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LITE: Dynamic Self-Reference in Peano Arithmetic

Abstract

The Look In The Eye (LITE) framework demonstrates how a system within Peano Arithmetic (PA) can exhibit a dynamic, evolving self-referential structure. By leveraging strong self-reference techniques (via the Recursion Theorem) and bounded proof searches (formalized within PA), LITE reveals a feedback mechanism in which an arithmetical function updates its values upon discovering new proofs or refutations about its own behavior. This iterative mechanism challenges the notion that formal arithmetic is merely static, showing instead that PA can support unfolding, adaptive processes guided by proof discoveries. LITE thus broadens our perspective on definability, self-reference, and how “mathematics can respond to itself” through a carefully orchestrated interplay of function definitions and proof predicates.

1. Introduction: Understanding LITE

1.1. The Traditional View: Formal Arithmetic as Static

Historically, many areas of mathematics—such as dynamical systems, ergodic theory, and chaos theory—have dealt extensively with evolving or changing phenomena. However, within the framework of formal arithmetic (like Peano Arithmetic), the standard approach has often been seen as “static”: once definitions, axioms, and theorems are established, their logical status does not spontaneously shift in response to newly discovered proofs about those very definitions. In other words, arithmetic does not typically include a built-in mechanism that causes the system itself to “recompute” or “update” function values based on fresh derivations.

By focusing on dynamic self-reference specifically in Peano Arithmetic, the LITE framework aims to show that we can incorporate a kind of “feedback loop” directly into the definition of an arithmetical function—without adding new axioms or stepping outside classical PA. This approach neither denies nor contradicts the broader history of mathematics involving evolving processes; rather, it highlights how even a traditionally “static” formal theory can be made to adapt its outputs when short proofs about its own behavior appear.

1.2. A Fresh Angle on Self-Reference

Classical Gödelian self-reference focuses on the existence of certain sentences (“I am not provable in PA”) whose truth and provability statuses intertwine in profound ways. LITE extends this tradition by allowing infinitely many self-referential statements—one for each natural number input—and by making the function’s value at each step sensitive to whether these statements (or their negations) are provably derivable within specific bounded lengths:

In bridging the power of the Recursion Theorem with bounded searches for proofs, LITE creates a genuinely adaptive structure. Instead of a single diagonal that remains inert, LITE’s approach is to generate a family of statements whose provability shapes the next stage of f’s definition.

1.3. What This Means for Arithmetic

LITE reveals that arithmetic can “listen” to whether a statement about itself is provable (or refutable) within specific short-proof bounds, then incorporate that discovery into its own next step. This viewpoint transforms our usual conception of arithmetic:

The sections below lay out how LITE is formally constructed, the mathematical details that ensure its consistency, and why it broadens the frontier of arithmetical definability and self-reference.

Beyond Static Self-Reference

Classical Approach: Self-referential statements often revolve around fixed claims, like “I am not provable.”

LITE’s Perspective: Multiple, input-indexed statements feed back into the value of a function within arithmetic, creating an ongoing interplay between proof discovery and numeric outputs.

Mathematical Foundation

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2. Mathematical Foundation

2.1. Peano Arithmetic and Gödel Encoding

LITE rests on classical Peano Arithmetic (PA). Its language comprises symbols for 0, successor (S(⋅)), addition, multiplication, and ordering, along with the usual logical machinery. Through Gödel numbering, every finite string (be it a formula or proof) is encoded as a natural number:

  1. Gödel Coding: A bijection ⟨·⟩: Σ* → ℕ assigns unique codes to expressions, enabling arithmetic to speak about syntactic objects numerically.
  2. Provability Predicate Prf(p, ⌈ψ⌉): Within PA, this relation asserts that p is the code of a valid proof of ψ. This is key for referencing “whether ψ can be proven in PA.”

With this apparatus, statements in arithmetic can self-reflect: they can contain references to codes of formulae and proofs, including statements about the function f we aim to define.

2.2. Bounded Proof Search via g(n)

Central to LITE is the notion of checking for a proof within a specific size bound, controlled by a fast-growing function g(n):

∃p ≤ g(n) Prf(p, ⌈ψ⌉).

Although potentially huge, g(n) remains finite for each n, so there is no paradox in “looking for all proofs up to length g(n).” If none exist, the search terminates without success; if a short proof does exist, it is eventually identified. This bounding technique makes the entire process definable inside PA and ensures that checking these proofs does not require stepping outside the system.

By coupling g(n) with the function we define, LITE effectively “asks” at each stage whether certain statements can be proven (or refuted) under that growing size limit. This local proof search is the engine of the dynamic update process.

2.3. The Recursion Theorem

An essential ingredient is the Recursion Theorem, which guarantees that a function can validly reference its own code without inconsistency. In simpler terms, it states that for any total computable operator Ψ(α,n), there is a code β such that the partial computable function ϕβ satisfies:

ϕβ(n) = Ψ(β, n).

By interpreting β as the “code of f,” we obtain a legitimate self-referential definition: f can talk about f. In LITE, this theorem underpins how ϕ(n) can explicitly mention f(n), enabling the dynamic interplay between proof discovery and numeric values.

3. Dynamic Self-Reference in LITE

3.1. Main Construction

At the heart of LITE is a function f: ℕ → ℕ defined via: 

f(n) =
  { n + H1(n), if ∃p ≤ g(n) Prf(p, ⌈ϕ(n)⌉);
    n + H2(n), if ∃p ≤ g(n) Prf(p, ⌈¬ϕ(n)⌉);
    n + 1,               otherwise. }

Here, ϕ(n) is a statement about f, referencing its own Gödel code. The functions H1(n) and H2(n) are chosen to ensure a noticeable “jump” if a proof is found. If neither ϕ(n) nor its negation is proven within size g(n), then f(n) = n + 1 by default.

Because of the Recursion Theorem, there is no contradiction in letting ϕ(n) refer to “the value of f(n),” even though f is simultaneously being defined by this piecewise rule. This yields a consistent, total function f inside PA that “listens” for short proofs about itself.

3.2. Shifting Values Based on Proof Discovery

The novelty lies in how f(n) is not predetermined by a simple recursive or closed-form expression. Instead, it reacts if a short proof of ϕ(n) or ¬ϕ(n) emerges within g(n)-bounded length. If it does, LITE “steers” the function to n + H1(n) or n + H2(n). Otherwise, it leaves f(n) at n + 1.

Because n + H1(n) or n + H2(n) can be relatively large jumps, subsequent statements ϕ(n+1), ϕ(n+2), etc., may become provable (or refutable) in new ways. Over time, f thus evolves under the influence of the system’s ability to locate relevant derivations, effectively “recording” the discovery of short proofs in its numeric output.

3.3. Feedback Over the Natural Numbers

By design, each n extends the conversation: “Is ϕ(n) provable up to length g(n)? Is ¬ϕ(n) provable up to length g(n)?” This ongoing check seeds an iterative feedback loop throughout the natural numbers. No single instance “locks in” the entire behavior forever; instead, each step can bring fresh shifts based on newly accessible derivations. This iterative nature is the cornerstone of LITE’s dynamic self-reference.

4. Mathematical Properties

4.1. Well-Defined and Total

4.2. Intrinsic Dynamism

Classic recursive functions remain fixed after definition, but f(n) here can shift in response to the discovery of short proofs. This “listening” for new derivations brings an element of adaptivity not typically associated with straightforward recursive definitions.

4.3. Rich Structure

Each point n can drastically affect future points n+1, n+2, etc., by changing the form or provability of statements involving f’s subsequent values. This can lead to sequences that remain stable for many steps, then undergo large jumps once a relevant proof or disproof appears.

5. Theoretical Significance

5.1. Extending Gödel’s Legacy

Gödel’s incompleteness opened the door to self-reference in arithmetic. LITE pushes this further, showcasing how an entire collection of self-referential statements—one per input—can collectively govern an arithmetical function’s evolving values.

5.2. Bridging Proof Discovery and Function Values

5.3. Deeper View of Arithmetic’s Expressive Power

By embedding a “self-rewriting” protocol into a single function, LITE underscores that arithmetic can exhibit forms of feedback and adaptation once we leverage self-reference carefully. This broadens conceptions of what “arithmetical definability” can achieve.

Mathematical Analysis

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6. Mathematical Analysis

6.1. Iterative Construction

  1. For each n: Check for proofs of ϕ(n) or ¬ϕ(n) up to length g(n).
  2. Branching: If one is found, pick n + H1(n) or n + H2(n). Otherwise, default to n + 1.
  3. Cascade: That output may alter the future geometry of ϕ(n+1) or beyond, letting new proofs arise down the line.

6.2. Potentially Complex Behavior

Although f is total, its path can exhibit surprising turns when proofs (or disproofs) become accessible. Long periods of quiescence may give way to sudden large changes if certain short proofs about f appear.

6.3. Governed by Proof Bounds

All stages remain anchored in finite proof searches up to g(n). As n grows, g(n) grows, allowing the system to glean new insights from previously inaccessible derivations. This interplay of “growing bounding function” and “iterative self-reference” is the essence of LITE’s dynamic flavor.

6.4. Concrete Example

Illustrative Setup: To make LITE more tangible, let us consider a simplified instance:

Step-by-Step Evolution (Hypothetical):

  1. n = 0: We check all proofs up to length g(0) = 2^(0+1) = 2. Likely no short proof can establish or refute ϕ(0) in just 2 steps, so f(0) = 0 + 1 = 1.
  2. n = 1: Now g(1) = 2^(1+1) = 4, allowing a bit more space for proofs. Still, if no proof or disproof of ϕ(1) is found within 4 steps, we get f(1) = 1 + 1 = 2.
  3. n = 2: g(2) = 8. Suppose ϕ(2) is provable in 8 steps, effectively establishing "f(2) ≥ 2 + 10." Then by definition, f(2) = 2 + 10 = 12. This "jump" might influence the subsequent form of ϕ(3).
  4. n = 3, 4, ... As n grows, g(n) grows exponentially. If at any point ¬ϕ(n) is provable, then we'd define f(n) = n + 20. If neither is provable, we default to n + 1.

This toy version highlights how bounded proof searches can yield a function that “listens” to whether certain statements about its own values are provable and then adapts accordingly. In a full formal treatment, ϕ(n) would be precisely Gödel-encoded to reference f’s own definition, ensuring self-reference by the Recursion Theorem.

7. Formal Structure

7.1. Recursion Theorem Implementation

To ensure ϕ(n) truly references f, one typically constructs a computable operator Ψ(α, n) that uses α as the code of a function. The Recursion Theorem guarantees a fixed point β with ϕβ(n) = Ψ(β, n). Interpreting β as the code of f completes the self-referential loop.

In essence, this means our function f can literally “read” its own description and incorporate references to its own values in the statement ϕ(n). Crucially, this is performed within PA, yielding a self-consistent definition that does not overstep the theory’s expressive capabilities.

7.2. The Proof Predicate

The definability of Prf(p, ⌈ψ⌉) in PA makes it possible for f(n) to incorporate checks of whether ψ is proven by a code p ≤ g(n). This tight coupling of code, proof, and bounding function ensures the entire construction is an internal process in arithmetic.

7.3. Harmony of Components

Combined, these yield LITE’s characteristic feedback loop: each numeric argument n spurs an inspection of short proofs about f(n), and the outcome decides the numeric output.

7.4. Core Theorem and Complete Proof

Theorem (Totality and Consistency): Within PA, the function f defined by LITE is total, and no contradiction arises from letting ϕ(n) reference f.

Complete Proof:

  1. Definition of the Operator Ψ:
    We first specify a total computable operator Ψ(α, n) that carries out the following procedure when given a code α: it interprets α as the description of a function φα, and then defines a value for Ψ(α, n) via the piecewise rule: 
            if (∃p ≤ g(n) such that Prf(p, ⌈"ϕ(n) under code α"⌉)) then output n + H₁(n)
        else if (∃p ≤ g(n) such that Prf(p, ⌈¬"ϕ(n) under code α"⌉)) then output n + H₂(n)
        else output n + 1
    where "ϕ(n) under code α" references the would-be value of φα(n). The existence of such a total computable operator Ψ follows from standard methods of representing finite searches and proof predicates in arithmetic.
  2. Application of the Recursion Theorem:
    By the Recursion Theorem (Kleene's Recursion Theorem), there exists a code β such that the partial computable function φβ satisfies:

    φβ(n) = Ψ(β, n).

    In other words, φβ is the self-referential fixed point: when β is interpreted as "the code for φβ," the definition of φβ(n) precisely mirrors the piecewise behavior of the operator Ψ.
  3. Identification of β with f:
    We then interpret β as the code of our desired function f. By construction:

    f(n) = φβ(n) = Ψ(β,n).

    Concretely, this means f can refer to its own code and check whether proofs of "ϕ(n)" or "¬ϕ(n)" exist within the bound g(n).
  4. Finite Proof Search Ensures a Well-Defined Output:
    For each n, verifying whether a proof of length ≤ g(n) exists is a finite (albeit possibly large) procedure. One of the following must hold:
    • A proof of "ϕ(n)" is found in ≤ g(n) steps.
    • A proof of "¬ϕ(n)" is found in ≤ g(n) steps.
    • Neither is found.
    This directly matches one of the branches in the piecewise definition of f(n). Thus, f(n) is always assigned a specific natural number, guaranteeing totality.
  5. Absence of Contradiction:
    A typical concern with self-reference is the possibility of creating a "Liar paradox" scenario. However, in this construction:
    • Each statement "ϕ(n)" is bounded by the search limit g(n). We do not assert unconditionally that "ϕ(n)" is provable or disprovable; instead, we let the existence (or non-existence) of a short proof decide f(n).
    • The Recursion Theorem framework ensures that referencing the code β of f is done consistently, without leading to an unformalizable or contradictory self-reference.
    • Since the definition relies on a finite (though unbounded in n) search, there is no infinite regress and no direct statement of self-contradiction. The function's value is simply determined by the outcome of these local checks.
    Consequently, no contradiction akin to "This statement is false" arises, preserving consistency within PA.
  6. Conclusion (Totality and Consistency):
    Putting these points together:
    • The search for proofs up to size g(n) is finite for each n, so exactly one of the three branches in the piecewise definition applies unambiguously.
    • The Recursion Theorem guarantees the self-referential code β yields a well-defined f that can mention its own values via φβ. No paradox emerges from this self-reference.
    • Hence, f is total (it has a well-defined natural number output for every n) and does not engender contradiction in PA.
    Therefore, f as defined by LITE is both total and consistent with PA, completing the proof.
BROADER PERSPECTIVE

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8. Broader Perspective

8.1. From Single Statement to Infinite Family

Traditional self-reference often hinges on a single statement (Gödel’s “I am not provable”). LITE extends this to a structured infinite family \(\{\varphi(n)\}\), each referencing a different input n, forging a more expansive notion of self-reference in arithmetic.

8.2. Iterative Updates

8.3. New Interpretations of Self-Reference

LITE demonstrates that self-reference need not be limited to self-negating or self-asserting statements. Instead, it becomes a tool for building adaptive structures: a function that “consults” the presence or absence of certain proofs about itself and reacts accordingly.

8.4. Possible Generalizations

8.5. Comparison with Classic Self-Reference Frameworks

Framework Style of Reference Static vs. Dynamic Typical Outcome
Gödel’s Single Sentence “I am not provable.” Static once formulated Establishes incompleteness
Rosser Variants Refines Gödel’s approach using disjunctions Still static Avoid certain assumptions; simpler formal incompleteness
Reflection Principles Global additions like “If φ is provable, then φ” Often extends the theory in larger steps Does not typically yield local iterative changes in one function
LITE Infinite family ϕ(n); bounded proof checks at each step Dynamic, indefinite updates Self-referential function evolves with newly found proofs

9. Conclusion

The LITE framework underscores a novel view of self-reference in which an arithmetical function f(n) can shift its values based on whether short proofs concerning ϕ(n) or its negation appear within a bound g(n). In contrast to single-shot Gödelian statements, LITE orchestrates a sequence of evolving self-referential checks, producing an unfolding chain of potential updates across the natural numbers. Each time new derivations are found, f is redefined accordingly, weaving a feedback loop into the very fabric of arithmetic.

By relying solely on standard tools—Gödel coding, the definable proof predicate Prf(p, ⌈ψ⌉), a fast-growing bound g(n), and the Recursion Theorem—this construction remains entirely within classical PA. It refutes the notion that arithmetic must always be a static discipline; instead, LITE reveals that PA can exhibit “liveliness,” adapting to new proof discoveries in an ongoing manner. The ramifications touch not just on self-reference theory, but on how we conceive the possible internal transformations a formal system can undergo. The boundary between “the system that states truths” and “the system that adapts with those truths” is shown to be far more flexible than typically assumed.

Ultimately, LITE pushes us to recognize that self-reference in arithmetic need not be static or singular. With careful constraints and bounding, we can build a function that truly “observes” its own proofs and adjusts itself accordingly—marking a major step forward in understanding the dynamic potentials that lie within arithmetic itself.

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