# Observer Frameworks in Logical Systems: Extending Gödel's Incompleteness Theorems

Emanuel J Thompson

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Emanuel J Thompson. Observer Frameworks in Logical Systems: Extending Gödel's Incompleteness Theorems. 2024. hal-04713224v2

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# OBSERVER FRAMEWORKS IN LOGICAL SYSTEMS: EXTENDING GÖDEL'S INCOMPLETENESS THEOREMS

EMANUEL J. THOMPSON

**ABSTRACT.** This paper examines the role of observer in the context of provability within logical systems and the inherent limitations imposed by Gödel's Incompleteness Theorems. With the law of the Excluded Middle (LEM) and the concept of undecidable propositions being separate entities, a sequence of logical systems {$s_n$} is constructed to represent the idea that every proposition may eventually be provable within some sufficiently strong system {$s_n+$} and assessing whether such model can overcome the fundamental limitations identified by Gödel. An hierarchy of observers {$O_n$}, each capable of perceiving statements that previous observers cannot, is introduced, and its relation to the provability of propositions is analyzed. Ultimately, undecidability remains an intrinsic feature of these observed systems.

## 1. INTRODUCTION

The Law of the Excluded Middle (LEM) asserts for any proposition $P$, either $P$ is true or its negation $\neg P$ is true:

$$P \lor \neg P$$

This principle is foundational in classical logic, representing many logical systems and arguments. Concurrently, the principle of bivalence maintains that every proposition possesses exactly one of two truth values: true or false. Together, LEM and bivalence establish a binary framework for understanding truth in logical propositions.

However, within formal systems, the existence of undecidable propositions challenges the statements of truth and provability within the LEM. Gödel's Incompleteness Theorems demonstrate that in any consistent, sufficiently expressive formal system, there exist true statements that are unprovable within that system.

Undecidable propositions are treated as statements for which current logical systems lack the necessary tools for proof. By constructing a sequence of increasingly powerful logical systems {$s_n$}. Investigating whether every proposition could eventually be provable within some system in this sequence. Furthermore, a hierarchy of observers {$O_n$} is introduced, each capable of perceiving statements beyond the reach of previous observers. This idea is then analyzed to understand its implications for the provability of propositions and the inherent limitations imposed by Gödel's Incompleteness Theorems and their relation to the Law of the Excluded Middle will be implied.

<footnote>Date: September 24, 2024.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 03B30, 03B40; Secondary 03D05, 03B55.</footnote>
**Expanding the Scope:** This idea extends beyond abstract logical systems to consider how observation and propositional truths apply to a spectrum of entities—from fundamental particles like atoms and molecules, through biological organisms, to complex beings like humans and artificial intelligences (AI). By examining these varied entities, the paper seeks to bridge the gap between formal logic and practical, observable phenomena, thereby modifying provability and the limitations of formal systems, with undecidability remaining an intrinsic feature of these observed systems.

## 2. SEQUENCES OF LOGICAL SYSTEMS AND PROVABILITY

**2.1. Definitions.** Let $P$ be the space of all logical propositions, encompassing both finite and infinite-length statements.

**Definition 2.1 (Space of Logical Systems).** The space of logical systems $S$ is defined as the set of all formal logical systems, each characterized by its own axioms, inference rules, and or capabilities. Formally, $S = \{s_1, s_2, s_3, \dots\}$, where each $s_n$ is a distinct system within $S$.

**Definition 2.2 (Sequence of Logical Systems).** Within $S$ being the Space of logical systems with $s_n$ is a sequence within this space:

$$S_n = \{s_n \mid n \in \mathbb{N}\},$$

where each $s_n$ is a formal logical system, and $s_{n+1}$ extends $s_n$, by potentially adding new axioms and or rules to prove previously undecidable propositions.

**Definition 2.3 (Propositions in reference).** In the context of this paper, a proposition is defined as either:

(1) A formal declarative statement that is either true or false within a given logical system.

(2) An operation to be performed, which may involve transforming propositions or influencing the system's state.

This duality allows for the exploration of both declarative and procedural aspects within logical systems as perceived by different observers.

**Examples:**

* **Formal Statement of Truth:** "For every natural number $n$, there exists a prime number greater than $n$."

* **Operation to be Performed:** "Compute the factorial of $n$, denoted as $n!$."

**Definition 2.4 (Provability Function).**

$$ \text{Provable}(x, s_n) = \begin{cases} \text{true} & \text{if } x \text{ is provable in } s_n, \\ \text{false} & \text{if } \neg x \text{ is provable in } s_n, \\ \text{undecidable} & \text{if neither } x \text{ nor } \neg x \text{ is provable in } s_n. \end{cases} $$

**Definition 2.5 (Infinite-Length Proposition).** An infinite-length proposition is a non-terminating sequence of symbols from a finite alphabet, that extends indefinitely without concluding. Formally, it can be represented as:

$$x = x_1 x_2 x_3 x_4 \dots$$
where each $x_i$ is a symbol from the referenced alphabet. It is important to note that this extension deviates from classical formal systems and embraces the idea of an infinite logical statement.

**Definition 2.6 (Observer and Observation Set).** An observer $O_n$ is an entity capable of perceiving a specific subset of propositions within $P$. The observation set $E_n$ represents all propositions that $O_n$ can observe. Each observer $O_n$ is associated with a logical system $S_n$, which can prove all propositions in $E_n$. No countable sequence of observations can fully cover an uncountable $P$.

**Examples of Observers:**

* **Atomic Observers:** Fundamental particles such as atoms and molecules, whose interactions and states can be described by propositions pertaining to quantum mechanics and chemistry.

* **Biological Observers:** Organisms which perceive their environment through sensory inputs and can process information based on biological and neurological propositions.

**Conscious Observers:** capable of varying levels of abstract reasoning, language, and complex problem-solving, thereby observing propositions across diverse domains such as mathematics, philosophy, and daily experiences.

* **Artificial Observers:** Artificial intelligences (AI), which process data and perform computations, thereby observing propositions related to mathematics, algorithmic logic, machine learning, and computational theory.

**Definition 2.7 (Perception Function).** For each observer $O_n$ within the space of propositions $P$, the **perception function** $\pi_n$ is defined as the characteristic function of $E_n$:

$$ \pi_n(p) = \begin{cases} 1 & \text{if } p \in E_n, \\ 0 & \text{if } p \notin E_n. \end{cases} $$

This function indicates whether observer $O_n$ can perceive and or prove the proposition $p$ within the logical system $S_n$.

### 3. OBSERVER FORMALIZATION

**Definition 3.1 (Observer Formalization).** An observation $O_n$ within the space $P$ is formally defined as a perception ($S_n, E_n$), where:

* $S_n$ is the $n^{th}$ logical system within $S$,

* $E_n$ is the set of propositions observable and provable within $S_n$.

**Example 3.2.** Consider an observer $O_1$ with perception ($S_1, E_1$), where $S_1$ is a logical system capable of proving all propositions related to basic arithmetic, and $E_1$ includes propositions such as $2+2=4$ and $\forall x(x+0=x)$.

**Definition 3.3 (Limit Observer).** The limit observer $O_\infty$ is an ideal observer capable of perceiving and proving all propositions within the space $P$, including those of infinite length. $O_\infty$ operates across an infinite hierarchy of logical systems $\{S_n\}$, thereby circumventing the limitations of any single system by accessing and integrating multiple logical frameworks.
### 3.1. Main Assertions.

**Lemma 3.4.** In any sequence of consistent, effectively axiomatizable formal systems {$s_n$} capable of expressing basic arithmetic, there exist propositions $x \in P$ such that:

$$ \forall n \in \mathbb{N}, \quad \text{Provable}(x, s_n) = \text{undecidable}. $$

*Proof.* This follows from Gödel's Incompleteness Theorems. For each system $s_n$, there exists a Gödel sentence $G_n$ that is undecidable within $s_n$. Extending the system to $s_{n+1}$ introduces new undecidable propositions. Therefore, there are propositions that remain undecidable in all systems $s_n$. $\square$

**Lemma 3.5.** For every proposition $x \in P$, there exists a formal system $s_n$, $n \in \mathbb{N}$, such that $x$ is either provable or its negation is provable in $s_n$. However, new undecidable propositions will emerge as systems are extended.

*Proof.* Starting with a base system $s_1$, we can extend it to $s_2$ to resolve some undecidable propositions by adding new axioms. This process can be continued, ensuring that for each $x \in P$, there exists some $s_n$ where $x$ is decidable. However, by Gödel's Incompleteness Theorems, each extension $s_{n+1}$ will introduce new undecidable propositions not resolvable by any finite sequence of extensions.

*Extending the System:*

To address the undecidability of $x$, we can extend $s_n$ to a larger system $s_{n+1}$ by adding $x$ or $\neg x$ as a new axiom. This creates a new system where $x$ is decidable:

$$ \text{Provable}(x, s_{n+1}) \neq \text{undecidable}. $$

$\square$

**Observation:**

Let $O$ represent the observer with access to all logical systems {$s_n$}.

**Observer's Limitations:**

$$ (1) \quad \exists x \in P, \quad \forall n \in \mathbb{N}, \quad \text{Provable}(x, s_n) = \text{undecidable}. $$

This means that the observer cannot prove $x$ within any system accessible to them.

**Existence of systems beyond the observer:**

$$ (2) \quad \exists \text{ system } s^* \notin \{s_n\}, \text{ such that } \text{Provable}(x, s^*) \neq \text{undecidable}. $$

However, such a system may not be effectively axiomatizable or may compromise consistency. It is with the existence of further logical systems that arise, which the observer cannot formalize or propose, that these limitations become apparent.

By examining the previous statements as the observer of this proposition, we must acknowledge that we are unable to prove this proposition given that we have all of the logic that can be obtained at our disposal. Therefore, there must exist some logical systems beyond the observer's view.
## 3.2. Limitations and Gödel's Theorems.

**Theorem 3.6 (Gödel's First Incompleteness Theorem).** In any consistent, effectively axiomatizable formal system capable of expressing arithmetic, there exist true propositions that are unprovable within that system.

*Note:* The Law of the Excluded Middle (LEM) asserts that every proposition has a definite truth value, either true or false. However, within the context of a logical system in-view, observed, or governed by an observer, certain propositions may become undecidable. This means that while LEM maintains the existence of truth values for all propositions, the observer's logical framework may lack the necessary tools to determine the truth value of some propositions, rendering them undecidable within that system.

**Definition 3.7 (Observer and Observation Set).** An observer $O_n$ is an entity capable of perceiving a specific subset of propositions within $P$. The observation set $E_n$ represents all propositions that $O_n$ can observe. Each observer $O_n$ is associated with a logical system $s_n$, which can prove all propositions in $E_n$.

Let $P$ be the space of all propositions.

Let $O$ be a sequence of **hierarchical observers**:

$$O = \{O_n \mid n \in \mathbb{N}\}.$$

The observer acts like a cover to some of the propositions within the space $P$.

* **Structure of advancing Observations:**

$$O_{n-1} \subset O_n, \quad \text{for all } n \geq 1.$$

* **Observation Sets:**

$$E_n = \{x \in P \mid \text{Observed}(x, O_n)\}.$$

* **Hierarchy of Observation Sets:**

$$E_{n-1} \subset E_n, \quad \text{for all } n \geq 1.$$

* **Set of Newly Observed Propositions:**

$$U_n = E_n \setminus E_{n-1},$$

with the assumption that:

$$U_n \neq \emptyset, \quad \text{for all } n \geq 1.$$

**Lemma 3.8.** For each $n \in \mathbb{N}$, $U_n = E_n \setminus E_{n-1} \neq \emptyset$.

*Proof.* Assume, for contradiction, that $U_n = \emptyset$ for some $n \in \mathbb{N}$. This would imply $E_n = E_{n-1}$, meaning observer $O_n$ does not observe any new propositions beyond those observed by $O_{n-1}$. However, by the definition of a strictly increasing sequence of observers, each new observer must perceive at least one new proposition. This contradicts the assumption that $U_n = \emptyset$. Therefore, $U_n \neq \emptyset$ for all $n \in \mathbb{N}$. □
**3.3. Properties of Observers.**

**Lemma 3.9.** The sequence {$s_n$} of logical systems is strictly increasing; that is, $S_{n-1} \subsetneq s_n$ for all $n \ge 1$.

*Proof.* From Lemma 3.8, we know that $U_n = E_n \setminus E_{n-1} \neq \emptyset$ for all $n \in \mathbb{N}$. This implies that observer $O_n$ observes at least one proposition $x \in U_n$ that was not observed by $O_{n-1}$.

To prove $x$, the logical system $s_n$ must include new axioms or inference rules not present in $S_{n-1}$. These additions are necessary to enable $s_n$ to prove propositions that $S_{n-1}$ cannot. Consequently, $s_n$ strictly contains $S_{n-1}$, establishing that $S_{n-1} \subsetneq s_n$.

Therefore, the sequence {$s_n$} is strictly increasing. $\square$

**Lemma 3.10.** *Assuming that $E_\infty = \bigcup_{n=1}^\infty E_n = P$, for each proposition $x \in P$, there exists an observer $O_N$ (for some $N \in \mathbb{N}$) such that $x \in E_N$.*

*Proof.* Since $E_\infty = \bigcup_{n=1}^\infty E_n = P$ by assumption, for every $x \in P$, there exists $N \in \mathbb{N}$ such that $x \in E_N$. $\square$

*Note:* This lemma relies on the assumption that $P$ is exhaustively covered by the union of all $E_n$. If $P$ is uncountable and each $E_n$ is countable, then $E_\infty$ (a countable union of countable sets) is also countable. This leads to a contradiction if $P$ is indeed uncountable. Consequently, Gödel's Incompleteness Theorem reinforces the notion that no countable sequence of observation sets can cover all propositions in $P$, as some will remain undecidable within any formal system associated with the referenced observers.

**3.4. Uncountability of P and the Limit Observer.** In standard formal systems, $P$, the set of all propositions, is countable due to the finite length of each proposition. However, by introducing **infinite-length** propositions, $P$ becomes **uncountable**, aligning with Cantor's diagonal argument. This transition necessitates an extension beyond classical formal systems into frameworks that accommodate infinite constructs, such as **infinite-logic** or **modal logic**.

Assuming the existence of a **limit observer** $O_\infty$, who has access to all propositions in $P$, including those of infinite length, we can explore the implications for provability and undecidability. The limit observer represents an idealized entity capable of perceiving every proposition.

**Theorem 3.11 (Uncountability of P).** *Assuming the inclusion of infinite-length propositions, the space of all propositions P is uncountable.*

*Proof.* By Cantor's diagonal argument, the set of all infinite-length sequences over a finite list and in this case an alphabet is uncountable. Since each infinite-length proposition corresponds to a unique infinite sequence of symbols, $P$ is uncountable when such propositions are included. Finite-length propositions alone form a countable set, but the addition of infinite-length propositions extends $P$ to an uncountable set. $\square$

**Implications:**

The uncountability of $P$ underpins the existence of propositions that transcend any countable hierarchy of observers {$O_n$}. Even with a limit observer $O_\infty$, who
theoretically perceives all propositions, the introduction of infinite-length propositions ensures that $P$ cannot be fully encompassed, thereby reinforcing Gödel's assertion of inherent limitations in formal systems.

**Theorem 3.12.** There does not exist a finite $N \in \mathbb{N}$ such that the observation set $E_N$ equals the space of all propositions $P$. No finite observer can perceive all propositions within $P$.

*Proof.* Assume, for contradiction, that such a finite $N$ exists. Then, by definition, $E_N = P$.

Consider any $n > N$. Since observers form a strictly increasing sequence as per Lemma 3.9, it follows that:

$$E_{n-1} \subsetneq E_n$$

Given that $N$ is finite and $E_N = P$, for all $n > N$, we would have:

$$E_n = E_N = P$$

This implies that:

$$E_{n-1} \subsetneq E_n = P$$

However, since $E_n = P$, this leads to:

$$E_{n-1} \subsetneq P$$

But $E_N = P$, so for $n > N$, $E_{n-1}$ would also equal $P$, which contradicts the strict inclusion established by Lemma 3.9. Therefore, our initial assumption that such a finite $N$ exists must be false.

Furthermore, by Cantor's diagonal argument, the set of all infinite-length sequences over a finite alphabet is uncountable. Since each infinite-length proposition corresponds to a unique infinite sequence of symbols, the space of all propositions $P$ is uncountable when such propositions are included.

**Implications:**

The uncountability of $P$ underpins the existence of propositions that transcend any finite hierarchy of observers $\{O_n\}$. Even with a limit observer $O_\infty$, who can theoretically perceives all propositions, the introduction of infinite-length propositions ensures that $P$ cannot be fully encompassed, thereby reinforcing the inherent limitations identified by Gödel's Incompleteness Theorems. □

## 4. DECENTRALIZED OBSERVERS

**Definition 4.1 (Decentralized Observers).** Decentralized observers are multiple observers $O_{n+1}$ and $O_{n+2}$ associated with an overall observation set $E_n$ such that:

$$E_{n+1} \subseteq E_n \text{ and } E_{n+2} \subseteq E_n,$$

where:

$$E_{n+1} \cap E_{n+2} = \emptyset.$$

This implies that $O_{n+1}$ and $O_{n+2}$ observe distinct subsets of propositions within $E_n$, operating under different logical systems $S_{n+1}$ and $S_{n+2}$ respectively.

**Proposition 4.2.** Given an overall observation set $E_n$, there exist at least two decentralized observers $O_{n+1}$ and $O_{n+2}$ such that:

$$E_{n+1} \subseteq E_n, \quad E_{n+2} \subseteq E_n, \quad \text{and} \quad E_{n+1} \cap E_{n+2} = \emptyset.$$
*Proof.* Consider the overall observation set $E_n$. To establish the existence of decentralized observers $O_{n+1}$ and $O_{n+2}$, we construct two non-overlapping subsets $E_{n+1}$ and $E_{n+2}$ of $E_n$.

**Construction:**

(1) Select a proposition $x \in E_n$ and assign it to $E_{n+1}$:

$$E_{n+1} = \{x\}.$$

(2) Select another proposition $y \in E_n$ distinct from $x$ and assign it to $E_{n+2}$:

$$E_{n+2} = \{y\}.$$

(3) Continue this process, ensuring that for each new observer $O_{n+k}$, the corresponding observation set $E_{n+k}$ contains propositions not included in any previous $E_{n+j}$ for $j < k$.

By construction, $E_{n+1} \cap E_{n+2} = \emptyset$, satisfying the condition for decentralized observers. Each observer $O_{n+k}$ operates under a distinct logical system $S_{n+k}$ tailored to prove the propositions within their respective observation sets $E_{n+k}$.

**Conclusion:** Thus, for any overall observation set $E_n$, it is always possible to construct at least two decentralized observers $O_{n+1}$ and $O_{n+2}$ with non-overlapping observation sets, thereby decentralizing the observation process within the logical framework. □

**Example 4.3.** Consider two decentralized observers $O_{n+1}$ and $O_{n+2}$ within the logical system $S_n$. Observer $O_{n+1}$ focuses on algebraic propositions, while $O_{n+2}$ specializes in geometric propositions. Through the perception of an overall observer from which $(O_{n+1}, O_{n+2}) \subset O_n$ and through some perception $\gamma_{n+1,n+2}$, resolving a previously undecidable proposition that intersects both algebra and geometry.

*Note:* Through the function $\gamma_{n+1,n+2}$, $O_{n+1}$ and $O_{n+2}$ can integrate their specialized perceptions within $O_n$ in order to approach and attempt to resolve propositions that span both algebraic and geometric domains, thereby enhancing $O_n$ overall analytical capabilities.

**4.1. Properties of Decentralized Observers.** Decentralized observers introduce a layer of complexity to the hierarchy of observers $\{O_n\}$. Unlike a singular observer whose observation set expands monotonically, decentralized observers can simultaneously observe distinct subsets of propositions, each governed by their own logical systems.

Decentralized observers can allow for the parallel exploration and proof of distinct propositions within the same overarching logical framework. Decentralized networks of observers may lead to a more robust and diverse approach to addressing undecidable propositions, as different logical systems can offer varied methodologies for proving or disproving specific statements.

However, this also introduces potential challenges:

* **Consistency Across Systems:** Ensuring that the multiple logical systems $\{S_n\}$ remain consistent with each other to prevent contradictions.

* **Inter-System Communication:** Facilitating the exchange of information or propositions between decentralized observers, which may require bridging logical gaps.

* **Scalability:** Managing an ever-increasing number of decentralized observers as new propositions emerge.
Despite these challenges, decentralized observers provide a promising avenue for expanding the capacity of formal systems to address a broader range of propositions, albeit within the constraints imposed by Gödel's Incompleteness Theorems.

# 5. ANALYZING THE LIMIT OBSERVER IN THE CONTEXT OF GÖDEL'S THEOREMS

## 5.1. Inherent Limitations.

**Theorem 5.1.** Even if $E_\infty = P$, there exist true propositions in $E_\infty$ that are unprovable within any consistent, effectively axiomatizable formal system associated with the observers.

*Proof.* By Gödel's Incompleteness Theorems, any consistent, effectively axiomatizable formal system capable of expressing basic arithmetic will have true but unprovable propositions. Associating observers with formal systems does not eliminate these limitations. Even the limit observer $O_\infty$, if associated with a formal system, cannot prove all truths in $P$ without encountering inconsistency or non-effectiveness. □

*note:* Observers equipped to handle infinite-length propositions utilize extended perception functions that can process non-terminating sequences, enabling the analysis and interpretation of such propositions within their respective logical systems.

## 5.2. Observers as Formal Systems.

**Definition 5.2 (Association of Observers and Formal Systems).** Associate each observer $O_n$ with a formal system $s_n$:

$$O_n \leftrightarrow s_n,$$

where observation corresponds to provability:

$$\text{Observed}(x, O_n) \leftrightarrow \text{Provable}(x, s_n).$$

**Lemma 5.3.** For each formal system $s_n$, there exist propositions that are undecidable within $s_n$.

*Proof.* By Gödel's First Incompleteness Theorem, any consistent, effectively axiomatizable formal system $s_n$ that is sufficiently expressive to encode basic arithmetic will contain propositions that are true but unprovable within $s_n$. Therefore, for each $s_n$, there exists at least one proposition $x \in P$ such that neither $x$ nor $\neg x$ is provable in $s_n$, making $x$ undecidable within $s_n$. □

**Theorem 5.4.** A sequence $\{s_n\}$ cannot collectively prove all truths in $P$.

*Proof.* Assume, for contradiction, that the sequence $\{s_n\}$ can collectively prove all propositions in $P$. This would imply that:

$$P = \bigcup_{n=1}^{\infty} \text{Provable}(P, s_n).$$

However, by Lemma 3.9, each $s_{n+1}$ is an extension of $s_n$ that resolves some undecidable propositions from $s_n$ by adding new axioms. According to Gödel's First Incompleteness Theorem, every sufficiently strong formal system $s_n$ contains propositions that are undecidable within it. When extending $s_n$ to $s_{n+1}$, while some undecidable propositions from $s_n$ become decidable in $s_{n+1}$, new undecidable propositions emerge in $s_{n+1}$.
This infinite process of extension does not result in a system that can prove all propositions in $P$, as there will always be new undecidable propositions at each stage. Therefore, the assumption that $\{s_n\}$ can collectively prove all truths in $P$ leads to a contradiction. Hence, the sequence $\{s_n\}$ cannot collectively prove all truths in $P$. □

**5.3. Observers Across Different Scales of Existence.** The concept of observers $\{O_n\}$ extends beyond abstract entities to encompass a wide spectrum of beings and systems, each operating at different scales and possessing varying capacities for perception and propositional processing.

**1. Atomic Observers:** At the most fundamental level, atoms and molecules interact based on the principles of quantum mechanics and chemistry. Propositions at this scale involve states of particles, reactions, and energy exchanges. Logical systems $S_n$ associated with these observers encapsulate the axioms and rules governing physical interactions and transformations ie(Human).

**2. Biological Observers:** Organisms such as ants and dogs perceive their environment through sensory mechanisms and process information based on neurological and biological propositions. The corresponding logical systems $S_n$ for these observers include biological laws, behavioral patterns, and ecological interactions.

**3. Conscious Observers:** ie(Humans) have advanced cognitive abilities, able to engage with a vast array of propositions spanning mathematics, philosophy, language, and everyday life. The logical systems $S_n$ linked to human observers are multifaceted, incorporating abstract reasoning, linguistic structures, and ethical frameworks.

**4. Artificial Observers:** Artificial intelligences (AI) process data and execute algorithms, perceiving propositions related to mathematics, computational logic, data analysis, and machine learning. The logical systems $S_n$ for AI observers encompass algorithmic rules, optimization strategies, and artificial neural networks.

Even by examining these observers, and the universality of logical systems, undecidability permeate various facets of reality.

## 6. TOPOLOGICAL PROPERTIES OF THE SPACE OF PROPOSITIONS

**Definition 6.1 (Space of Propositions).** Let $P$ be the space of all logical propositions, encompassing both finite and infinite-length statements. $P$ consists of all logical propositional statements expressible within the language of the logical systems under consideration. $P$ is defined as:

$$P = \Sigma^* \cup \Sigma^\omega$$

where:

* $\Sigma$ is a finite or countably infinite alphabet of symbols.

* $\Sigma^*$ represents all finite-length strings (propositions).

* $\Sigma^\omega$ represents all infinite-length strings (propositions).

**Definition 6.2 (Topology on Space of Propositions).** The topology on the space of propositions $P$ is defined by using the product topology, where each symbol in $\Sigma$ is endowed with a discrete topology. Specifically from the reference of the observer:

* **Finite Propositions ($\Sigma^*$):** Treated as isolated points in the topology, forming a discrete subspace.
* **Infinite Propositions** ($\Sigma^\omega$): Form a Cantor space when $\Sigma$ is finite with at least two symbols, characterized by being compact, perfect, and totally disconnected.

**Theorem 6.3 (Cardinality of the Space of Propositions).** (1) If $\Sigma$ is finite with $|\Sigma| \ge 2$, then:

$$|\Sigma^*| = \aleph_0 \quad \text{and} \quad |\Sigma^\omega| = 2^{\aleph_0}$$

(2) Therefore, the overall space $P$ is:

$$|P| = |\Sigma^* \cup \Sigma^\omega| = 2^{\aleph_0}$$

*Proof.* (1) Finite Propositions ($\Sigma^*$): Each finite-length string can be mapped bijectively to natural numbers, establishing $|\Sigma^*| = \aleph_0$.

(2) Infinite Propositions ($\Sigma^\omega$): Cantor's diagonal argument demonstrates that $|\Sigma^\omega| = 2^{\aleph_0}$.

(3) Union: Since $2^{\aleph_0} > \aleph_0$, the union $\Sigma^* \cup \Sigma^\omega$ has cardinality $2^{\aleph_0}$.

□

**Proposition 6.4 (Topological Properties of P).** (1) Second Countable: The space $P$ is second countable if and only if $P$ is countable. Since $P$ includes $\Sigma^\omega$ which is uncountable, $P$ is not second countable when infinite-length propositions are included.

(2) Compactness: The space $\Sigma^\omega$, under the product topology, is compact by Tychonoff's theorem. Consequently, $P$ is not compact since $\Sigma^*$ is a discrete, non-compact subspace.

(3) Perfectness and Total Disconnectedness: The infinite part $\Sigma^\omega$ is perfect (no isolated points) and totally disconnected.

## 7. OBSERVER HIERARCHY AND LOGICAL SYSTEMS

**Definition 7.1 (Observer Hierarchy in Topological Space).** Consider the observer hierarchy $\{O_n\}$ where each observer $O_n$ is associated with a logical system $S_n$. Each observer $O_n$ perceives a subset $E_n \subseteq P$ such that:

$$E_n \subseteq E_{n+1} \subseteq \dots \subseteq P$$

This forms an **increasing chain** of subsets in the topological space $P$.

**Proposition 7.2 (Limitations of Second Countable Spaces).** In a second countable space, every open cover has a countable subcover. However, since $P$ is uncountable and not second countable when infinite-length propositions are included, no countable observer hierarchy $\{O_n\}$ can cover the entire space $P$.

*Proof.* Since $P$ is uncountable and lacks a countable base, any countable observer hierarchy $\{O_n\}$ with $\bigcup_{n=1}^\infty E_n \subseteq P$ cannot exhaustively cover all propositions, especially those in $\Sigma^\omega$. □

**Definition 7.3 (Infinite Logic and Extended Observer Hierarchy).** By incorporating **infinite logic**, which allows for reasoning about and within infinite-length propositions, an extended observer hierarchy $\{O_n\}$ can be conceptualized. This hierarchy operates at **higher-order cardinalities**, enabling observers to handle increasingly complex and uncountable subsets of $P$ as countable sets.
**7.1. Extended Observer Hierarchy with Infinite Logic.** To address the expanded space $P$, an extended observer hierarchy employing infinite logic is conceptualized. This hierarchy operates at varying cardinalities, allowing observers to perceive and potentially prove propositions within increasingly complex logical systems. However, the inherent uncountability and topological properties of $P$ ensure that:

**Definition 7.4 (Extended Observer Hierarchy).** Consider an extended observer hierarchy {$O_κ$}, where each observer $O_κ$ is associated with a logical system $S_κ$ capable of handling propositions up to cardinality $κ$. Each observer manages a countable subset $E_κ ⊆ P$, such that:

$$E_κ ⊆ P \quad \text{and} \quad |E_κ| = ℵ_0$$

The hierarchy is well-ordered by cardinality, meaning that for any two observers $O_κ$ and $O_λ$, if $κ < λ$, then $E_κ ⊂ E_λ$.

**Theorem 7.5 (Coverage of the Space of Propositions).** *The extended observer hierarchy {$O_κ$} covers the space of propositions $P$ if:*

$$\bigcup_{κ} E_κ = P$$

*However, due to the uncountable nature of $P$, this union requires an **uncountable indexing set** for $κ$, ensuring that each $E_κ$ contributes a countable portion to the overall coverage.*

*Proof.* Since $P$ is uncountable when $Σ^ω$ is included, to cover $P$ entirely, an uncountable number of countable subsets $E_κ$ are necessary. Each $O_κ$ handles a distinct $E_κ$, and the union of all $E_κ$ across the uncountable hierarchy equals $P$. □

* **Inherent Incompleteness Remains:** Despite leveraging infinite logic, Gödel's Incompleteness Theorems persist, as no single, consistent, and effectively axiomatizable logical system can be both complete and sound when dealing with sufficiently complex propositions.

* **Persistent Emergence of New Undecidable Propositions:** By extending logical systems to cover more propositions will invariably introduce new undecidable propositions, maintaining the perpetual state of incompleteness within the observer hierarchy.

**Proposition 7.6 (Topological Implications of Extended Hierarchy).** *In the topological space $P$, the extended observer hierarchy {$O_κ$} with each $E_κ$ being countable ensures that:*

(1) Each $E_κ$ is discrete within its subspace.

(2) The union $\bigcup_{κ} E_κ = P$ maintains the perfectness and total disconnectedness of $Σ^ω$.

(3) No single observer $O_κ$ can cover an uncountable portion of $P$, preserving the inherent incompleteness of each $S_κ$.

*Proof.* (1) Each $E_κ$ is countable and, under the discrete topology, each point is isolated.

(2) The union preserves the topological properties of $P$, as $Σ^ω$ remains perfect and totally disconnected.
(3) Since $P$ is uncountable, no finite or countable union of countable $E_κ$ can cover it without requiring an uncountable hierarchy. □

## 8. EXPLORATION OF THE TRANSFINITE OF $P$

**Definition 8.1** (Transfinite Space of Propositions). The **transfinite space of propositions** $P$ is an extension of the previously defined space $P$, incorporating propositions indexed by ordinals beyond $ω$. Formally:

$$P = \bigcup_{\alpha < \Theta} \Sigma^{\alpha}$$

where:

* $\Sigma$ is a finite or countably infinite alphabet.

* $\Sigma^\alpha$ denotes the set of propositions of length $\alpha$, with $\alpha$ being an ordinal.

* $\Theta$ represents the supremum of the ordinals considered in $P$, potentially extending into higher cardinalities.

**Definition 8.2** (Higher Cardinality Observer). A **higher cardinality observer** $O_κ$ is an observer within the hierarchy $\{O_κ\}$ associated with a logical system $S_κ$. This observer is capable of managing propositions $E_κ \subseteq P$ indexed by ordinals $\alpha < \kappa$, where $κ$ is a transfinite cardinal.

**Definition 8.3** (Extended Observer Hierarchy with Ordinals). The **extended observer hierarchy** $\{O_κ\}$ consists of observers indexed by transfinite cardinals $κ$. Each observer $O_κ$ is responsible for a countable subset $E_κ \subseteq P$, where $E_κ$ contains propositions indexed by ordinals $\alpha < \kappa$.

**Theorem 8.4** (Comprehensive Coverage of $P$). The extended observer hierarchy $\{O_κ\}$ achieves a **comprehensive coverage** of the space of propositions $P$ if:

$$\bigcup_{\kappa \text{ transfinite}} E_κ = P$$

Given that $P$ is indexed by ordinals up to $\Theta$, the hierarchy must include observers $O_κ$ for each $κ$ corresponding to the cardinalities of the ordinals in $P$.

*Proof.* Since each observer $O_κ$ handles propositions indexed by ordinals $\alpha < \kappa$, and $P$ encompasses propositions indexed by all ordinals up to $\Theta$, the union of all $E_κ$ across the transfinite hierarchy $\{O_κ\}$ covers $P$. This assumes that $\Theta$ is the supremum of the cardinals used to index the observers. □

**Proposition 8.5** (Topological Properties with Higher Ordinals). When the space of propositions $P$ includes propositions indexed by ordinals beyond $ω$:

(1) $P$ inherits the order topology from the indexing ordinals.

(2) Limit Points: Propositions indexed by limit ordinals act as accumulation points, impacting properties like compactness and connectedness.

(3) Separation Properties: Depending on $\Theta$, $P$ may exhibit enhanced separation properties, facilitating clearer distinctions between different observer subsets.

*Proof.* (1) The order topology is naturally induced by the well-ordered set of ordinals indexing $P$.
(2) Limit ordinals introduce accumulation points, as every neighborhood around a proposition indexed by a limit ordinal contains propositions indexed by smaller ordinals approaching it.

(3) Enhanced separation properties, such as normality or Hausdorffness, depend on the specific ordinals included and their arrangement within P. □

**Theorem 8.6** (Cardinality of the Transfinite Space of Propositions). (1) If $\Sigma$ is finite with $|\Sigma| \ge 2$, then for each ordinal $\alpha$:

$$|\Sigma^\alpha| = |\Sigma|^\alpha$$

(2) Specifically, for $\alpha = \omega$:

$$|\Sigma^\omega| = 2^{\aleph_0}$$

(3) For $\alpha$ uncountable (e.g., $\omega_1$):

$$|\Sigma^\alpha| = 2^{|\alpha|}$$

(4) Therefore, the overall space P has cardinality:

$$|P| = \sup_{\alpha<\Theta} |\Sigma^\alpha| = 2^{|\Theta|}$$

*Proof.* (1) For each ordinal $\alpha$, $|\Sigma^\alpha|$ is determined by the power set of $\alpha$, given that each position in the proposition can be filled by any symbol in $\Sigma$.

(2) For $\alpha = \omega$, $|\Sigma^\omega| = 2^{\aleph_0}$ when $|\Sigma| \ge 2$.

(3) For uncountable $\alpha$, $|\Sigma^\alpha| = 2^{|\alpha|}$, following the general cardinal exponentiation rules.

(4) The supremum over all $|\Sigma^\alpha|$ up to $\Theta$ determines the total cardinality of P. □

## 9. EXPANDING THE ANALYSIS: THE SPACE OF LOGICAL SYSTEMS S VERSUS THE SPACE OF PROPOSITIONS P

In this section, we explore the relationship between the space of logical systems S and the space of propositions P. With the intent to explore the potential size of S relative to P.

### 9.1. Elaborating on the Space of Logical Systems S.

**Definition 9.1** (Space of Logical Systems). Let $\Sigma$ be a finite or countably infinite alphabet of symbols. The space of logical systems S is defined as the set of all formal logical systems that can be constructed using $\Sigma$. Each logical system $s \in S$ is characterized by:

* A set of **axioms** $A_s \subseteq \Sigma^*$.

* A set of **inference rules** $R_s$, which are finite or countably infinite sequences of transformations applied to propositions within P.

* A **proof procedure** $\Pi_s$, which defines how proofs are constructed within S.

Formally,

$$S = \{s \mid s = (A_s, R_s, \Pi_s)\}.$$
**Theorem 9.2 (Cardinality of the Space of Logical Systems).** Assuming $\Sigma$ is a finite enumeration with $|\Sigma| \ge 2$, the space of logical systems $S$ has a cardinality of:

$$|S| = 2^{2^{\aleph_0}}.$$

*Proof.* By construction $|S| = 2^{2^{\aleph_0}}$.

(1) **Space of Propositions P:**

The set of all finite strings over $\Sigma$ is denoted $\Sigma^*$, and the set of all infinite strings is $\Sigma^\omega$. The total space of propositions is:

$$P = \Sigma^* \cup \Sigma^\omega.$$

The cardinalities are:

$$|\Sigma^*| = \aleph_0, \quad |\Sigma^\omega| = 2^{\aleph_0}, \quad |P| = \max\{\aleph_0, 2^{\aleph_0}\} = 2^{\aleph_0}.$$

(2) **Axioms $A_s$:**

Each logical system's axioms are subsets of $P$:

$$A_s \subseteq P.$$

The number of possible sets of axioms is given by the power set of $P$:

$$|A_s| = |\mathcal{P}(P)| = 2^{|P|} = 2^{2^{\aleph_0}}.$$

(3) **Inference Rules $R_s$:**

Inference rules can be considered as subsets of $P^{<\omega} \times P$, where $P^{<\omega}$ denotes the set of all finite sequences of enumerations over $P$:

$$R_s \subseteq P^{<\omega} \times P.$$

Since $|P^{<\omega}| = 2^{\aleph_0}$, the cardinality of the set of all possible inference rules is:

$$|R_s| = |\mathcal{P}(P^{<\omega} \times P)| = 2^{|P^{<\omega} \times P|} = 2^{2^{\aleph_0}}.$$

(4) **Proof Procedures $\Pi_s$:**

Proof procedures can be any function from finite sequences of propositions to propositions:

$$\Pi_s = \{f \mid f : P^{<\omega} \to P\}.$$

The number of such functions is:

$$|\Pi_s| = |P|^{2^{<\omega}}} = (2^{\aleph_0})^{2^{\aleph_0}} = 2^{2^{\aleph_0}}.$$

(5) **Combined Logical Systems $S$:**

Each logical system $s$ can be a tuple $(A_s, R_s, \Pi_s)$. The total number of logical systems in this configuration:

$$|S| = |A_s| \times |R_s| \times |\Pi_s| = (2^{2^{\aleph_0}}) \times (2^{2^{\aleph_0}}) \times (2^{2^{\aleph_0}}).$$

In cardinal arithmetic, multiplying cardinals of the same size will yield the same cardinality:

$$2^{2^{\aleph_0}} \times 2^{2^{\aleph_0}} \times 2^{2^{\aleph_0}} = 2^{2^{\aleph_0}}.$$

Therefore,

$$|S| = 2^{2^{\aleph_0}}.$$

□
## 9.2. Topological Properties of S Compared to P.

**Definition 9.3 (Topology on the Space of Logical Systems).** The topology on S is defined as a product topology:

$$ \tau_S = \tau_{A_s} \times \tau_{R_s} \times \tau_{\Pi_s}, $$

where each $\tau_{A_s}$, $\tau_{R_s}$, and $\tau_{\Pi_s}$ is the discrete topology on their respective components.

**Theorem 9.4 (Cardinality of S and P).** Given that $|S| = 2^{2^{\aleph_0}}$ and $|P| = 2^{\aleph_0}$, it follows that:

$$ |S| > |P|. $$

*Proof.* By Cantor's theorem, for any set X, the power set $P(X)$ has a strictly greater cardinality than X itself. Since $P = \Sigma^* \cup \Sigma^\omega$ has a cardinality of $|P| = 2^{\aleph_0}$, and S involves constructing tuples from subsets of P (axioms, inference rules, and proof procedures,) the cardinality of S becomes $|S| = 2^{2^{\aleph_0}}$. Therefore, $|S| > |P|$. □

## 9.3. Bounds of Corollaries. 
With the cardinality of the space of logical systems S being defined for a tuple, we can infer a few corollaries.

**Corollary 9.5.** The space of logical systems S has the same cardinality as the power set of the space of propositions P:

$$ |S| = |\mathcal{P}(P)| = 2^{|P|} = 2^{2^{\aleph_0}}. $$

*Proof.* From the theorem, $|S| = 2^{2^{\aleph_0}}$. Since $|P| = 2^{\aleph_0}$, the power set $\mathcal{P}(P)$ has cardinality:

$$ |\mathcal{P}(P)| = 2^{|P|} = 2^{2^{\aleph_0}}. $$

$$ \text{Therefore, } |S| = |\mathcal{P}(P)|. \quad \square $$

**Corollary 9.6.** No countable set can enumerate all logical systems in S.

*Proof.* The cardinality of any countable set is $\aleph_0$. Since $|S| = 2^{2^{\aleph_0}}$ is uncountable and significantly larger than $\aleph_0$, it is impossible for a countable set to enumerate all elements of S. □

**Corollary 9.7.** The set of all possible axiomatizations $A_s$ of logical systems has cardinality $|A_s| = 2^{2^{\aleph_0}}$.

*Proof.* As shown in the proof of the theorem 9.4, the number of possible sets of axioms is:

$$ |A_s| = |\mathcal{P}(P)| = 2^{|P|} = 2^{2^{\aleph_0}}. \quad \square $$

## 9.4. Implications for Observer Hierarchies.

**Theorem 9.8 (Observer Hierarchy Coverage).** Given that $|S| = 2^{2^{\aleph_0}}$ and $|P| = 2^{\aleph_0}$, no observer hierarchy $\{O_n\}$ can simultaneously encompass all logical systems in S while fully covering P.

*Proof.* • Size Disparity: Since $|S| > |P|$, there exist logical systems in S that are not mapped to any proposition in P.
* **Hierarchy Limitation:** An observer hierarchy {$O_n$} can at most associate each observer $O_n$ with a unique logical system $s_n$. However, the sheer number of logical systems in $S$ exceeds the number of propositions in $P$, making it impossible for any countable or even uncountable hierarchy (depending on $P$'s cardinality) to cover all of $S$.

* **Conclusion:** Therefore, observer hierarchies must necessarily be **incomplete** with respect to the space of logical systems $S$.

☐

**Theorem 9.9 (Enhanced Undecidability).** The significant size of $S$ relative to $P$ reinforces **Gödelian incompleteness**, as each logical system introduces its own set of undecidable propositions, thereby expanding the landscape of undecidability beyond what a single or finite observer hierarchy can manage.

*Proof.* Each logical system $s \in S$ possesses unique axioms and inference rules, leading to distinct sets of provable propositions $\phi(s)$. Given $|S| > |P|$, the introduction of new logical systems continuously generates new undecidable propositions that cannot all be encompassed within any finite or countable observer hierarchy, thereby reinforcing and expanding the scope of Gödel's incompleteness. ☐

**Corollary 9.10 (Observer Limitations).** Observers within any hierarchy {$O_n$} are inherently limited by both the **propositional coverage** and the **logical system diversity**. As $S$ grows, observers cannot simultaneously maintain coverage over all $P$ without encountering **inherent inconsistencies** or **unmanageable complexity**.

## 9.5. Formalizing the Relationship Between $S$ and $P$.

**Definition 9.11 (Provability Mapping).** A provability mapping $\phi: S \to 2^P$ where:

$$ \phi(s) = \{p \in P \mid p \text{ is provable in logical system } s\}. $$

**Theorem 9.12 (Properties of the Provability Mapping $\phi$).** The mapping $\phi: S \to 2^P$ has the following properties:

(1) **Non-Injective:** Multiple logical systems can prove the same set of propositions, leading to $\phi(s_1) = \phi(s_2)$ for distinct $s_1, s_2 \in S$.

(2) **Non-Surjective:** Not all subsets of $P$ correspond to a provability set of some logical system $s \in S$, especially given the constraints imposed by Gödel's Incompleteness Theorems.

(3) **Partial Order Preservation:** If $s_1$ is a subsystem of $s_2$ (i.e., $s_2$ extends $s_1$ with additional axioms or rules), then $\phi(s_1) \subseteq \phi(s_2)$.

*Proof.* (1) **Non-Injective:** Consider two logical systems $s_1$ and $s_2$ that differ only by non-essential axioms that do not affect the provability of any proposition in $P$. Hence, $\phi(s_1) = \phi(s_2)$ despite $s_1 \neq s_2$.

(2) **Non-Surjective:** By Gödel's Incompleteness Theorems, no consistent, effectively axiomatizable logical system can have $\phi(s) = P$. Additionally, certain subsets of $P$ may not align with any coherent set of provable propositions in $S$.

(3) **Partial Order Preservation:** If $s_2$ extends $s_1$, then any proposition provable in $s_1$ is also provable in $s_2$, ensuring $\phi(s_1) \subseteq \phi(s_2)$. ☐

☐
**Corollary 9.13** (Limitations of Observer Hierarchies in Mapping *S* to *P*). Given the properties of *φ*, any observer hierarchy {$O_n$} aiming to map logical systems to propositions cannot achieve a complete and injective correspondence between *S* and *P*. This inherently limits the hierarchy's ability to fully represent the probability landscape within *P*.

## 10. ON THE POSSIBILITY OF AN INITIALLY EMPTY PROPOSITION SPACE

The theoretical scenario where the space of propositions *P* is initially assumed to be empty, *P* = ∅, and assessing whether logical systems *S* can introduce propositions into *P*, effectively "forcing" propositions onto *P*, and analyze the coherence and feasibility of this concept within a formal logic.

### 10.1. Theoretical Framework.

**Definition 10.1** (Empty Proposition Space). Let $P = \Sigma^* \cup \Sigma^\omega$ represent the space of propositions, where $\Sigma$ is a finite or countably infinite alphabet of symbols, $\Sigma^*$ denotes finite-length strings, and $\Sigma^\omega$ denotes infinite-length strings. We consider the scenario where $P$ is initially empty:

$$P = \emptyset.$$

### 10.2. Logical Systems and Proposition Generation.

**Definition 10.2** (Logical System). A logical system $s \in S$ can be defined as a tuple:

$$s = (A_s, R_s, \Pi_s),$$

where:

* $A_s \subseteq \Sigma^*$ is the set of **axioms**.

* $R_s$ is the set of **inference rules**.

* $\Pi_s$ is the **proof procedure**.

### 10.3. Introducing Propositions into an Empty P.
Assuming $P = \emptyset$, suppose logical systems *S* can introduce propositions into *P*. Consider the following:

**Definition 10.3** (Existence Proposition). A proposition $p_0$ as:

$p_0$ : "The space of propositions *P* exists."

**Proposition 10.4** (Populating *P* with $p_0$). Introducing $p_0$ into *P* populates *P* such that:

$$P = \{p_0\}.$$

*Proof.* By defining $p_0$ as the existence of *P*, we add $p_0$ to *P*, thereby making *P* non-empty. □

### 10.4. Circularity and Logical Consistency.
However, this approach introduces a circular dependency:

* Circularity Issues: The proposition $p_0$ asserts the existence of *P*, which is contingent on $p_0$ being an element of *P*.

* Logical Inconsistency: If *P* is empty, there is no mechanism to introduce $p_0$ into *P*. Conversely, introducing $p_0$ requires *P* to already contain propositions.
**Lemma 10.5 (Circular Dependency).** It is impossible to introduce a proposition into an initially empty P without violating logical consistency.

*Proof.* Assume $P = \emptyset$. To introduce $p_0$, we must have:

$$P = P \cup \{p_0\}.$$

However, $p_0$ is defined as asserting the existence of P, which requires P to contain at least $p_0$, leading to a circular reference. □

**10.5. Implications for Observer Hierarchies.** Given the circularity issue, observer hierarchies {$O_n$} cannot operate meaningfully if P starts as empty. Observers rely on existing propositions to perform operations such as proving or deducing new propositions. Without an initial set of propositions, observers lack the foundational elements necessary for logical operations.

**Corollary 10.6 (Observer Hierarchy Limitation).** An observer hierarchy {$O_n$} cannot function if the space of propositions P is initially empty.

*Proof.* Since observers require propositions to interact with, an empty P provides no content for observers to engage with, rendering the hierarchy non-operational. □

**10.6. Conclusion.** The exploration of an initially empty proposition space P reveals fundamental limitations within formal logical systems. Logical systems S depend on the existence of propositions to function, and introducing propositions into an empty P leads to circular dependencies that undermine logical consistency. Consequently, for observer hierarchies to operate effectively, P must be non-empty from the outset, ensuring that logical systems have the necessary propositions to work with.

# 11. INTRODUCING A VIRTUAL FUNCTION BEYOND FORMAL SYSTEMS

In formal logic frameworks, the space of propositions P serves as the foundation upon which logical systems operate. Propositions are manipulated through axioms, inference rules, and proofs within logical systems S. However, the scenario where P is initially empty ($P = \emptyset$) poses a challenge: without propositions, logical systems lack the necessary content for meaningful operation. To address this, the concept of a **Virtual Proposition Injector** — is a hypothetical function that operates beyond formal logical systems, and is capable of introducing propositions into P even when P is empty.

## 11.1. Definition of the Virtual Proposition Injector.

**Definition 11.1 (Virtual Proposition Injector).** A Virtual Proposition Injector V is a hypothetical function defined as:

$$V : \mathcal{U} \to P$$

where $\mathcal{U}$ represents an abstract universe of potential propositions not confined within any formal logical system. The function V can introduce new propositions $p \in P$ into the proposition space P, thereby populating P from an initially empty state.
11.2. **Operational Mechanism of V.** The Virtual Proposition Injector V operates independently of any formal logical system $s \in S$. Its mechanisms are conceptualized as follows:

(1) **An External Existence:** V exist externally, outside the boundaries of formal systems, to introduce new propositions.

(2) **Proposition Generation:** Upon an invocation, V selects or generates a proposition $p$ from $U$.

(3) **Insertion into P:** The generated proposition $p$ is then inserted into P, changing the state of P from $\emptyset$ to $\{p\}$.

(4) **Subsequent Operations:** With P now containing propositions, formal logical systems $s \in S$ can commence operations upon observation such as proving, deducing, or deriving new propositions based on the introduced elements.

11.3. **Addressing Circularity and Logical Consistency.** The introduction of V presents a challenge of circularity:

* Circular Dependency: The proposition $p$ introduced by V could be self-referential, such as "the space of propositions P exists." This creates a circular dependency, as $p$'s truth relies on the existence of P, which in turn relies on $p$.

* Consistency Constraints: To prevent inconsistencies, V must operate under strict constraints, ensuring that the introduced propositions do not violate the foundational axioms or rules of any existing logical systems.

**Lemma 11.2 (Non-Circular Proposition Introduction).** *The Virtual Proposition Injector V cannot introduce self-referential propositions that assert the existence of P without causing logical inconsistency.*

*Proof.* Suppose V introduces a proposition $p_0$ defined as:

$p_0$: "The space of propositions P exists."

If P is initially empty, the introduction of $p_0$ by V populates P such that:

$$P = \{p_0\}$$

However, $p_0$ asserts the existence of P, which is contingent upon $p_0$ being in P. This creates a circular dependency, rendering the proposition's truth status undefined within the system, thereby leading to logical inconsistency. □

11.4. **Implications for Observer Hierarchies.** The introduction of V has significant implications for observer hierarchies {$O_n$}:

**Theorem 11.3 (Observer Hierarchy Limitation with a Virtual Injector).** *The existence of a Virtual Injector V imposes complexity on observer hierarchies {$O_n$}, restricting their ability to fully capture the provability landscape within P due to the external and or potentially unbounded introduction of propositions.*

*Proof.* Observers within the hierarchy {$O_n$} rely on the propositions within P to perform logical operations. The external introduction of propositions via V can lead to:

* Unpredictable Expansion: V's operations are not governed by the hierarchy's internal rules, leading to an unpredictable and potentially unbounded expansion of P.
* Incomplete Mapping: Since $V$ operates outside the formal systems, observers cannot account for propositions introduced by $V$, resulting in gaps in the provability mappings.

* Inconsistency Risks: As demonstrated in the previous lemma, $V$ can introduce propositions that undermine the consistency of $P$, complicating observers' tasks in maintaining logical coherence.

Therefore, observer hierarchies must contend with propositions introduced externally, limiting their capacity to provide a **comprehensive** and **consistent** provability landscape. ☐

*Note: Now that we have defined some of the spaces of use, lets construct the space measure of provable functions*

## 12. CONSTRUCTING THE CARTESIAN MEASURE OF PROVABLE FUNCTIONS $P \times S$

**Theorem 12.1** (Constructing a Measure Space on Finite Subsets of $S \times P$). Let $S$ be a finite set of logical systems, and $P$ be a finite set of propositions. Let the Cartesian product $\Omega = S \times P$. We can construct a measure space $(\Omega, \mathcal{F}, \mu)$, where:

(1) $\mathcal{F} = 2^\Omega$ is the sigma-algebra of all subsets of $\Omega$.

(2) The measure $\mu: \mathcal{F} \to [0, \infty)$ is defined for any $A \in \mathcal{F}$ by

$$ \mu(A) = \sum_{(s,p) \in A} \chi(s,p), $$

where

$$ \chi(s,p) = \begin{cases} 1, & \text{if logical system s provides True } p, \\ 2, & \text{if logical system s provides False } p, \\ 3, & \text{if } p \text{ is Undecidable in } s, \\ 0, & \text{if } (s,p) \text{ is not in consideration.} \end{cases} $$

*Note: The measure can be thought of as a function of the Observer.*

Then $(\Omega, \mathcal{F}, \mu)$ is a finite measure space, and the associated measure algebra is non-homogeneous.

*Proof.* To verify that $(\Omega, \mathcal{F}, \mu)$ is a finite measure space, we need to confirm that $\mu$ satisfies the properties of a measure and that the total measure $\mu(\Omega)$ is finite.

### 1. Non-negativity:

For any $A \in \mathcal{F}$,

$$ \mu(A) = \sum_{(s,p) \in A} \chi(s,p) \geq 0, $$

since $\chi(s,p) \in \{0, 1, 2, 3\}$.

### 2. Null Empty Set:

$$ \mu(\emptyset) = \sum_{(s,p) \in \emptyset} \chi(s,p) = 0. $$

### 3. Countable Additivity:

Let $\{A_i\}_{i=1}^\infty$ be a countable collection of disjoint sets in $\mathcal{F}$. Then,

$$ \mu\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{(s,p) \in \bigcup_{i=1}^{\infty} A_i} \chi(s,p) = \sum_{i=1}^{\infty} \sum_{(s,p) \in A_i} \chi(s,p) = \sum_{i=1}^{\infty} \mu(A_i). $$
Since $\Omega$ is finite, the infinite sums reduce to finite sums, ensuring that countable additivity holds.

**4. Finite Total Measure:**

The total measure is

$$ \mu(\Omega) = \sum_{(s,p) \in \Omega} \chi(s,p) \leq |\Omega| \times 3 = |S| \times |P| \times 3 < \infty. $$

Thus, $\mu(\Omega)$ is finite.

**Non-Homogeneity of the Measure Algebra:**

To show that the measure algebra is non-homogeneous, we observe that there exist atoms of different measures, which cannot be mapped onto each other via measure-preserving automorphisms.

Consider the subsets:

$$ A_1 = \{(s,p) \in \Omega \mid \chi(s,p) = 1\}, \quad A_2 = \{(s,p) \in \Omega \mid \chi(s,p) = 2\}, \quad A_3 = \{(s,p) \in \Omega \mid \chi(s,p) = 3\}. $$

Each singleton set $\{(s,p)\}$ where $\chi(s,p) > 0$ is an atom in the measure algebra with measure $\mu(\{(s,p)\}) = \chi(s,p)$.

- Atoms in $A_1$ have measure 1. - Atoms in $A_2$ have measure 2. - Atoms in $A_3$ have measure 3.

Suppose, for contradiction, that the measure algebra is homogeneous. Then there exists a measure-preserving automorphism $\phi$ of the measure algebra such that it can map any atom onto any other atom.

However, since the atoms have different measures, this is impossible. A measure-preserving automorphism must map atoms to atoms of the same measure to preserve the measure.

Therefore, the measure algebra is non-homogeneous because it contains atoms of different measures, and there are limited measure-preserving automorphisms that can map atoms of one measure onto atoms of a different measure. $\square$

**Corollary 12.2 (Nonexistence of Measure-Preserving Automorphisms Between Different Truth Values).** In the measure algebra associated with the measure space $(\Omega, \mathcal{F}, \mu)$, there are no measure-preserving automorphisms that can map sets consisting of atoms corresponding to one truth value onto sets consisting of atoms corresponding to a different truth value. Specifically, atoms of measure 1 ("True" propositions) cannot be mapped onto atoms of measure 2 ("False" propositions) or measure 3 ("Undecidable" propositions), and vice versa.

*Proof.* Let $\Omega = S \times P$, where S is a finite set of logical systems and P is a finite set of propositions. The sigma-algebra $\mathcal{F} = 2^{\Omega}$ is the power set of $\Omega$, and the measure $\mu: \mathcal{F} \to [0, \infty)$ is defined by

$$ \mu(A) = \sum_{(s,p) \in A} \chi(s,p), $$

where

$$ \chi(s,p) = \begin{cases} 1, & \text{if logical system } s \text{ provides True } p, \\ 2, & \text{if logical system } s \text{ provides False } p, \\ 3, & \text{if } p \text{ is Undecidable in } s, \\ 0, & \text{if } (s,p) \text{ is not in consideration.} \end{cases} $$
In the associated measure algebra, atoms are the singleton sets $\{(s, p)\}$ with $\chi(s, p) > 0$, each having measure $\mu(\{(s, p)\}) = \chi(s, p)$.

Suppose, for contradiction, that there exists a measure-preserving automorphism $\phi$ of the measure algebra such that it maps an atom $A$ of measure $m$ to an atom $B$ of measure $n$, where $m \neq n$.

Without loss of generality, let $m = 1$ (an atom corresponding to a "True" proposition) and $n = 2$ (an atom corresponding to a "False" proposition).

Since $\phi$ is measure-preserving, it must satisfy:

$$\mu(\phi(A)) = \mu(A).$$

But:

$$\mu(\phi(A)) = \mu(B) = n = 2 \neq m = 1 = \mu(A),$$

which contradicts the measure-preserving property.

Therefore, no such automorphism $\phi$ exists, and atoms of measure 1 cannot be mapped onto atoms of measure 2. The same argument applies to atoms of measure 3. Thus, measure-preserving automorphisms cannot map atoms corresponding to one truth value onto atoms corresponding to a different truth value. $\square$

**Example.** Suppose:

* $S = \{s_1, s_2\}$ (two logical systems).

* $P = \{p_1, p_2\}$ (two propositions).

Define $\chi(s, p)$ as:

<table><thead><tr><th>(s, p)</th><th>χ(s, p)</th><th>Interpretation</th></tr></thead><tbody><tr><td>(s<sub>1</sub>, p<sub>1</sub>)</td><td>1</td><td>s<sub>1</sub> provides True p<sub>1</sub></td></tr><tr><td>(s<sub>1</sub>, p<sub>2</sub>)</td><td>2</td><td>s<sub>1</sub> provides False p<sub>2</sub></td></tr><tr><td>(s<sub>2</sub>, p<sub>1</sub>)</td><td>3</td><td>p<sub>1</sub> is Undecidable in s<sub>2</sub></td></tr><tr><td>(s<sub>2</sub>, p<sub>2</sub>)</td><td>1</td><td>s<sub>2</sub> provides True p<sub>2</sub></td></tr></tbody></table>

**Atoms and Their Measures:**

* Atom $A_1 = \{(s_1, p_1)\}$ has measure $\mu(A_1) = 1$.

* Atom $A_2 = \{(s_1, p_2)\}$ has measure $\mu(A_2) = 2$.

* Atom $A_3 = \{(s_2, p_1)\}$ has measure $\mu(A_3) = 3$.

* Atom $A_4 = \{(s_2, p_2)\}$ has measure $\mu(A_4) = 1$.

**Measure-Preserving Automorphisms:**

* Possible between $A_1$ and $A_4$ (both measure 1).

* Not possible between $A_1$ and $A_2$ or $A_3$ due to differing measures.

This example illustrates that atoms of different measures cannot be mapped onto each other via measure-preserving automorphisms, supporting Corollary 1.1.

note:

*Analysis of Non-Homogeneity.* In this measure space:

- **Atoms:** The atoms are the singleton sets $\{(s, p)\}$ where $\chi(s, p) > 0$. Each atom has a measure equal to $\chi(s, p)$.

- **Differing Measures:** Atoms have measures of 1, 2, or 3, corresponding to the logical outcome of $p$ in $s$.

- **Limited Automorphisms:** Measure-preserving automorphisms must map atoms to atoms of the same measure to preserve the measure. Thus, automorphisms are limited to permutations within each class of atoms with the same measure.
**- No Cross-Measure Automorphisms:** There are no measure-preserving automorphisms that map atoms of one measure onto atoms of a different measure.

This structural rigidity leads to the **non-homogeneity** of the measure algebra, as it lacks the uniformity required for homogeneity. Homogeneity would require that any two non-zero elements could be mapped onto each other via a measure-preserving automorphism, which is not possible in this example here.

*Implications.* - **Measure Algebra Classification of this set:** The measure algebra cannot be classified as homogeneous and may be described using multiple Maharam types corresponding to the different atom measures.

- **Structure of Automorphism Groups:** The automorphism group of the measure algebra is the direct product of the symmetric groups acting on the atoms of each measure class.

## 13. THE OBSERVER NETWORK

(1) **Proposition Generation:** $O$ formulates a new proposition $p$ that addresses the identified limitation or extends the logical system's capabilities.

(2) **Injection:** $O$ injects $p$ into $P$, expanding the proposition space and providing new material for logical systems to operate upon.

(3) **Integration:** Logical systems $s \in S$ can now utilize $p$ within their axioms, inference rules, or proofs, effectively integrating the newly introduced proposition into the logical framework.

Building upon the analysis of the space of logical systems $S$ and the space of propositions $P$, we introduce an advanced concept where observers within the hierarchy $\{O_n\}$ can function as **virtual injectors**. This mechanism allows observers to dynamically introduce new propositions into $P$, thereby addressing the limitations and incompleteness inherent in formal logical systems as described by Gödel's Incompleteness Theorems.

### 13.1. Observers as Virtual Injectors.

**Definition 13.1 (OVI).** An **Observer Virtual Injector** $O$ is an observer within the hierarchy $\{O_n\}$ that possesses the capability to introduce new propositions $p \in P$ into the proposition space $P$. This injection operates under specific rules or mechanisms that govern how and when propositions are added, ensuring **relative consistency** and coherence within the logical framework.

### 13.2. Formalizing the Injection Process.

**Definition 13.2 (Injection Function).** An injection function $\iota: O \to P$ where each observer $O_n \in \{O_n\}$ can map to a new proposition $p_n \in P$. The function operates as:

$$ \iota(O_n) = p_n $$

ensuring that $p_n$ is consistent with existing propositions in $P$.

**Theorem 13.3 (Properties of the Injection Function).** The injection function $\iota$ possesses the following properties:

(1) **Consistency Preservation:** For each $p_n = \iota(O_n)$, the addition of $p_n$ to $P$ maintains the consistency of the logical systems in $S$.

(2) **Non-Circularity:** The propositions introduced via $\iota$ do not create circular dependencies within $P$.
(3) Controlled Expansion: The injection process is governed by predefined rules or constraints to prevent arbitrary or contradictory proposition additions.

*Proof.* (1) **Consistency Preservation:** Observers can analyze the current state of *P* and generate propositions that extend the logical system without introducing inconsistencies. Each *p*<sub>*n*</sub> can be formulated based on existing axioms and inference rules, ensuring compatibility.

(2) **Non-Circularity:** The injection function $\iota$ can avoid introducing self-referential propositions (e.g., $p_n$ asserting the existence of $P$) by adhering to logical constraints during proposition generation.

(3) Controlled Expansion: The rules governing $\iota$ are defined, potentially involving meta-logical criteria or consensus among multiple observers to approve and validate new propositions before injection. $\square$

**13.3. Mechanism of Observer Injection.** The Observer Virtual Injector *O* interacts with *P* through the following steps:

(1) **Initiation:** Observer $O_n$ identifies a limitation or an undecidable proposition within the current logical system.

(2) **Proposition Generation:** $O_n$ formulates a new proposition $p_n$ aimed at addressing the identified limitation or extending the system's capabilities.

(3) **Injection:** Utilizing the injection function $\iota$, $O_n$ introduces $p_n$ into *P*, thereby expanding the proposition space.

(4) **Integration:** Logical systems $s \in S$ incorporate $p_n$ into their axioms, inference rules, or proofs, enhancing their operational scope.

**13.4. Implications for the Observer Hierarchy Framework.**

**Theorem 13.4 (Observer Hierarchy Enhancement).** *The ability of observers to function as virtual injectors {$O_n$} enhances a observer hierarchy's capacity to dynamically adapt to limitations within logical systems *S*, thereby extending the coverage and mitigating some effects of Gödel's Incompleteness Theorems.*

*Proof.* By enabling observers to introduce new propositions $p_n$ into $P$, an observer hierarchy can address specific undecidable propositions or extend the logical system's expressiveness. This dynamic adaptation allows the hierarchy to evolve alongside $S$, providing new avenues for provability and reducing the immediate impact of incompleteness by expanding the proposition space. $\square$

**Corollary 13.5 (Limitations Persist Despite Injection).** *While observers acting as virtual injectors {$O_n$} can mitigate certain limitations, the inherent vastness of *S* relative to *P* ensures that *incompleteness and undecidability persist, as the injection mechanism cannot fully encapsulate all logical systems within *S*.***

*Proof.* Given $|S| = 2^{2^{\aleph_0}}$ and $|P| = 2^{\aleph_0}$, no injection mechanism, even distributed among observers, can cover all logical systems. Therefore, while observers can extend $P$, they cannot eliminate the foundational limitations imposed by Gödel's Incompleteness Theorems. $\square$
**13.5. Ensuring Logical Consistency.** To maintain logical consistency during the injection process, the following measures may be implemented:

(1) **Validation Mechanism:** Before a proposition $p_n$ is injected into $P$, it undergoes a validation process to ensure it does not introduce contradictions or inconsistencies within existing logical systems.

(2) **Consensus Among Observers:** Introducing a consensus mechanism where multiple observers must agree on the validity and necessity of a new proposition before it is injected.

(3) **Meta-Logical Rules:** Define meta-logical rules that govern the injection process, providing a formal framework for how and when observers can introduce new propositions.

**13.6. Example Scenario.**

**Example 13.6 (Observer Injection in Action).** Consider a logical system $s_1 \in S$ that encounters an undecidable proposition $p_u$ within $P$. Observer $O_1$ identifies the limitation and formulates a new proposition $p_n$ to address it:

$p_n$: "There exists a proposition in $P$ that resolves $p_u$."

Utilizing the injection function $\iota$, $O_1$ introduces $p_n$ into $P$:

$$ \iota(O_1) = p_n \implies P = P \cup \{p_n\} $$

Logical system $s_1$ can now incorporate $p_n$ into its axioms or inference rules, thereby extending its capacity to resolve or analyze $p_u$.

**13.7. Philosophical and Practical Considerations.** Philosophical Implications:

- **Epistemological Expansions:** Observers as virtual injectors represent an **epistemological expansion**, allowing the logical framework to grow beyond its initial axiomatic boundaries.

- **Possible Autonomy of Observers:** Granting observers the authority to introduce propositions challenges the **objectivity** and **autonomy** within formal systems, by introducing elements of **agency** and or **intervention**.

**Practical Considerations:**

- **Complexity of implementation:** Designing observers with the capability to act as virtual injectors introduces **complexities** in the logical framework, necessitating robust mechanisms to manage injection processes.

- **Risk of Inconsistency:** Without stringent controls, the injection process risks introducing **inconsistencies**, undermining the reliability of the logical systems.

*Note: The concept of observers as virtual injectors significantly compli-cates the observer hierarchy framework. By enabling dynamic and controlled ex-pansion of $P$. This mechanism while it offers a pathway to address limitations imposed by Gödel's Incompleteness Theorems, it also allowing logical systems $S$ to evolve in response to encountered undecidability; greatly increasing The vastness of $S$ relative to $P$ ensuring that limitations persist, reinforcing the necessity for continued exploration of logical frameworks and consistency-preserving mechanisms.*

**Theorem 13.7 (Gödel's Incompleteness in Extended Hierarchy).** (1) When referencing the Second Countable Space $P = \Sigma^*$: In this framework, each logical system $S_\kappa$ associated with observer $O_\kappa$ is subject to Gödel's First
*Incompleteness Theorem*, asserting that $S_κ$ cannot be both complete and consistent if it is sufficiently expressive.

(2) *Uncountable Space P = Σ*<sup>*∞*</sup> ∪ Σ<sup>ω</sup> ∪ Σ<sup>ω+1</sup> ∪ ...: *The introduction of propositions indexed by higher ordinals amplifies the scope of undecidability. Each observer O<sub>κ</sub> managing higher ordinals still cannot achieve completeness, as new undecidable propositions emerge within each extended logical system S<sub>κ</sub>.*

*Proof.* (1) Second Countable Space: Gödel's First Incompleteness Theorem applies to any consistent, effectively axiomatizable, and sufficiently expressive formal system. Each $S_κ$ falls under this category, hence containing undecidable propositions.

(2) Uncountable Space: The addition of higher ordinal-indexed propositions ensures that the set of propositions remains uncountable. Despite each $O_κ$ handling a countable subset $E_κ$, the overall hierarchy cannot eliminate decidability, as each $S_κ$ introduces new limitations consistent with Gödel's theorems.

<figure><img src="image_3.png" /></figure>

**Proposition 13.8 (Observer Hierarchy's Perpetual Incompleteness).** The observer hierarchy {$O_κ$}, even when extended to handle higher cardinalities through infinite logic, cannot achieve complete coverage of the space of propositions P due to the uncountable nature of P and the inherent limitations imposed by Gödel's Incompleteness Theorems.

*Proof.* Each observer $O_κ$ handles a countable subset $E_κ$, and while the uncountable hierarchy {$O_κ$} aims to cover P, Gödel's Incompleteness Theorems ensure that no single logical system $S_κ$ can be both complete and consistent. Therefore, as the hierarchy expands, new undecidable propositions continually emerge, preventing the attainment of complete coverage. ☐

<figure><img src="image_4.png" /></figure>

13.8. **Collective Capacity of Interacting Observers.**

**Definition 13.9 (Interacting Observer).** An **interacting observer** $O_i$ is defined as an observer with a specific observation set $E_i \subseteq P$, where $E_i$ consists of propositions within a particular operational domain. Each interacting observer is capable of perceiving and proving propositions within its defined capacity but is limited by the inherent cardinality of $E_i$, typically countable ($|E_i| = \aleph_0$).

**Definition 13.10 (Network of Interacting Observers).** A **network of interacting observers** is a collection {$O_{i1}, O_{i2}, \dots, O_{in}$} where each $O_{ij}$ is an interacting observer. The interaction is facilitated through a communication protocol or shared logical framework, enabling the network to collectively perceive and prove propositions beyond the capacity of any individual observer.

**Proposition 13.11.** *The network of interacting observers {$O_{i1}, O_{i2}, \dots, O_{in}$} can collectively conceptualize mathematical truths operating on a higher cardinality than any individual observer within the network, provided that the interactions allow for the integration of their observation sets and logical systems.*

*Proof.* Each interacting observer $O_{ij}$ has an observation set $E_{ij} \subseteq P$, with $|E_{ij}| = \aleph_0$. Individually, each observer is limited to handling propositions within their specific operational domain.
When these interacting observers form a network, their combined observation set becomes:

$$E_{\text{network}} = \bigcup_{j=1}^{n} E_{ij}$$

Assuming that the communication protocol facilitates the aggregation of observation sets and the synchronization of their logical systems, $E_{\text{network}}$ can encompass a higher cardinality set of propositions. For instance, if each $E_{ij}$ addresses distinct operational domains (e.g., arithmetic, algebra, calculus), their union can cover a broader and more complex spectrum of mathematical truths.

Furthermore, through collaborative reasoning and shared logical frameworks, the network can handle propositions that span multiple domains, thereby accessing higher-order mathematical truths that exceed the capacity of any single interacting observer.

Thus, the network transcends the individual limitations of its constituent interacting observers, enabling the conceptualization and proof of mathematical propositions of higher cardinality. □

**Example 13.12.** Consider a set of interacting observers {$O_{i1}, O_{i2}, O_{i3}$}, each specialized in different areas of operational propositions for example: arithmetic ($O_{i1}$), algebra ($O_{i2}$), and calculus ($O_{i3}$). Individually, each observer can only handle propositions and their logical systems within their specialization. However, when interacting within a network, they can collaboratively address propositions and access logical systems that span multiple domains, such as the consistency of certain algebraic structures within calculus. This collaborative approach enables them to handle a higher cardinality that none could individually.

**Theorem 13.13.** *Even with a network of interacting observers capable of conceptualizing higher cardinality propositions, there exist propositions in P that remain undecidable within the referenced network, thereby preserving the inherent limitations imposed by Gödel's Incompleteness Theorems.*

*Proof.* By Proposition 13.11, a network of interacting observers can collectively handle a higher cardinality of propositions than any individual observer. However, Gödel's First Incompleteness Theorem asserts that any consistent, sufficiently expressive formal system contains true propositions that are unprovable within that system.

Applying this to the network, even though the collective observation set $E_{\text{network}}$ is larger, the network itself constitutes a formal system with its own set of axioms and inference rules. Consequently, there will exist propositions within P that are true but remain unprovable within $E_{\text{network}}$, ensuring that undecidability persists.

Thus, the network does not eliminate undecidability but merely shifts its scope, maintaining the intrinsic limitations identified by Gödel. □

**Remark 13.14.** Theorem 13.13 highlights that while networks of interacting observers can enhance the collective capacity to handle complex mathematical truths, they do not circumvent the fundamental limitations imposed by Gödel's Incompleteness Theorems. Undecidable propositions remain an intrinsic feature of formal logical systems, regardless of their hierarchical or networked structures.
## 13.9. Equivalence of Observation Networks and Higher Observers.

**Definition 13.15 (Higher Observer).** A higher observer $O_κ$ is an idealized observer capable of perceiving and proving all propositions within its referenced countable space within $P$, including those of infinite length. This observer operates within an extended logical system $S_κ$ that integrates the capacities of multiple interacting observers, effectively transcending the limitations of any individual observer.

**Definition 13.16 (Observation Network).** An observation network $\mathcal{N}$ is defined as a collection of interacting observers $\{O_{i1}, O_{i2}, \dots, O_{in}\}$ where each $O_{ij}$ is an interacting observer with its own observation set $E_{ij} \subseteq P$. The network operates through a shared communication protocol or logical framework, enabling the collective handling of propositions across different domains.

**Theorem 13.17. Equivalence of Observation Networks and Higher Observers:** An observation network $\mathcal{N} = \{O_{i1}, O_{i2}, \dots, O_{in}\}$ is equivalent to a higher observer $O_κ$. Specifically, the collective perception of the network $\mathcal{N}$ is equal to the perception of $O_κ$ when the network's combined observation sets cover the entirety of a referenced countable set within $P$ relating to some cardinality:

$$ \bigcup_{j=1}^{n} E_{ij} = P_{o} = P_{κ} $$

*note: $P_o$ is a countable space in reference to a the observation within the space of $P$*

*Proof.* To establish the equivalence between an observation network $\mathcal{N}$ and a higher observer $O_κ$, we proceed as follows:

1. **Alignment of previous definitions:** - The higher observer $O_κ$ has the observation set $E_κ = P_o$, capable of perceiving all propositions within the referenced countable set within space $P$. - The observation network $\mathcal{N}$ comprises multiple interacting observers $\{O_{i1}, O_{i2}, \dots, O_{in}\}$, each with observation sets $E_{ij} \subseteq P_o$.

2. **Collective Observation Set:** - The collective observation set of the network is:

$$ E_{\mathcal{N}} = \bigcup_{j=1}^{n} E_{ij} $$

- By the theorem's premise, $E_κ = P_o$, meaning the network collectively perceives all propositions beyond each of its members.

3. **Logical System Integration:** - The network $\mathcal{N}$ operates under a idealized combined logical system $S_κ$ that integrates the logical frameworks $S_{ij}$ of each observer $O_{ij}$. - This integration means the network can prove any proposition that any individual observer $O_{ij}$ can prove within their respective systems.

4. **Establishing Equivalence:** - Since $E_κ = P_o$, the network $\mathcal{N}$ can perceive and prove all propositions that $O_κ$ can. - Conversely, the higher observer $O_κ$ can simulate the network $\mathcal{N}$ by encapsulating the combined logical capabilities of all $O_{ij}$.

5. **Conclusion:** - Therefore, the observation network $\mathcal{N}$ and the higher observer $O_κ$ are equivalent in terms of their collective perception and provability within the space $P$.

□
**Example 13.18.** Consider an observation network $\mathcal{N} = \{O_{i1}, O_{i2}, O_{i3}, O_{i4}\}$ composed of four interacting observers:

* $O_{i1}$: Specializes in arithmetic propositions, with $E_{i1} = \{\text{ie. addition, subtraction, etc.}\}$.

* $O_{i2}$: Focuses on algebraic propositions, with $E_{i2} = \{\text{ie. algebraic structures, operations, etc.}\}$.

* $O_{i3}$: Deals with calculus propositions, with $E_{i3} = \{\text{ie. differential and integral calculus.}\}$.

* $O_{i4}$: Deals with philosophical propositions, with $E_{i4} = \{\text{ie. questions of the truth value of statements.}\}$.

Assume that the combined observation sets cover all mathematical propositions within $P$:

$$E_{\mathcal{N}} = E_{i1} \cup E_{i2} \cup E_{i3} \cup E_{i4} = P_o \subset P$$

Under these conditions, the network $\mathcal{N}$ can collectively perceive and prove any proposition within $P_o$, effectively functioning as the higher observer of this set $O_\kappa$. This demonstrates that the network's collective capacity matches that of an individual higher observer relative to the individual, establishing their equivalence as per Theorem 13.17.

*Foundation for Advanced Logical Systems:*

This equivalence can set a basis for developing more sophisticated logical systems that leverage networks of interacting observers. Future research can explore how these networks interact with alternative and or adaptive logical frameworks to manage undecidability and incompleteness.

## 14. DISCUSSION

### 14.1. Limitations of Infinite Extensions.
The idea of an infinite sequence of observers or logical systems suggests that we can progressively resolve undecidable propositions. However, through Gödel's Incompleteness Theorems undecidability is established to be inherent within a logical system and cannot be fully eliminated through such extensions. This remains true regardless of the scale or complexity of the observer.

### 14.2. The Role of Observation Across Scales.
The ability of the Observer—whether natural or artificial—to conceptualize propositions beyond the scope of the logical systems it constructs does not equate to the formal provability of the truths within those systems. Observers at different scales interact with propositions relevant to their domains. Therefore, to represent its own observation entirely as a set of logical statements would necessitate a system that transcends the observation itself. For instance:

* **Atomic Observers:** Limited to propositions governed by physical laws, unable to perceive or prove abstract mathematical truths.

* **Biological Observers:** Constrained by biological and neurological frameworks, facing challenges in formalizing certain cognitive and or abstract propositions.

* **Human Observers:** While capable of complex reasoning, humans are still subject to the inherent limitations of formal logical systems created, as evidenced by Gödelian undecidability.

* **Artificial Observers:** AI systems, despite speed of computations cannot transcend the fundamental limitations imposed by the logical systems they are built upon.
**14.3. Decentralized Observers and Their Implications.** Decentralized observers {$O_{n+1}, O_{n+2}, \dots$} can introduce a multifaceted approach to addressing undecidable propositions. By having multiple observers each with distinct, non-overlapping observable sets, the overall net framework can allow for parallel and specialized methods of proving propositions. This decentralization can enhance the overall capacity to handle a wider array of propositions, leveraging the strengths of different logical systems {$S_n$}.

**Possible advantages of localized Decentralization:**

* **Parallel Processing:** Multiple observers can simultaneously address different propositions, potentially increasing the efficiency of the system.

* **Specialization** Logical systems {$S_n$} can be tailored to specific domains or types of propositions, enhancing their effectiveness.

* **Resilience** Decentralized systems are generally more resilient to failures, as the compromise or limitation of one observer does not directly impact the others.

**Challenges and Considerations:**

* **Consistency loops:** Ensuring that the multiple logical systems {$S_n$} remain consistent with each other to prevent contradictions.

* **Inter-System Communication:** Facilitating the exchange of information or propositions between decentralized observers may require bridging logical gaps.

* **Scalability** Mapping an ever-increasing number of observers.

**14.4. Implications of Infinite-Length Propositions on Gödel's Theorems.** Infinite-length propositions expand the landscape of $P$ beyond the confines of countable enumeration and or alphabetization. By embracing infinite logical statements, the set $P$ becomes uncountable, as demonstrated by Cantor's diagonal argument. This expansion necessitates an extension beyond classical formal systems into frameworks that accommodate infinite constructs, for example infinite logic or modal logic.

However, even with the inclusion of infinite-length propositions, Gödel's Incompleteness Theorems retain their potency. The uncountability of $P$ ensures that no formal system, even when augmented with a limit observer $O_\infty$, can encapsulate all truths, there will invariably exist propositions that remain unprovable within any given system. This reinforces the inherent limitations identified by Gödel, highlighting the inescapable nature of undecidability in formal logical frameworks.

*Note:* Observations refers to the inherent ability of the observer to conceptualize propositional truths.

**14.5. Gödel's Theorems.** Even with an exhaustive observation set $S_\infty = X$, Gödel's Incompleteness Theorem remains a fundamental theorem. There exist true propositions that are unprovable within any consistent, effectively axiomatizable formal system, including those associated with our observers.

## 15. CONCLUSION

By constructing a hierarchy of logical systems {$s_n$} and observers {$O_n$}, this paper expands the set of provable or observed propositions beyond any finite observer's capability. Despite these infinite extensions or observations, the limitations
imposed by Gödel's Incompleteness Theorems and undecidability remain an intrinsic feature of formal systems.

Introducing varying magnitudes of infinite-length propositions renders the space of all propositions *P* uncountable, further reinforcing the impossibility of any finite observer encompassing the entirety of *P*. This illustrates that undecidability transcends specific logical systems and permeates throughout various scales of existence.

**Future Work:** Future research could explore the integration of non-standard logical frameworks, such as infinitary logic or modal logic, to better represent infinite-length propositions. Additionally, developing more sophisticated collaborative mechanisms between decentralized observers could enhance the framework's capacity to handle complex, interdisciplinary propositions.

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