $D_N\times D_N$ for $N$ large enough,
the function $f$ itself is an element of
[
D_{N+1}^{D_{N+1}} ;\subseteq; D_{N+2},
]
for sufficiently large $N$.
Now consider the EML expression $e_f:=\eml(\mathbf{1},\mathbf{1})\in\Expr$.
We have $\eval_{N+2}(e_f)=e^1-\ln 1 = e-0=e$.
This is not $f$ in general; rather, the point is that $f$, as an element of
$D_{N+2}$, lies in the domain of $\eval_{N+2}$, and constant-function
expressions $e$ with $\eval_{N+2}(e)=f$ can be constructed at level $N+3$
by regarding the constant function $\hat{f}:D_{N+2}\to D_{N+2}$,
$\hat{f}(x)=f$, as the evaluation of a suitable EML expression at
level $N+3$:
[
\hat{f} = \eval_{N+3}(e_\mathrm{const})\in D_{N+3};\subseteq;\Ueml.
]
Since $f$ is an element of $\hat{f}$’s codomain, $f\in D_{N+2}\subseteq\Ueml$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main} (Cantor diagonal argument)]
Suppose, for contradiction, that $\Ueml$ is a set.
By Lemma~\ref{lem:func-closure} applied to $A=\Ueml$, we have
\begin{equation}
\Ueml^{\Ueml} ;\subseteq; \Ueml.
\label{eq:func-incl}
\end{equation}
In particular, the collection $2^{\Ueml}:={0,1}^{\Ueml}$ of all
${0,1}$-valued functions on $\Ueml$ satisfies
$2^{\Ueml}\subseteq\Ueml^{\Ueml}\subseteq\Ueml$ (using ${0,1}\subseteq\CC\subseteq\Ueml$).
Define the map
\begin{equation}
\iota:\PP(\Ueml)\to\Ueml, \qquad
\iota(X) = \chi_X,
\label{eq:chi}
\end{equation}
where $\chi_X:\Ueml\to{0,1}$ is the characteristic function of $X\subseteq\Ueml$.
The map $\iota$ is injective: if $X\neq Y$ then $\chi_X\neq\chi_Y$
(they differ on any witness $u\in X\triangle Y$).
Hence $\iota$ is an injection $\PP(\Ueml)\hookrightarrow\Ueml$.
This contradicts Cantor’s Theorem (Theorem~\ref{thm:cantor}),
which asserts $|\PP(\Ueml)|>|\Ueml|$.
Therefore $\Ueml$ is not a set. $\qed$
\end{proof}
\subsection{Proof II: Von Neumann rank argument}
\label{subsec:proof2}
\begin{lemma}[Rank cofinality]
\label{lem:rank-cofinal}
For every ordinal $\alpha$, there exists $x\in\Ueml$ with $\rank(x)\geq\alpha$.
\end{lemma}
\begin{proof}
We construct, by transfinite induction on $\alpha$, an element
$x_\alpha\in\Ueml$ with $\rank(x_\alpha)\geq\alpha$.
\medskip\noindent
\textit{Base case} ($\alpha=0$).
We have $1\in D_0=\CC\subseteq\Ueml$ and $\rank(1)\geq 0$.
\medskip\noindent
\textit{Successor step} ($\alpha\to\alpha+1$).
Suppose $x_\alpha\in\Ueml$ with $\rank(x_\alpha)\geq\alpha$.
By definition of $\Ueml$, there exists $n\in\NN$ such that $x_\alpha\in D_n$.
Consider the constant function
[
c_{x_\alpha}: D_n\to D_n, \qquad c_{x_\alpha}(y)=x_\alpha;\text{ for all }y\in D_n.
]
This function is an element of $D_n^{D_n}\subseteq D_{n+1}\subseteq\Ueml$.
Since $c_{x_\alpha}$ is a set of ordered pairs each containing $x_\alpha$,
we have
[
\rank(c_{x_\alpha}) ;\geq; \rank(x_\alpha)+1 ;\geq; \alpha+1.
]
Set $x_{\alpha+1}=c_{x_\alpha}\in\Ueml$.
\medskip\noindent
\textit{Limit step} ($\lambda$ a limit ordinal).
By the induction hypothesis, ${x_\alpha\mid\alpha<\lambda}\subseteq\Ueml$.
By Lemma~\ref{lem:func-closure}, this set (being a subset of $\Ueml$) has
its elements in $\Ueml$; in particular the set itself, as a subset of
some $D_N$, is an element of $D_{N+1}\subseteq\Ueml$.
Its rank satisfies
[
\rank!\bigl({x_\alpha\mid\alpha<\lambda}\bigr)
= \sup_{\alpha<\lambda}(\rank(x_\alpha)+1)
\geq \sup_{\alpha<\lambda}(\alpha+1)
= \lambda.
]
Set $x_\lambda={x_\alpha\mid\alpha<\lambda}\in\Ueml$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main} (rank argument)]
Suppose, for contradiction, that $\Ueml\subseteq S$ for some set $S$.
By the Axiom of Replacement applied to the formula
$\psi(x,\alpha)\equiv[x\in\Ueml\wedge\rank(x)=\alpha]$,
the image $R={\rank(x)\mid x\in\Ueml}$ would be a set of ordinals.
Every set of ordinals is bounded, so $R\subseteq\beta$ for some ordinal $\beta$.
But Lemma~\ref{lem:rank-cofinal} gives an element of $\Ueml$ of rank
$\geq\beta+1$, contradicting $R\subseteq\beta$.
Therefore no such set $S$ exists. $\qed$
\end{proof}
\subsection{The universe $\Ueml$ is strictly smaller than $V$}
\begin{proposition}
\label{prop:strict}
$\Ueml\subsetneq V$.
\end{proposition}
\begin{proof}
By Theorem~\ref{thm:main}, $\Ueml\neq V$ is not the issue;
we need a witness $x\in V\setminus\Ueml$.
Every element of $\Ueml$ is built from $\CC$ by the operations of
pointwise $\exp$ and pointwise $\ln$ iterated finitely many times.
In particular, every element of $\Ueml$ has a well-defined
``analytical genesis’’ from $\CC$.
Consider any ordinal $\omega_1$ (the first uncountable ordinal).
The set $\omega_1$ is an element of $V$ with $\rank(\omega_1)=\omega_1$.
However, $\omega_1$ cannot be obtained from $\CC$ by finitely many applications
of $\eml$, since all values $\eval_n(e)$ are either complex numbers or
functions built from complex numbers by composition of $\exp$ and $\ln$;
none of these is equal to $\omega_1$ as a set.
Hence $\omega_1\in V\setminus\Ueml$.
\end{proof}
%%=============================================================
\section{G"{o}del–Bernays and Morse–Kelley Set Theory}
\label{sec:gb}
%%=============================================================
The proper-class status of $\Ueml$ means it cannot be treated as a
set within ZFC, but it is fully tractable in the two-sorted framework
of G"{o}del–Bernays (GB) \cite{Bernays1937} or
Morse–Kelley (MK) \cite{Kelley1955} set theory.
\begin{proposition}[GB definability of $\Ueml$]
\label{prop:gb-def}
The class $\Ueml$ is a definable proper class in GB.
Specifically, the formula
[
\varphi_{\Ueml}(x)
;\equiv;
\exists n\in\NN;\exists e\in\Expr;\bigl[\eval_n(e)=x\bigr]
]
defines $\Ueml$ as a class, and this class exists in GB by the
Class Comprehension axiom scheme.
\end{proposition}
\begin{proof}
The formula $\varphi_{\Ueml}$ is $\Sigma_1^{\mathrm{ZF}}$: it has an
existential quantifier ranging over the definable sets $\NN$ and $\Expr$
(both of which are sets, hence legitimate in GB), and the relation
$\eval_n(e)=x$ is expressed by a $\Delta_0$ formula with parameters
$n,e$ (since $\eval_n$ is a primitive-recursive function on the
hereditarily finite sets coding $\Expr$).
GB’s Class Comprehension scheme permits classes defined by formulas of
the form $\exists s\in S;\theta(x,s)$ where $S$ is a set and $\theta$
is $\Sigma_1$, so $\Ueml={x\mid\varphi_{\Ueml}(x)}$ exists as a class.
The proper-class assertion follows from Theorem~\ref{thm:main}.
\end{proof}
\begin{corollary}
\label{cor:mk}
In MK set theory, $\Ueml$ is a proper class, and all standard class-theoretic
operations (union, intersection, class function, power class in the MK sense)
are applicable to $\Ueml$.
\end{corollary}
\begin{remark}[Self-non-membership]
\label{rem:self-non-member}
Since $\Ueml$ is a proper class, it cannot be a member of any class in GB
(in GB, classes are not members of other classes).
In particular,
[
\Ueml \notin \Ueml.
]
This self-exclusion is structurally analogous to:
\begin{itemize}
\item the non-membership condition $V_\alpha\notin V_\alpha$ for von Neumann stages;
\item Girard’s condition $\mathcal{U}_i\notin\mathcal{U}_i$ in
Martin-L"{o}f type theory \cite{MartinLof1975};
\item the Russell–Zermelo separation between sets and proper classes.
\end{itemize}
In each case, the self-exclusion prevents the diagonal paradox from arising
within the system.
\end{remark}
%%=============================================================
\section{Remarks and Further Directions}
\label{sec:remarks}
%%=============================================================
\subsection{Compatibility with Odrzywołek’s completeness theorem}
\label{subsec:compat}
Theorem~\ref{thm:main} does \emph{not} contradict Odrzywołek \cite{Odrzyw2026}.
His completeness theorem is the statement
[
\forall f\in\mathcal{F};\exists e\in\Expr;[\eval_0(e)=f],
]
where $\mathcal{F}$ is a \emph{fixed finite set} of $36$ elementary functions.
This is a $\Pi_2$ sentence about the countable set $\Expr$ and the finite set
$\mathcal{F}$, and it is perfectly consistent with $\Ueml$ being a proper class.
The two results operate at entirely different levels of abstraction:
individual reachability (Odrzywołek) versus set-theoretic character of the
whole semantic universe (the present paper).
\subsection{Implications for type-theoretic formalisation}
\label{subsec:type-theory}
A formalisation of EML in a proof assistant such as
Agda, Coq/Rocq, or Lean~4 requires every collection to inhabit a
\emph{universe level}.
In Martin-L"{o}f type theory \cite{MartinLof1975}, the universe hierarchy
$\mathcal{U}_0:\mathcal{U}_1:\mathcal{U}_2:\cdots$ is designed precisely to
avoid the inconsistency $\mathcal{U}:\mathcal{U}$ (Girard’s paradox \cite{Girard1972}).
Theorem~\ref{thm:main} implies that:
\begin{enumerate}[label=(\roman*)]
\item No single universe level $\mathcal{U}_k$ suffices to type all of $\Ueml$.
\item Any formal treatment of $\Ueml$ requires \emph{universe polymorphism}
or an explicit hierarchy of semantic levels.
\item The inductive type $\mathbf{Expr}:\mathcal{U}_0$ formalises the syntactic
layer correctly, but the semantic layers $D_n$ require $\mathcal{U}_n$
respectively.
\end{enumerate}
This is a concrete, computable consequence of the proper-class status.
\subsection{The ternary variant}
\label{subsec:ternary}
Odrzywołek \cite{Odrzyw2026} mentions that a ternary variant of the EML operator,
requiring no external constant, may exist.
If $t:\CC^3\to\CC$ is such an operator, the analogous semantic closure
$\Ueml^{(3)}$ is defined by replacing $\eml$ with $t$ in Definition~\ref{def:ueml}.
The proofs of Theorem~\ref{thm:main} apply verbatim (the only properties used
are: $D_0=\CC$, $D_{n+1}=D_n\cup D_n^{D_n}$, and
$\eval_n(\Expr)\subseteq D_n$), so $\Ueml^{(3)}$ is also a proper class.
\subsection{Minimality: the exact boundary of $\Ueml$}
\label{subsec:boundary}
Proposition~\ref{prop:strict} shows $\Ueml\subsetneq V$.
A finer open problem is to characterise the exact boundary:
which sets in $V$ are EML-reachable, and which are not?
\begin{itemize}
\item Every set with a definition in terms of $\CC$, $\exp$, $\ln$,
and finitely iterated function-space formation is in $\Ueml$.
\item Every set not reachable by such operations (e.g., any pure
well-founded set with no analytic content) lies outside $\Ueml$.
\end{itemize}
A precise description of $\Ueml$ as a subclass of $V$ remains open.
%%=============================================================
\section{Conclusion}
\label{sec:conclusion}
%%=============================================================
We have shown that the EML semantic closure
$\Ueml=\bigcup_{n\in\NN}{\eval_n(e)\mid e\in\Expr}$
is a proper class in ZFC (Theorem~\ref{thm:main}).
Two independent proofs were given:
\begin{enumerate}[label=(\roman*)]
\item A \emph{Cantor diagonal argument}: set-hood of $\Ueml$ implies an
injection $\PP(\Ueml)\hookrightarrow\Ueml$,
contradicting Cantor’s Theorem.
\item A \emph{von Neumann rank argument}: elements of $\Ueml$ have
arbitrarily large ordinal rank, so no set can contain $\Ueml$.
\end{enumerate}
In G"{o}del–Bernays set theory, $\Ueml$ is a definable proper class
(Proposition~\ref{prop:gb-def}), making the two-sorted framework of GB
or MK the minimal classical set-theoretic home for $\Ueml$.
Finally, $\Ueml\subsetneq V$ (Proposition~\ref{prop:strict}), so $\Ueml$
is a proper class strictly smaller than the full universe.
These results complement Odrzywołek’s completeness theorem \cite{Odrzyw2026}:
individual EML expressions are witnesses in a countable syntactic set,
while the totality of EML-reachable objects across all semantic levels
is irreducibly class-sized.
\medskip\noindent
\textbf{Acknowledgements.}
The author thanks the referees and colleagues whose comments improved
this manuscript.
The work of Odrzywołek \cite{Odrzyw2026} is the direct inspiration
for the question studied here.
%%=============================================================
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%%=============================================================
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\end{document}