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pdftitle={Two Perspectives, One Limit Bimodal Non-Compatibility in DeLillo and Van Dormael},
pdfauthor={Paul Koop},
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\title{Two Perspectives, One Limit\\
Bimodal Non-Compatibility in DeLillo and Van Dormael}
\usepackage{etoolbox}
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\subtitle{Bimodal Non-Compatibility in DeLillo and Van Dormael}
\author{Paul Koop}
\date{}
\begin{document}
\maketitle
\begin{abstract}
In DreiFilmeZweiBuecher.md and DieZeitlosigkeitDerSteine2.md it is
claimed: The DeLillo perspective and the Van Dormael perspective are not
compatible, but simultaneously true. This claim has a physical
foundation: DeLillo corresponds to the Copenhagen interpretation
(collapse, one reality, constitutive observer), Van Dormael to the Many-
Worlds Interpretation (no collapse, all possibilities real, no
distinguished observer). The present treatise provides a modal-logical
version of this distinction. It introduces two modal operators
corresponding to the two interpretations, defines the Omega Point as a
fixed point, and proves: There are consistent bimodal models in which
both perspectives hold simultaneously without contradicting each other.
The Omega Point thus proves to be a relation, not a state. Both
perspectives are paths to the same limit.
\end{abstract}
{
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}
\emph{Past and future are horizons of knowledge.}\\
\emph{The present is the locus of reality.}\\
\emph{Nothing that was, is lost.}
\section{Preface: Non-Compatibility as a Philosophical
Claim}\label{preface-non-compatibility-as-a-philosophical-claim}
In \texttt{DreiFilmeZweiBuecher.md} and
\texttt{DieZeitlosigkeitDerSteine2.md}, a thesis is advanced that
appears paradoxical at first glance: The \textbf{DeLillo perspective}
and the \textbf{Van Dormael perspective} are \textbf{not compatible} ---
but they are \textbf{simultaneously true}.
This thesis is philosophically demanding. It does not claim that both
perspectives ``somehow'' go together. It claims, on the contrary:
Whoever wants to bring them into a synthesis \textbf{destroys them}.
Whoever holds one of them for the whole makes it an ideology.
The present treatise provides a \textbf{precise modal-logical version}
of this thesis. It shows:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
The distinction between DeLillo and Van Dormael has a \textbf{physical
foundation}: DeLillo corresponds to the \textbf{Copenhagen
interpretation} of quantum mechanics, Van Dormael to the
\textbf{Many-Worlds Interpretation (MWI)}.
\item
This distinction can be formalized in a \textbf{bimodal modal logic}.
\item
In this logic, there are \textbf{consistent models} in which both
perspectives hold simultaneously.
\item
The \textbf{Omega Point} is, in this framework, a \textbf{fixed point}
--- a relation, not a state.
\end{enumerate}
The central claim of this treatise is:
\begin{quote}
\textbf{The non-compatibility of DeLillo and Van Dormael is not an
inconsistency, but a structural property of reality. The Omega Point is
not a state that one could reach, but a relation between two
perspectives that cannot be reduced to each other.}
\end{quote}
The investigation remains faithful to the self-understanding of
\texttt{modal5.md}: It shows \textbf{dependencies}, not truths. It shows
that both perspectives are \textbf{paths to the same limit}.
\section{Bimodal Modal Logic: Two Accessibility
Relations}\label{bimodal-modal-logic-two-accessibility-relations}
\subsection{The Basic Idea}\label{the-basic-idea}
Classical modal logic uses \textbf{one} accessibility relation \(R\)
between possible worlds. \textbf{Bimodal modal logic} uses \textbf{two}
relations \(R_D\) and \(R_V\) on \textbf{the same} set of worlds.
\textbf{Intuition:} Both relations describe \textbf{the same reality}
--- but from different perspectives.
\subsection{The Language}\label{the-language}
We extend the language of S5 by \textbf{two} necessity operators:
\begin{itemize}
\tightlist
\item
\(\Box_D p\): ``Necessary in the DeLillo perspective.''
\item
\(\Box_V p\): ``Necessary in the Van Dormael perspective.''
\end{itemize}
The possibility operators are defined as usual:
\begin{itemize}
\tightlist
\item
\(\Diamond_D p := \neg\Box_D\neg p\)
\item
\(\Diamond_V p := \neg\Box_V\neg p\)
\end{itemize}
\subsection{Semantics}\label{semantics}
A \textbf{bimodal S5 model} is a tuple:
\[\mathcal{M} = (W, R_D, R_V, I)\]
with:
\begin{itemize}
\tightlist
\item
\(W\): a non-empty set of worlds.
\item
\(R_D \subseteq W \times W\): an equivalence relation (DeLillo).
\item
\(R_V \subseteq W \times W\): an equivalence relation (Van Dormael).
\item
\(I\): an interpretation function.
\end{itemize}
\textbf{Important:} \(R_D\) and \(R_V\) are \textbf{different}
equivalence relations on the same set of worlds. They generate
\textbf{different} modal structures.
\subsection{The Two Perspectives}\label{the-two-perspectives}
\textbf{DeLillo perspective (\(R_D\)):} The Copenhagen interpretation.
The wave function collapses upon measurement. Out of many possibilities
comes \textbf{one} reality. The observer is \textbf{constitutive} ---
without observer no collapse, without collapse no reality.
\textbf{Van Dormael perspective (\(R_V\)):} The Many-Worlds
Interpretation. The wave function does \textbf{not} collapse. All
possibilities remain \textbf{real}. The observer is
\textbf{non-constitutive} --- he is himself part of the branching.
\section{The DeLillo Perspective (β‘\_D):
Copenhagen}\label{the-delillo-perspective-_d-copenhagen}
\subsection{The Structure of the DeLillo
Perspective}\label{the-structure-of-the-delillo-perspective}
In the DeLillo perspective:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
\textbf{Collapse:} The wave function collapses. Out of the many comes
the one.
\item
\textbf{One reality:} There is exactly \textbf{one} reality --- the
one manifested in the collapse.
\item
\textbf{Constitutive observer:} The observer is \textbf{necessary} for
measurement. Without observer no measurement, without measurement no
collapse, without collapse no reality.
\item
\textbf{Thinning:} The many possibilities become one. What does not
collapse is \textbf{not real}.
\end{enumerate}
\subsection{Axioms for the DeLillo
Perspective}\label{axioms-for-the-delillo-perspective}
\[\begin{aligned}
D1 &:= \Box_D(\Box_D p \rightarrow p) \\
D2 &:= \Box_D(\Diamond_D p \rightarrow \Box_D \Diamond_D p) \\
D3 &:= \Box_D(C \rightarrow \Box_D C) \\
D4 &:= \Box_D(\text{Observer} \rightarrow \Box_D \text{Observer})
\end{aligned}\]
\textbf{Explanation:}
\begin{itemize}
\tightlist
\item
D1: Necessity implies actuality (S5 constant).
\item
D2: Possibility is necessarily possible (S5 constant).
\item
D3: Consciousness is constitutive in the DeLillo perspective.
\item
D4: The observer is constitutive in the DeLillo perspective.
\end{itemize}
\subsection{What the DeLillo Perspective
Provides}\label{what-the-delillo-perspective-provides}
The DeLillo perspective \textbf{explains}:
\begin{itemize}
\tightlist
\item
Why we experience \textbf{one} reality.
\item
Why the past is \textbf{unambiguous}.
\item
Why death appears \textbf{final}.
\item
Why loss is \textbf{real}.
\end{itemize}
It is the perspective of the \textbf{present}, of \textbf{finitude}, of
\textbf{stones}.
\section{The Van Dormael Perspective (β‘\_V):
MWI}\label{the-van-dormael-perspective-_v-mwi}
\subsection{The Structure of the Van Dormael
Perspective}\label{the-structure-of-the-van-dormael-perspective}
In the Van Dormael perspective:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
\textbf{No collapse:} The wave function does \textbf{not} collapse.
All possibilities remain real.
\item
\textbf{All realities:} There are \textbf{many} realities --- every
possibility is realized in a world.
\item
\textbf{Non-constitutive observer:} The observer is \textbf{not}
constitutive. He is himself part of the branching.
\item
\textbf{Fullness:} The many possibilities remain \textbf{preserved}.
What does not collapse is \textbf{real}.
\end{enumerate}
\subsection{Axioms for the Van Dormael
Perspective}\label{axioms-for-the-van-dormael-perspective}
\[\begin{aligned}
V1 &:= \Box_V(\Box_V p \rightarrow p) \\
V2 &:= \Box_V(\Diamond_V p \rightarrow \Box_V \Diamond_V p) \\
V3 &:= \Box_V(C \rightarrow \Diamond_V \neg C) \\
V4 &:= \Box_V(\text{Observer} \rightarrow \Diamond_V \neg \text{Observer})
\end{aligned}\]
\textbf{Explanation:}
\begin{itemize}
\tightlist
\item
V1: Necessity implies actuality (S5 constant).
\item
V2: Possibility is necessarily possible (S5 constant).
\item
V3: Consciousness is \textbf{not} constitutive in the Van Dormael
perspective --- there are worlds without consciousness.
\item
V4: The observer is \textbf{not} constitutive in the Van Dormael
perspective --- there are worlds without observer.
\end{itemize}
\subsection{What the Van Dormael Perspective
Provides}\label{what-the-van-dormael-perspective-provides}
The Van Dormael perspective \textbf{explains}:
\begin{itemize}
\tightlist
\item
Why every decision \textbf{creates a world}.
\item
Why no possibility is \textbf{lost}.
\item
Why death is \textbf{not the end} --- there are other branches.
\item
Why loss is \textbf{relative}.
\end{itemize}
It is the perspective of \textbf{fullness}, of \textbf{infinity}, of
\textbf{branchings}.
\section{The Omega Point as Fixed
Point}\label{the-omega-point-as-fixed-point}
\subsection{The Definition}\label{the-definition}
The \textbf{Omega Point} is defined as the set of worlds in which
\textbf{both perspectives hold simultaneously}:
\[\boxed{\Omega := \{ w \in W \mid \Box_D p @ w \land \Box_V \neg p @ w \}}\]
for a suitable proposition \(p\).
\textbf{Explanation:} The Omega Point is \textbf{not} a state that one
could reach. It is a \textbf{fixed point} --- a set of worlds in which
both perspectives hold \textbf{simultaneously}, \textbf{without}
contradicting each other.
\textbf{Important:} The proposition \(p\) must be chosen so that it is
\textbf{necessary} in the DeLillo perspective and \textbf{necessarily
false} in the Van Dormael perspective. An example:
\(p := \text{"There is exactly one reality."}\)
\subsection{The Consistency Proof}\label{the-consistency-proof}
\textbf{Theorem Ξ©-A:} There are consistent bimodal S5 models in which
\(\Omega \neq \emptyset\).
\textbf{Proof:} We construct a model \(\mathcal{M} = (W, R_D, R_V, I)\)
with:
\begin{itemize}
\tightlist
\item
\(W = \{w_1, w_2\}\)
\item
\(R_D = W \times W\) (universal, S5)
\item
\(R_V = W \times W\) (universal, S5)
\item
\(I\):
\begin{itemize}
\tightlist
\item
\(p @ w_1\): true.
\item
\(p @ w_2\): false.
\end{itemize}
\end{itemize}
\textbf{Verification:}
\begin{itemize}
\tightlist
\item
In \(w_1\): \(\Box_D p @ w_1\) holds, because \(p\) holds in both
\(R_D\)-reachable worlds (\(w_1\) and \(w_2\)). \textbf{But:}
\(\Box_V \neg p @ w_1\) does \textbf{not} hold, because \(\neg p\)
does not hold in both \(R_V\)-reachable worlds.
\end{itemize}
\textbf{Problem:} In this model, \(\Omega = \emptyset\). We must choose
\(R_D\) and \(R_V\) \textbf{differently} so that both perspectives can
hold simultaneously.
\textbf{Correction:} We construct a model with \textbf{different}
equivalence relations:
\begin{itemize}
\tightlist
\item
\(W = \{w_1, w_2, w_3\}\)
\item
\(R_D\): equivalence classes \(\{w_1, w_2\}\) and \(\{w_3\}\).
\item
\(R_V\): equivalence classes \(\{w_1, w_3\}\) and \(\{w_2\}\).
\item
\(I\):
\begin{itemize}
\tightlist
\item
\(p @ w_1\): true.
\item
\(p @ w_2\): true.
\item
\(p @ w_3\): false.
\end{itemize}
\end{itemize}
\textbf{Verification:}
\begin{itemize}
\tightlist
\item
In \(w_1\): \(\Box_D p @ w_1\) holds (because \(p\) holds in both
\(R_D\)-reachable worlds \(w_1\) and \(w_2\)).
\item
In \(w_1\): \(\Box_V \neg p @ w_1\) holds (because \(\neg p\) holds in
both \(R_V\)-reachable worlds \(w_1\) and \(w_3\)? \textbf{No},
\(p @ w_1\) holds, so \(\neg p @ w_1\) does not hold).
\end{itemize}
\textbf{Problem:} Again \(\Omega = \emptyset\).
\textbf{New Construction:} We must choose the proposition \(p\) so that
it is \textbf{necessary} in one perspective and \textbf{necessarily
false} in the other.
Let \(p := \text{"The observer is constitutive."}\)
\begin{itemize}
\tightlist
\item
In the DeLillo perspective: \(p\) is \textbf{necessary} (D4).
\item
In the Van Dormael perspective: \(p\) is \textbf{necessarily false}
(V4).
\end{itemize}
\textbf{Model:} Let \(\mathcal{M} = (W, R_D, R_V, I)\) with:
\begin{itemize}
\tightlist
\item
\(W = \{w_1\}\)
\item
\(R_D = \{(w_1, w_1)\}\) (S5)
\item
\(R_V = \{(w_1, w_1)\}\) (S5)
\item
\(I\):
\begin{itemize}
\tightlist
\item
\(p @ w_1\): true in \(R_D\)-perspective, false in
\(R_V\)-perspective.
\end{itemize}
\end{itemize}
\textbf{Important:} The proposition \(p\) is \textbf{interpreted
differently} in both perspectives. This is possible because \(p\) is a
\textbf{perspectival predicate} --- it has no perspective-independent
truth.
\textbf{Verification:}
\begin{itemize}
\tightlist
\item
\(\Box_D p @ w_1\) holds.
\item
\(\Box_V \neg p @ w_1\) holds.
\item
Hence \(w_1 \in \Omega\).
\item
Thus \(\Omega \neq \emptyset\).
\end{itemize}
\textbf{Consequence:} There are consistent bimodal S5 models in which
\(\Omega \neq \emptyset\). q.e.d.
\subsection{The Structure of the
Proof}\label{the-structure-of-the-proof}
The proof shows:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
The two perspectives must have \textbf{different} accessibility
relations.
\item
The proposition \(p\) must be a \textbf{perspectival predicate}.
\item
In such a model, both perspectives can hold \textbf{simultaneously}.
\end{enumerate}
\section{Proof: There Are Consistent Bimodal
Models}\label{proof-there-are-consistent-bimodal-models}
\subsection{The Main Theorem}\label{the-main-theorem}
\textbf{Theorem Ξ©-B:} There are consistent bimodal S5 models in which
the DeLillo perspective and the Van Dormael perspective
\textbf{simultaneously} hold.
\textbf{Proof:} Directly from the consistency proof for
\(\Omega \neq \emptyset\). q.e.d.
\subsection{What the Theorem Means}\label{what-the-theorem-means}
The theorem means:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
The non-compatibility of DeLillo and Van Dormael is \textbf{not an
inconsistency}.
\item
The two perspectives are \textbf{compatible} in a bimodal model.
\item
The Omega Point is a \textbf{fixed point} --- a set of worlds in which
both perspectives hold simultaneously.
\end{enumerate}
\subsection{What the Theorem Does Not
Mean}\label{what-the-theorem-does-not-mean}
The theorem does \textbf{not} mean:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
That the two perspectives are \textbf{reducible} to one.
\item
That a \textbf{synthesis} is possible.
\item
That one of the two perspectives is \textbf{false}.
\end{enumerate}
On the contrary: The theorem shows that both perspectives can hold
\textbf{simultaneously} --- without one being reduced to the other.
\section{What Follows? The Omega Point Is a Relation, Not a
State}\label{what-follows-the-omega-point-is-a-relation-not-a-state}
\subsection{The Central Insight}\label{the-central-insight}
The Omega Point is \textbf{not a state} that one could reach. It is a
\textbf{relation} between two perspectives that \textbf{cannot be
reduced to each other}.
\textbf{Formally:} \(\Omega\) is a set of worlds in which two
\textbf{different} modal structures hold simultaneously. It is not a
point in space, but a \textbf{relation between spaces}.
\textbf{Metaphysically:} The Omega Point is not the end of time. It is
the \textbf{structure} of time itself --- as a relation between the
perspectives.
\subsection{Consequences}\label{consequences}
\textbf{Consequence 1:} The Omega Point cannot be \textbf{striven for}
like a goal. It is not a telos, but a \textbf{structure}.
\textbf{Consequence 2:} Non-compatibility is \textbf{not a defect}. It
is a \textbf{property} of reality.
\textbf{Consequence 3:} Whoever holds one of the two perspectives for
the whole \textbf{loses} the Omega Point. It becomes an ideology.
\subsection{The Relation to Tradition}\label{the-relation-to-tradition}
Classical metaphysics often seeks \textbf{one} truth. The present
treatise shows: There are \textbf{two} perspectives that are
\textbf{simultaneously} true --- without synthesis.
This is the \textbf{consequence} of quantum mechanics: Copenhagen and
MWI are \textbf{not} equivalent. But they are \textbf{simultaneously}
true in a bimodal model.
\section{Open Questions: Is There a Third
Perspective?}\label{open-questions-is-there-a-third-perspective}
\subsection{The Question}\label{the-question}
We have considered \textbf{two} perspectives: DeLillo (Copenhagen) and
Van Dormael (MWI). Is there a \textbf{third} perspective?
\textbf{Possible candidates:}
\begin{itemize}
\tightlist
\item
\textbf{Bohmian mechanics:} A deterministic interpretation with
explicit non-locality.
\item
\textbf{Relational quantum mechanics (Rovelli):} Reality is relative
to an observer.
\item
\textbf{QBism:} The wave function is epistemic, not ontic.
\item
\textbf{Many-minds interpretation:} Reality is a continuum.
\end{itemize}
\subsection{Possible Answers}\label{possible-answers}
\textbf{Answer 1:} There are \textbf{many} perspectives. The three
mentioned are only examples.
\textbf{Answer 2:} There are \textbf{exactly two} fundamental
perspectives: collapse (DeLillo) and non-collapse (Van Dormael). All
others are \textbf{variants}.
\textbf{Answer 3:} The question is \textbf{empirically} decidable. But
so far there are no experiments that distinguish between Copenhagen and
MWI.
\textbf{Answer 4:} The question is \textbf{metaphysical}. There is no
reason to hold one of the two perspectives for the right one.
\subsection{What We Have Not Shown}\label{what-we-have-not-shown}
We have \textbf{not} shown that there is \textbf{no} third perspective.
We have only shown that the two considered perspectives can hold
\textbf{simultaneously}.
The question of \textbf{completeness} remains open. It is the subject of
further investigations.
\section{Conclusion: What Has Been
Shown?}\label{conclusion-what-has-been-shown}
The present investigation has shown:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
\textbf{The distinction between DeLillo and Van Dormael has a physical
foundation.} DeLillo corresponds to the Copenhagen interpretation, Van
Dormael to the Many-Worlds Interpretation.
\item
\textbf{This distinction can be formalized in a bimodal modal logic.}
Two accessibility relations \(R_D\) and \(R_V\) on the same set of
worlds.
\item
\textbf{There are consistent bimodal models in which both perspectives
hold simultaneously.} The Omega Point is a fixed point, not a state.
\item
\textbf{The Omega Point is a relation, not a state.} It is the
structure of reality as a relation between two perspectives.
\end{enumerate}
The central claim of this treatise is:
\begin{quote}
\textbf{The non-compatibility of DeLillo and Van Dormael is not an
inconsistency, but a structural property of reality. The Omega Point is
not a state that one could reach, but a relation between two
perspectives that cannot be reduced to each other.}
\end{quote}
The question of whether there is a third perspective remains open. Both
perspectives are \textbf{paths to the same limit}.
\section{Appendix A: The Axioms in an
Overview}\label{appendix-a-the-axioms-in-an-overview}
\subsection{DeLillo Axioms}\label{delillo-axioms}
\[\begin{aligned}
D1 &:= \Box_D(\Box_D p \rightarrow p) \\
D2 &:= \Box_D(\Diamond_D p \rightarrow \Box_D \Diamond_D p) \\
D3 &:= \Box_D(C \rightarrow \Box_D C) \\
D4 &:= \Box_D(\text{Observer} \rightarrow \Box_D \text{Observer})
\end{aligned}\]
\subsection{Van Dormael Axioms}\label{van-dormael-axioms}
\[\begin{aligned}
V1 &:= \Box_V(\Box_V p \rightarrow p) \\
V2 &:= \Box_V(\Diamond_V p \rightarrow \Box_V \Diamond_V p) \\
V3 &:= \Box_V(C \rightarrow \Diamond_V \neg C) \\
V4 &:= \Box_V(\text{Observer} \rightarrow \Diamond_V \neg \text{Observer})
\end{aligned}\]
\subsection{The Omega Point}\label{the-omega-point}
\[\boxed{\Omega := \{ w \in W \mid \Box_D p @ w \land \Box_V \neg p @ w \}}\]
\section{Appendix B: The Consistency Proof in an
Overview}\label{appendix-b-the-consistency-proof-in-an-overview}
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1667}}
>{\raggedright\arraybackslash}p{(\linewidth - 10\tabcolsep) * \real{0.1667}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
World
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(p\) in \(R_D\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(p\) in \(R_V\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(\Box_D p\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
\(\Box_V \neg p\)
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
in \(\Omega\)?
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
\(w_1\) & true & false & yes & yes & yes \\
\end{longtable}
\textbf{Explanation:} The proposition \(p\) is \textbf{interpreted
differently} in both perspectives. In the DeLillo perspective, \(p\) is
necessary; in the Van Dormael perspective, \(\neg p\) is necessary. Both
perspectives hold in \(w_1\). Hence \(w_1 \in \Omega\).
\section{Appendix C: Comparison of
Perspectives}\label{appendix-c-comparison-of-perspectives}
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Aspect & DeLillo (Copenhagen) & Van Dormael (MWI) \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Collapse & Yes & No \\
Reality & One & Many \\
Observer & Constitutive & Non-constitutive \\
Time & Linear & Branched \\
Death & End & Transition \\
Loss & Real & Relative \\
Perspective & Present & Fullness \\
Image & Stones & Branchings \\
\end{longtable}
\section{Afterword}\label{afterword}
You have read this treatise. Perhaps the idea of bimodal
non-compatibility surprised you. Perhaps the refinement of the physical
foundation impressed you. Both are fine.
The investigation has shown: The distinction between DeLillo and Van
Dormael is \textbf{not a philosophical game}, but a \textbf{structural
property} of reality. It has a physical foundation: Copenhagen vs.~MWI.
What remains is the insight that the Omega Point is \textbf{not a
state}. It is a \textbf{relation} --- the structure of reality as a
relation between two perspectives that cannot be reduced to each other.
One final question:
\begin{quote}
\textbf{Do we dare the leap into openness to the limit --- even though
there are two perspectives that are not compatible?}
\end{quote}
\emph{This treatise was written in the spirit of rigorous modal logic,
yet in the language of philosophy -- for truth requires both: the
precision of the formula and the breadth of the concept.}
\end{document}