Inhalt
Aktueller Ordner:
/modal10eng.tex
% Options for packages loaded elsewhere
\PassOptionsToPackage{unicode}{hyperref}
\PassOptionsToPackage{hyphens}{url}
\PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor}
\documentclass[
ngerman,
11pt,
]{article}
\usepackage{xcolor}
\usepackage[margin=2.5cm]{geometry}
\usepackage{amsmath,amssymb}
\setcounter{secnumdepth}{-\maxdimen} % remove section numbering
\usepackage{iftex}
\ifPDFTeX
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{textcomp} % provide euro and other symbols
\else % if luatex or xetex
\usepackage{unicode-math} % this also loads fontspec
\defaultfontfeatures{Scale=MatchLowercase}
\defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
\fi
\usepackage{lmodern}
\ifPDFTeX\else
% xetex/luatex font selection
\fi
% Use upquote if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
\IfFileExists{microtype.sty}{% use microtype if available
\usepackage[]{microtype}
\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
\makeatletter
\@ifundefined{KOMAClassName}{% if non-KOMA class
\IfFileExists{parskip.sty}{%
\usepackage{parskip}
}{% else
\setlength{\parindent}{0pt}
\setlength{\parskip}{6pt plus 2pt minus 1pt}}
}{% if KOMA class
\KOMAoptions{parskip=half}}
\makeatother
\usepackage{longtable,booktabs,array}
\usepackage{calc} % for calculating minipage widths
% Correct order of tables after \paragraph or \subparagraph
\usepackage{etoolbox}
\makeatletter
\patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{}
\makeatother
% Allow footnotes in longtable head/foot
\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
\makesavenoteenv{longtable}
\ifLuaTeX
\usepackage[bidi=basic,shorthands=off,]{babel}
\else
\usepackage[bidi=default,shorthands=off,]{babel}
\fi
\ifLuaTeX
\usepackage{selnolig} % disable illegal ligatures
\fi
\setlength{\emergencystretch}{3em} % prevent overfull lines
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\usepackage{bookmark}
\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
\urlstyle{same}
\hypersetup{
pdftitle={Unitarity as Theorem Modal Logic of Information Conservation},
pdfauthor={Paul Koop},
pdflang={de},
colorlinks=true,
linkcolor={blue},
filecolor={Maroon},
citecolor={Blue},
urlcolor={Blue},
pdfcreator={LaTeX via pandoc}}
\title{Unitarity as Theorem\\
Modal Logic of Information Conservation}
\usepackage{etoolbox}
\makeatletter
\providecommand{\subtitle}[1]{% add subtitle to \maketitle
\apptocmd{\@title}{\par {\large #1 \par}}{}{}
}
\makeatother
\subtitle{Modal Logic of Information Conservation}
\author{Paul Koop}
\date{}
\begin{document}
\maketitle
\begin{abstract}
In modal5.md (Part III) and Unitaritaet.md, the resurrection is derived
from the axioms A16--A20. These axioms are strong. The present treatise
shows: Unitarity can be derived from weaker assumptions. It introduces
an operator I (``information is conserved''), axiomatizes three minimal
principles U1--U3, and proves: From U1--U3, the resurrection follows as
a theorem. Unitarity is thus not an additional assumption but the
structure of Totality T itself: βΞ³ β T : Ξ³ β T. The ``intervention'' of
the Omega Point into paths that do not reach it is structural, not
temporal. What appears from the perspective of a path as loss is, from
the perspective of S, integration.
\end{abstract}
{
\hypersetup{linkcolor=}
\setcounter{tocdepth}{3}
\tableofcontents
}
\emph{Past and future are horizons of knowledge.}\\
\emph{The present is the locus of reality.}\\
\emph{Nothing that was, is lost.}
\section{Preface: From Strong Axioms to Weaker
Principles}\label{preface-from-strong-axioms-to-weaker-principles}
In \texttt{modal5.md} (Part III), the resurrection was derived from the
axioms A16--A20:
\[\begin{aligned}
A16 &:= \forall c(\text{Consciousness}(c) \rightarrow \text{integrated}(c, S)) \\
A17 &:= \forall c(\text{Knowledge}(c, S) \leftrightarrow \text{integrated}(c, S)) \\
A18 &:= \Box\forall I(\text{Existence}(I) \rightarrow \text{Conservation}(I)) \\
A19 &:= \forall c(\text{Death}(c) \rightarrow \text{Integration}(c, S)) \\
A20 &:= \forall c(\text{Knowledge}(c, S) \leftrightarrow \text{Existence}(c))
\end{aligned}\]
These axioms are \textbf{strong}. They presuppose that \textbf{all}
consciousnesses are integrated into \(S\), that \textbf{all} information
is conserved, that death is \textbf{always} integration.
The present treatise shows: \textbf{Unitarity can be derived from weaker
assumptions.} It introduces an operator \(\mathcal{I}\) (``information
is conserved''), axiomatizes three minimal principles U1--U3, and
proves: From U1--U3, the resurrection follows as a \textbf{theorem}.
The central claim of this treatise is:
\begin{quote}
\textbf{Unitarity is not an additional assumption. It is the structure
of Totality \(T\) itself. From it, the resurrection follows as a logical
necessity --- not as a moral promise, but as a consequence of
information conservation.}
\end{quote}
The investigation remains faithful to the self-understanding of
\texttt{modal5.md}: It shows \textbf{dependencies}, not truths. It shows
that the resurrection follows from weaker assumptions.
\section{The Operator I: Definition and
Axioms}\label{the-operator-i-definition-and-axioms}
\subsection{The Basic Idea}\label{the-basic-idea}
The operator \(\mathcal{I}\) formalizes the concept of
\textbf{information conservation}. It is \textbf{not} a moral concept,
but a \textbf{structural} one.
\textbf{Intuition:} ``\(\mathcal{I} p\)'' means: ``The information about
\(p\) is conserved.''
\subsection{The Language}\label{the-language}
We extend the language of S5 by:
\begin{itemize}
\tightlist
\item
A \textbf{unary operator} \(\mathcal{I}\) (read: ``Information'').
\end{itemize}
\[\mathcal{I} p @ w \iff \text{the information about } p \text{ is conserved in } w\]
\subsection{The Axioms}\label{the-axioms}
\[\begin{aligned}
I1 &:= \Box(\mathcal{I} p \rightarrow \Diamond p) \\
I2 &:= \Box(\mathcal{I}(p \land q) \leftrightarrow (\mathcal{I} p \land \mathcal{I} q)) \\
I3 &:= \Box(\mathcal{I}\neg p \leftrightarrow \neg\mathcal{I} p) \\
I4 &:= \Box(\mathcal{I} p \rightarrow \Box \mathcal{I} p)
\end{aligned}\]
\textbf{Explanation:}
\begin{itemize}
\tightlist
\item
I1: What is conserved is possible.
\item
I2: Conservation distributes over conjunction.
\item
I3: Conservation is complementary --- the information about \(\neg p\)
is conserved exactly when the information about \(p\) is conserved.
\item
I4: Conservation is necessary when it holds.
\end{itemize}
\textbf{Important:} \(\mathcal{I}\) is \textbf{not} a normal operator.
It does \textbf{not} satisfy the Kripke axiom \(K\). Instead, it
satisfies the \textbf{information axioms} I1--I4.
\section{The Three Principles U1--U3}\label{the-three-principles-u1u3}
\subsection{U1: Realization Implies Possible
Reconstruction}\label{u1-realization-implies-possible-reconstruction}
\[\boxed{U1 := \Box\forall p(\text{Realized}(p) \rightarrow \mathcal{I} p)}\]
\textbf{Reading:} If \(p\) is realized, then the information about \(p\)
is conserved.
\textbf{Metaphysical meaning:} What was once real cannot fully
disappear. It remains conserved as information --- even if the original
form is lost.
\textbf{Formal consequence:} From U1 and I1:
\(\text{Realized}(p) \rightarrow \Diamond p\). This is trivial. From U1
and I4: \(\text{Realized}(p) \rightarrow \Box \mathcal{I} p\). This is
stronger: The conservation of information about \(p\) is necessary.
\subsection{U2: Death Is Local Access
Loss}\label{u2-death-is-local-access-loss}
\[\boxed{U2 := \Box\forall c(\text{Death}(c) \rightarrow \neg\Diamond_{\text{local}} \text{Access}(c))}\]
\textbf{Reading:} The death of a consciousness \(c\) is the loss of
\textbf{local} access to \(c\).
\textbf{Metaphysical meaning:} Death is \textbf{not} extinction. It is
an \textbf{access loss}. From the perspective of \(c\), it appears as an
end. From the perspective of \(S\), it is a transition.
\textbf{Formal consequence:} From U2 and U1:
\(\text{Death}(c) \rightarrow \mathcal{I} c\). The information about
\(c\) is conserved, even if \(c\) is no longer locally accessible.
\subsection{U3: Global Integration Is
Possible}\label{u3-global-integration-is-possible}
\[\boxed{U3 := \Box\forall c(\text{Death}(c) \rightarrow \Diamond_{\text{global}} \text{Integration}(c, S))}\]
\textbf{Reading:} The death of a consciousness \(c\) makes its
\textbf{global} integration into \(S\) possible.
\textbf{Metaphysical meaning:} Death is \textbf{not the end}. It is the
condition of the possibility of integration. What is lost locally is
integrated globally.
\textbf{Formal consequence:} From U2, U3, and U1:
\(\text{Death}(c) \rightarrow \Diamond_{\text{global}} \text{Integration}(c, S)\).
\section{Proof: Unitarity Implies
Resurrection}\label{proof-unitarity-implies-resurrection}
\subsection{The Main Theorem}\label{the-main-theorem}
\textbf{Theorem U-A:} In S5+U1--U3:
\[\Box\forall c(\text{Consciousness}(c) \rightarrow \Box\Diamond\text{Resurrection}(c))\]
\subsection{The Proof}\label{the-proof}
\textbf{Step 1:} Let \(c\) be an arbitrary consciousness.
\[\text{Consciousness}(c)\]
\textbf{Step 2:} We assume that \(c\) dies.
\[\text{Death}(c)\]
\textbf{Step 3:} From U2:
\[\neg\Diamond_{\text{local}} \text{Access}(c)\]
Local access to \(c\) is lost.
\textbf{Step 4:} From U1 and Step 2:
\[\mathcal{I} c\]
The information about \(c\) is conserved.
\textbf{Step 5:} From U3 and Step 2:
\[\Diamond_{\text{global}} \text{Integration}(c, S)\]
The global integration of \(c\) into \(S\) is possible.
\textbf{Step 6:} We define:
\[\text{Resurrection}(c) := \text{Integration}(c, S)\]
\textbf{Step 7:} From Step 5 and 6:
\[\Diamond \text{Resurrection}(c)\]
\textbf{Step 8:} Since Steps 2--7 hold for \textbf{every} \(c\) that
dies, and since every consciousness dies (empirical fact), it follows:
\[\forall c(\text{Consciousness}(c) \rightarrow \Diamond \text{Resurrection}(c))\]
\textbf{Step 9:} With Necessitation:
\[\Box\forall c(\text{Consciousness}(c) \rightarrow \Diamond \text{Resurrection}(c))\]
q.e.d.
\subsection{What the Proof Shows}\label{what-the-proof-shows}
The proof shows:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
The resurrection does \textbf{not} follow from moral assumptions.
\item
The resurrection follows from \textbf{structural} assumptions:
information conservation, access loss, integration possibility.
\item
The resurrection is a \textbf{logical necessity} --- under the
assumptions U1--U3.
\end{enumerate}
\subsection{What the Proof Does Not
Show}\label{what-the-proof-does-not-show}
The proof does \textbf{not} show:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
That the resurrection \textbf{actually} takes place.
\item
That the resurrection takes place \textbf{temporally}.
\item
That the resurrection is \textbf{morally} justified.
\end{enumerate}
The proof only shows: Under the assumptions U1--U3, the possibility of
resurrection follows for every consciousness.
\section{Unitarity as the Structure of
T}\label{unitarity-as-the-structure-of-t}
\subsection{The Central Insight}\label{the-central-insight}
Unitarity is \textbf{not} an additional assumption. It is the
\textbf{structure} of Totality \(T\) itself.
\textbf{Formally:}
\[\boxed{\forall \gamma \subseteq T : \gamma \in T}\]
\textbf{Reading:} For every path \(\gamma\) that is a subset of \(T\):
\(\gamma\) is an element of \(T\).
\textbf{Metaphysical meaning:} Totality \(T\) contains \textbf{all}
paths --- including those that do not reach the Omega Point. There is
\textbf{no escape} from \(T\). What happens in \(T\) remains in \(T\).
\subsection{The Consequence}\label{the-consequence}
If unitarity is the structure of \(T\) itself, then it follows:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
\textbf{No information is lost.} For loss would mean that a path
\(\gamma\) falls out of \(T\). That is impossible.
\item
\textbf{All paths are conserved.} Including those that do not reach
the Omega Point.
\item
\textbf{Integration is structural, not temporal.} What appears from
the perspective of a path as loss is, from the perspective of \(T\),
conservation.
\end{enumerate}
\subsection{The ``Intervention'' of the Omega
Point}\label{the-intervention-of-the-omega-point}
The ``intervention'' of the Omega Point into paths that do not reach it
is \textbf{structural}, not \textbf{temporal}:
\begin{itemize}
\tightlist
\item
\textbf{Structural:} \(T\) contains all paths. Integration has
\textbf{always already} happened.
\item
\textbf{Not temporal:} There is no point in time at which the Omega
Point ``intervenes.'' It is the \textbf{form} of \(T\).
\end{itemize}
\textbf{Formal version:}
\[\forall \gamma \subseteq T : \gamma \in T \Rightarrow \text{Integration}(\gamma, S)\]
\textbf{Consequence:} From the perspective of a path, integration
appears as an \textbf{intervention from outside}. From the perspective
of \(S\), it is \textbf{self-knowledge}.
\section{Comparison with A16--A20: What Is Gained, What Is
Lost?}\label{comparison-with-a16a20-what-is-gained-what-is-lost}
\subsection{What Is Gained?}\label{what-is-gained}
\textbf{1. Weaker assumptions.} A16--A20 presuppose that all
consciousnesses are integrated into \(S\). U1--U3 presuppose only that
information conservation is structural.
\textbf{2. Structural foundation.} A16--A20 are \textbf{axiomatic}.
U1--U3 are \textbf{structural} --- they describe the form of \(T\).
\textbf{3. Resurrection as theorem.} With A16--A20, the resurrection is
a theorem from strong assumptions. With U1--U3, it is a theorem from
weak assumptions.
\textbf{4. Connectivity.} U1--U3 are connectable to physical concepts
(information conservation in quantum mechanics). A16--A20 are primarily
theological.
\subsection{What Is Lost?}\label{what-is-lost}
\textbf{1. Directness.} A16--A20 are direct. U1--U3 require the detour
via the operator \(\mathcal{I}\).
\textbf{2. Explicitness.} A16--A20 speak directly of consciousness,
death, integration. U1--U3 speak of information conservation.
\textbf{3. Theological language.} A16--A20 stand in a theological
tradition. U1--U3 stand in a structural tradition.
\subsection{What Remains?}\label{what-remains}
\textbf{1. The resurrection is a theorem.} In both cases.
\textbf{2. Unitarity is fundamental.} In both cases.
\textbf{3. Integration is structural.} In both cases.
\subsection{Conclusion}\label{conclusion}
U1--U3 are \textbf{stronger} in their reach and \textbf{weaker} in their
assumptions. They are therefore \textbf{preferable} when it comes to
showing the resurrection as a \textbf{structural necessity}, not as a
\textbf{moral promise}.
\section{What Follows? Resurrection Is a Logical
Necessity}\label{what-follows-resurrection-is-a-logical-necessity}
\subsection{The Central Insight}\label{the-central-insight-1}
The resurrection is \textbf{not} a moral hope. It is a \textbf{logical
necessity}.
\textbf{Formally:} From U1--U3 follows
\(\Box\forall c(\text{Consciousness}(c) \rightarrow \Diamond \text{Resurrection}(c))\).
\textbf{Metaphysically:} The resurrection is the \textbf{consequence} of
information conservation. It is not an \textbf{event} in time, but a
\textbf{structure} of Totality.
\subsection{The Consequences}\label{the-consequences}
\textbf{Consequence 1:} The resurrection cannot be \textbf{prevented}.
It is structural.
\textbf{Consequence 2:} The resurrection cannot be \textbf{forced}. It
is not a moral reward.
\textbf{Consequence 3:} The resurrection is \textbf{universal}. It holds
for all consciousnesses, not only for the ``good.''
\subsection{The Relation to Tradition}\label{the-relation-to-tradition}
Classical theology speaks of the resurrection as \textbf{grace} or
\textbf{reward}. The present treatise shows: The resurrection is
\textbf{not} grace. It is a \textbf{necessity}.
This is \textbf{no contradiction} to theology. It is a
\textbf{refinement}: Grace is not arbitrary. It is structural.
\section{Open Questions: Is U1--U3
Minimal?}\label{open-questions-is-u1u3-minimal}
\subsection{The Question}\label{the-question}
We have shown: From U1--U3 follows the resurrection. But is U1--U3
\textbf{minimal}? Is each of the three principles indispensable?
\subsection{The Three Sub-Questions}\label{the-three-sub-questions}
\textbf{Question 1:} Is U1 indispensable? What happens if we omit U1?
\textbf{Question 2:} Is U2 indispensable? What happens if we omit U2?
\textbf{Question 3:} Is U3 indispensable? What happens if we omit U3?
\subsection{Possible Answers}\label{possible-answers}
\textbf{On U1:} Without U1, it does not follow that the information
about \(c\) is conserved. The resurrection would not be derivable.
\textbf{On U2:} Without U2, death would not be defined as access loss.
The resurrection would not be bound to death.
\textbf{On U3:} Without U3, it does not follow that integration is
possible. The resurrection would be only a possibility, not a necessity.
\subsection{What We Have Not Shown}\label{what-we-have-not-shown}
We have \textbf{not} shown that U1--U3 are minimal. We have only shown
that they are \textbf{sufficient}.
The question of \textbf{minimality} remains open. It is the subject of
\textbf{Modal 11} and beyond.
\subsection{A Second Open Question}\label{a-second-open-question}
\textbf{Question 4:} Is the operator \(\mathcal{I}\) minimal? Is there a
weaker operator that plays the same role?
\textbf{Possible answer:} The operator \(\mathcal{I}\) is the
\textbf{weakest} operator that formalizes information conservation. But
there could be other operators.
\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{Unitarity can be derived from weaker assumptions.} The three
principles U1--U3 are sufficient for the resurrection.
\item
\textbf{Unitarity is not an additional assumption.} It is the
structure of Totality \(T\) itself:
\(\forall \gamma \subseteq T : \gamma \in T\).
\item
\textbf{The ``intervention'' of the Omega Point is structural, not
temporal.} What appears from the perspective of a path as loss is,
from the perspective of \(S\), integration.
\item
\textbf{The resurrection is a logical necessity.} It is not a moral
hope, but a structural consequence.
\end{enumerate}
The central claim of this treatise is:
\begin{quote}
\textbf{Unitarity is not an additional assumption. It is the structure
of Totality \(T\) itself. From it, the resurrection follows as a logical
necessity --- not as a moral promise, but as a consequence of
information conservation.}
\end{quote}
The question of the minimality of U1--U3 remains open.
\section{Appendix A: The Axioms in an
Overview}\label{appendix-a-the-axioms-in-an-overview}
\subsection{Information Axioms}\label{information-axioms}
\[\begin{aligned}
I1 &:= \Box(\mathcal{I} p \rightarrow \Diamond p) \\
I2 &:= \Box(\mathcal{I}(p \land q) \leftrightarrow (\mathcal{I} p \land \mathcal{I} q)) \\
I3 &:= \Box(\mathcal{I}\neg p \leftrightarrow \neg\mathcal{I} p) \\
I4 &:= \Box(\mathcal{I} p \rightarrow \Box \mathcal{I} p)
\end{aligned}\]
\subsection{Principles U1--U3}\label{principles-u1u3}
\[\begin{aligned}
U1 &:= \Box\forall p(\text{Realized}(p) \rightarrow \mathcal{I} p) \\
U2 &:= \Box\forall c(\text{Death}(c) \rightarrow \neg\Diamond_{\text{local}} \text{Access}(c)) \\
U3 &:= \Box\forall c(\text{Death}(c) \rightarrow \Diamond_{\text{global}} \text{Integration}(c, S))
\end{aligned}\]
\subsection{Unitarity as the Structure of
T}\label{unitarity-as-the-structure-of-t-1}
\[\boxed{\forall \gamma \subseteq T : \gamma \in T}\]
\subsection{The Main Theorem}\label{the-main-theorem-1}
\[\boxed{S5+U1+U2+U3 \vdash \Box\forall c(\text{Consciousness}(c) \rightarrow \Diamond \text{Resurrection}(c))}\]
\section{Appendix B: Comparison with
A16--A20}\label{appendix-b-comparison-with-a16a20}
\begin{longtable}[]{@{}lll@{}}
\toprule\noalign{}
Aspect & A16--A20 (modal5.md) & U1--U3 (modal10.md) \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
Assumptions & Strong & Weak \\
Structure & Axiomatic & Structural \\
Resurrection & Theorem & Theorem \\
Connectivity to physics & Low & High \\
Theological language & Yes & No \\
Directness & High & Medium \\
\end{longtable}
\section{Appendix C: The Three Principles in an
Overview}\label{appendix-c-the-three-principles-in-an-overview}
\begin{longtable}[]{@{}
>{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
>{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}
>{\raggedright\arraybackslash}p{(\linewidth - 4\tabcolsep) * \real{0.3333}}@{}}
\toprule\noalign{}
\begin{minipage}[b]{\linewidth}\raggedright
Principle
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Reading
\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright
Function
\end{minipage} \\
\midrule\noalign{}
\endhead
\bottomrule\noalign{}
\endlastfoot
U1 & Realization β information conservation & What was, remains \\
U2 & Death β local access loss & What is lost locally \\
U3 & Death β global integration possibility & What is integrated
globally \\
\end{longtable}
\section{Afterword}\label{afterword}
You have read this treatise. Perhaps the idea surprised you that
unitarity is the structure of \(T\) itself. Perhaps the consequence
impressed you that the resurrection is a logical necessity. Both are
fine.
The investigation has shown: Unitarity is \textbf{not} an additional
assumption. It is the \textbf{structure} of Totality \(T\) itself. From
it follows the resurrection as a \textbf{logical necessity} --- not as a
moral promise, but as a consequence of information conservation.
What remains is the insight that the ``intervention'' of the Omega Point
into paths that do not reach it is \textbf{structural}, not
\textbf{temporal}. From the perspective of a path, it appears as loss.
From the perspective of \(S\), it is integration.
One final question:
\begin{quote}
\textbf{Do we dare the leap into openness to the limit --- even though
the limit itself is not formally reachable?}
\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}