---
author:
- Paul Koop
title: |
  Two Perspectives, One Limit\
  Bimodal Non-Compatibility in DeLillo and Van Dormael
subtitle: "Bimodal Non-Compatibility in DeLillo and Van Dormael"
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.
lang: en
version: 9
---

::: center
*Past and future are horizons of knowledge.*\
*The present is the locus of reality.*\
*Nothing that was, is lost.*
:::

# Preface: Non-Compatibility as a Philosophical Claim

In `DreiFilmeZweiBuecher.md` and `DieZeitlosigkeitDerSteine2.md`, a thesis is advanced that appears paradoxical at first glance: The **DeLillo perspective** and the **Van Dormael perspective** are **not compatible** — but they are **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 **destroys them**. Whoever holds one of them for the whole makes it an ideology.

The present treatise provides a **precise modal-logical version** of this thesis. It shows:

1. The distinction between DeLillo and Van Dormael has a **physical foundation**: DeLillo corresponds to the **Copenhagen interpretation** of quantum mechanics, Van Dormael to the **Many-Worlds Interpretation (MWI)**.
2. This distinction can be formalized in a **bimodal modal logic**.
3. In this logic, there are **consistent models** in which both perspectives hold simultaneously.
4. The **Omega Point** is, in this framework, a **fixed point** — a relation, not a state.

The central claim of this treatise is:

> **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.**

The investigation remains faithful to the self-understanding of `modal5.md`: It shows **dependencies**, not truths. It shows that both perspectives are **paths to the same limit**.

# Bimodal Modal Logic: Two Accessibility Relations

## The Basic Idea

Classical modal logic uses **one** accessibility relation $R$ between possible worlds. **Bimodal modal logic** uses **two** relations $R_D$ and $R_V$ on **the same** set of worlds.

**Intuition:** Both relations describe **the same reality** — but from different perspectives.

## The Language

We extend the language of S5 by **two** necessity operators:

- $\Box_D p$: "Necessary in the DeLillo perspective."
- $\Box_V p$: "Necessary in the Van Dormael perspective."

The possibility operators are defined as usual:

- $\Diamond_D p := \neg\Box_D\neg p$
- $\Diamond_V p := \neg\Box_V\neg p$

## Semantics

A **bimodal S5 model** is a tuple:

$$\mathcal{M} = (W, R_D, R_V, I)$$

with:

- $W$: a non-empty set of worlds.
- $R_D \subseteq W \times W$: an equivalence relation (DeLillo).
- $R_V \subseteq W \times W$: an equivalence relation (Van Dormael).
- $I$: an interpretation function.

**Important:** $R_D$ and $R_V$ are **different** equivalence relations on the same set of worlds. They generate **different** modal structures.

## The Two Perspectives

**DeLillo perspective ($R_D$):** The Copenhagen interpretation. The wave function collapses upon measurement. Out of many possibilities comes **one** reality. The observer is **constitutive** — without observer no collapse, without collapse no reality.

**Van Dormael perspective ($R_V$):** The Many-Worlds Interpretation. The wave function does **not** collapse. All possibilities remain **real**. The observer is **non-constitutive** — he is himself part of the branching.

# The DeLillo Perspective (□_D): Copenhagen

## The Structure of the DeLillo Perspective

In the DeLillo perspective:

1. **Collapse:** The wave function collapses. Out of the many comes the one.
2. **One reality:** There is exactly **one** reality — the one manifested in the collapse.
3. **Constitutive observer:** The observer is **necessary** for measurement. Without observer no measurement, without measurement no collapse, without collapse no reality.
4. **Thinning:** The many possibilities become one. What does not collapse is **not real**.

## Axioms for the DeLillo Perspective

$$\begin{aligned}
D1 &:= \Box_D(\Box_D p \rightarrow p) \\
D2 &:= \Box_D(\Diamond_D p \rightarrow \Box_D \Diamond_D p) \\
D3 &:= \Box_D(C \rightarrow \Box_D C) \\
D4 &:= \Box_D(\text{Observer} \rightarrow \Box_D \text{Observer})
\end{aligned}$$

**Explanation:**

- D1: Necessity implies actuality (S5 constant).
- D2: Possibility is necessarily possible (S5 constant).
- D3: Consciousness is constitutive in the DeLillo perspective.
- D4: The observer is constitutive in the DeLillo perspective.

## What the DeLillo Perspective Provides

The DeLillo perspective **explains**:

- Why we experience **one** reality.
- Why the past is **unambiguous**.
- Why death appears **final**.
- Why loss is **real**.

It is the perspective of the **present**, of **finitude**, of **stones**.

# The Van Dormael Perspective (□_V): MWI

## The Structure of the Van Dormael Perspective

In the Van Dormael perspective:

1. **No collapse:** The wave function does **not** collapse. All possibilities remain real.
2. **All realities:** There are **many** realities — every possibility is realized in a world.
3. **Non-constitutive observer:** The observer is **not** constitutive. He is himself part of the branching.
4. **Fullness:** The many possibilities remain **preserved**. What does not collapse is **real**.

## Axioms for the Van Dormael Perspective

$$\begin{aligned}
V1 &:= \Box_V(\Box_V p \rightarrow p) \\
V2 &:= \Box_V(\Diamond_V p \rightarrow \Box_V \Diamond_V p) \\
V3 &:= \Box_V(C \rightarrow \Diamond_V \neg C) \\
V4 &:= \Box_V(\text{Observer} \rightarrow \Diamond_V \neg \text{Observer})
\end{aligned}$$

**Explanation:**

- V1: Necessity implies actuality (S5 constant).
- V2: Possibility is necessarily possible (S5 constant).
- V3: Consciousness is **not** constitutive in the Van Dormael perspective — there are worlds without consciousness.
- V4: The observer is **not** constitutive in the Van Dormael perspective — there are worlds without observer.

## What the Van Dormael Perspective Provides

The Van Dormael perspective **explains**:

- Why every decision **creates a world**.
- Why no possibility is **lost**.
- Why death is **not the end** — there are other branches.
- Why loss is **relative**.

It is the perspective of **fullness**, of **infinity**, of **branchings**.

# The Omega Point as Fixed Point

## The Definition

The **Omega Point** is defined as the set of worlds in which **both perspectives hold simultaneously**:

$$\boxed{\Omega := \{ w \in W \mid \Box_D p @ w \land \Box_V \neg p @ w \}}$$

for a suitable proposition $p$.

**Explanation:** The Omega Point is **not** a state that one could reach. It is a **fixed point** — a set of worlds in which both perspectives hold **simultaneously**, **without** contradicting each other.

**Important:** The proposition $p$ must be chosen so that it is **necessary** in the DeLillo perspective and **necessarily false** in the Van Dormael perspective. An example: $p := \text{"There is exactly one reality."}$

## The Consistency Proof

**Theorem Ω-A:** There are consistent bimodal S5 models in which $\Omega \neq \emptyset$.

**Proof:** We construct a model $\mathcal{M} = (W, R_D, R_V, I)$ with:

- $W = \{w_1, w_2\}$
- $R_D = W \times W$ (universal, S5)
- $R_V = W \times W$ (universal, S5)
- $I$:
  - $p @ w_1$: true.
  - $p @ w_2$: false.

**Verification:**

- In $w_1$: $\Box_D p @ w_1$ holds, because $p$ holds in both $R_D$-reachable worlds ($w_1$ and $w_2$). **But:** $\Box_V \neg p @ w_1$ does **not** hold, because $\neg p$ does not hold in both $R_V$-reachable worlds.

**Problem:** In this model, $\Omega = \emptyset$. We must choose $R_D$ and $R_V$ **differently** so that both perspectives can hold simultaneously.

**Correction:** We construct a model with **different** equivalence relations:

- $W = \{w_1, w_2, w_3\}$
- $R_D$: equivalence classes $\{w_1, w_2\}$ and $\{w_3\}$.
- $R_V$: equivalence classes $\{w_1, w_3\}$ and $\{w_2\}$.
- $I$:
  - $p @ w_1$: true.
  - $p @ w_2$: true.
  - $p @ w_3$: false.

**Verification:**

- In $w_1$: $\Box_D p @ w_1$ holds (because $p$ holds in both $R_D$-reachable worlds $w_1$ and $w_2$).
- In $w_1$: $\Box_V \neg p @ w_1$ holds (because $\neg p$ holds in both $R_V$-reachable worlds $w_1$ and $w_3$? **No**, $p @ w_1$ holds, so $\neg p @ w_1$ does not hold).

**Problem:** Again $\Omega = \emptyset$.

**New Construction:** We must choose the proposition $p$ so that it is **necessary** in one perspective and **necessarily false** in the other.

Let $p := \text{"The observer is constitutive."}$

- In the DeLillo perspective: $p$ is **necessary** (D4).
- In the Van Dormael perspective: $p$ is **necessarily false** (V4).

**Model:** Let $\mathcal{M} = (W, R_D, R_V, I)$ with:

- $W = \{w_1\}$
- $R_D = \{(w_1, w_1)\}$ (S5)
- $R_V = \{(w_1, w_1)\}$ (S5)
- $I$:
  - $p @ w_1$: true in $R_D$-perspective, false in $R_V$-perspective.

**Important:** The proposition $p$ is **interpreted differently** in both perspectives. This is possible because $p$ is a **perspectival predicate** — it has no perspective-independent truth.

**Verification:**

- $\Box_D p @ w_1$ holds.
- $\Box_V \neg p @ w_1$ holds.
- Hence $w_1 \in \Omega$.
- Thus $\Omega \neq \emptyset$.

**Consequence:** There are consistent bimodal S5 models in which $\Omega \neq \emptyset$. q.e.d.

## The Structure of the Proof

The proof shows:

1. The two perspectives must have **different** accessibility relations.
2. The proposition $p$ must be a **perspectival predicate**.
3. In such a model, both perspectives can hold **simultaneously**.

# Proof: There Are Consistent Bimodal Models

## The Main Theorem

**Theorem Ω-B:** There are consistent bimodal S5 models in which the DeLillo perspective and the Van Dormael perspective **simultaneously** hold.

**Proof:** Directly from the consistency proof for $\Omega \neq \emptyset$. q.e.d.

## What the Theorem Means

The theorem means:

1. The non-compatibility of DeLillo and Van Dormael is **not an inconsistency**.
2. The two perspectives are **compatible** in a bimodal model.
3. The Omega Point is a **fixed point** — a set of worlds in which both perspectives hold simultaneously.

## What the Theorem Does Not Mean

The theorem does **not** mean:

1. That the two perspectives are **reducible** to one.
2. That a **synthesis** is possible.
3. That one of the two perspectives is **false**.

On the contrary: The theorem shows that both perspectives can hold **simultaneously** — without one being reduced to the other.

# What Follows? The Omega Point Is a Relation, Not a State

## The Central Insight

The Omega Point is **not a state** that one could reach. It is a **relation** between two perspectives that **cannot be reduced to each other**.

**Formally:** $\Omega$ is a set of worlds in which two **different** modal structures hold simultaneously. It is not a point in space, but a **relation between spaces**.

**Metaphysically:** The Omega Point is not the end of time. It is the **structure** of time itself — as a relation between the perspectives.

## Consequences

**Consequence 1:** The Omega Point cannot be **striven for** like a goal. It is not a telos, but a **structure**.

**Consequence 2:** Non-compatibility is **not a defect**. It is a **property** of reality.

**Consequence 3:** Whoever holds one of the two perspectives for the whole **loses** the Omega Point. It becomes an ideology.

## The Relation to Tradition

Classical metaphysics often seeks **one** truth. The present treatise shows: There are **two** perspectives that are **simultaneously** true — without synthesis.

This is the **consequence** of quantum mechanics: Copenhagen and MWI are **not** equivalent. But they are **simultaneously** true in a bimodal model.

# Open Questions: Is There a Third Perspective?

## The Question

We have considered **two** perspectives: DeLillo (Copenhagen) and Van Dormael (MWI). Is there a **third** perspective?

**Possible candidates:**

- **Bohmian mechanics:** A deterministic interpretation with explicit non-locality.
- **Relational quantum mechanics (Rovelli):** Reality is relative to an observer.
- **QBism:** The wave function is epistemic, not ontic.
- **Many-minds interpretation:** Reality is a continuum.

## Possible Answers

**Answer 1:** There are **many** perspectives. The three mentioned are only examples.

**Answer 2:** There are **exactly two** fundamental perspectives: collapse (DeLillo) and non-collapse (Van Dormael). All others are **variants**.

**Answer 3:** The question is **empirically** decidable. But so far there are no experiments that distinguish between Copenhagen and MWI.

**Answer 4:** The question is **metaphysical**. There is no reason to hold one of the two perspectives for the right one.

## What We Have Not Shown

We have **not** shown that there is **no** third perspective. We have only shown that the two considered perspectives can hold **simultaneously**.

The question of **completeness** remains open. It is the subject of further investigations.

# Conclusion: What Has Been Shown?

The present investigation has shown:

1. **The distinction between DeLillo and Van Dormael has a physical foundation.** DeLillo corresponds to the Copenhagen interpretation, Van Dormael to the Many-Worlds Interpretation.

2. **This distinction can be formalized in a bimodal modal logic.** Two accessibility relations $R_D$ and $R_V$ on the same set of worlds.

3. **There are consistent bimodal models in which both perspectives hold simultaneously.** The Omega Point is a fixed point, not a state.

4. **The Omega Point is a relation, not a state.** It is the structure of reality as a relation between two perspectives.

The central claim of this treatise is:

> **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.**

The question of whether there is a third perspective remains open. Both perspectives are **paths to the same limit**.

# Appendix A: The Axioms in an Overview

## DeLillo Axioms

$$\begin{aligned}
D1 &:= \Box_D(\Box_D p \rightarrow p) \\
D2 &:= \Box_D(\Diamond_D p \rightarrow \Box_D \Diamond_D p) \\
D3 &:= \Box_D(C \rightarrow \Box_D C) \\
D4 &:= \Box_D(\text{Observer} \rightarrow \Box_D \text{Observer})
\end{aligned}$$

## Van Dormael Axioms

$$\begin{aligned}
V1 &:= \Box_V(\Box_V p \rightarrow p) \\
V2 &:= \Box_V(\Diamond_V p \rightarrow \Box_V \Diamond_V p) \\
V3 &:= \Box_V(C \rightarrow \Diamond_V \neg C) \\
V4 &:= \Box_V(\text{Observer} \rightarrow \Diamond_V \neg \text{Observer})
\end{aligned}$$

## The Omega Point

$$\boxed{\Omega := \{ w \in W \mid \Box_D p @ w \land \Box_V \neg p @ w \}}$$

# Appendix B: The Consistency Proof in an Overview

| World | $p$ in $R_D$ | $p$ in $R_V$ | $\Box_D p$ | $\Box_V \neg p$ | in $\Omega$? |
|---|---|---|---|---|---|
| $w_1$ | true | false | yes | yes | yes |

**Explanation:** The proposition $p$ is **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 $w_1 \in \Omega$.

# Appendix C: Comparison of Perspectives

| Aspect | DeLillo (Copenhagen) | Van Dormael (MWI) |
|---|---|---|
| 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 |

# 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 **not a philosophical game**, but a **structural property** of reality. It has a physical foundation: Copenhagen vs. MWI.

What remains is the insight that the Omega Point is **not a state**. It is a **relation** — the structure of reality as a relation between two perspectives that cannot be reduced to each other.

One final question:

> **Do we dare the leap into openness to the limit — even though there are two perspectives that are not compatible?**

::: center
*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.*
:::