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\label{thm:four-paradox-diagnostic}
The following four paradoxes are instances of
$\Elayer{2} \to \Elayer{3}$ boundary crossings
(Definition~\ref{def:e2-e3-boundary-crossing}):
\begin{enumerate}
    \item \textbf{Cantor's Diagonal.}
          The ``set of all sets'' is an $\Elayer{2}$ code space
          whose cardinality is host-level.
          The diagonal forces self-modelling of size.
    \item \textbf{Russell's Paradox.}
          $R = \{x : x \notin x\}$
          is a self-referential $\Elayer{2}$ code
          whose self-membership is host-level.
          The definition forces self-modelling of the predicate.
    \item \textbf{G\"odel's Incompleteness.}
          The sentence~$G$
          is a self-referential $\Elayer{2}$ code
          whose derivability is host-level.
          The encoding forces self-modelling of inference.
    \item \textbf{Turing's Halting Problem.}
          The diagonaliser~$D$
          is a self-referential $\Elayer{2}$ code
          whose halting is host-level.
          The diagonalisation forces self-modelling of execution.
\end{enumerate}
In each case, self-reference ($\Elayer{2}$) suffices
to \emph{pose} the question
but not to \emph{answer} it.
The answer requires self-modelling ($\Elayer{3}$).
The paradox is the error message generated by the boundary.