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A Fibonacci-Based Gödel Numbering: $Δ_0$ Semantics Without Exponentiation

Milan Rosko

TL;DR

The paper addresses the challenge of reproducing Gödel-style self-reference and incompleteness within a strictly bounded, additive framework. It introduces Zeckendorf-based representations and a Typed Carryless Pairing that encode syntax and proofs entirely on finite carriers, enabling Δ0-definable encoding, substitution, and provability without multiplication or unbounded search. The core results include a fully Δ0-definable diagonal function and an additive diagonal lemma, yielding a Gödelian incompleteness within a predicative setting and a detailed comparison to traditional multiplicative encodings. The approach provides a principled, locally bounded foundation for metamathematics with potential applications in data encoding and bounded computation.

Abstract

This paper develops a fully additive account of Incompleteness based on finite supports of Fibonacci indices and Zeckendorf representations. "Carryless Pairing" provides an injective, reversible encoding of tuples, with evaluation and inversion confined to finite index domains. Using this framework, we obtain $Δ_0$-definable encodings of terms, formulas, proofs, and a substitution operator, and we formalize the provability predicate entirely within bounded arithmetic. The Diagonal Lemma and Gödel's First Incompleteness Theorem are then recovered without multiplication or unbounded search. The resulting system isolates a structure sufficient for self-reference and is grounded in finite-support recursion.

A Fibonacci-Based Gödel Numbering: $Δ_0$ Semantics Without Exponentiation

TL;DR

The paper addresses the challenge of reproducing Gödel-style self-reference and incompleteness within a strictly bounded, additive framework. It introduces Zeckendorf-based representations and a Typed Carryless Pairing that encode syntax and proofs entirely on finite carriers, enabling Δ0-definable encoding, substitution, and provability without multiplication or unbounded search. The core results include a fully Δ0-definable diagonal function and an additive diagonal lemma, yielding a Gödelian incompleteness within a predicative setting and a detailed comparison to traditional multiplicative encodings. The approach provides a principled, locally bounded foundation for metamathematics with potential applications in data encoding and bounded computation.

Abstract

This paper develops a fully additive account of Incompleteness based on finite supports of Fibonacci indices and Zeckendorf representations. "Carryless Pairing" provides an injective, reversible encoding of tuples, with evaluation and inversion confined to finite index domains. Using this framework, we obtain -definable encodings of terms, formulas, proofs, and a substitution operator, and we formalize the provability predicate entirely within bounded arithmetic. The Diagonal Lemma and Gödel's First Incompleteness Theorem are then recovered without multiplication or unbounded search. The resulting system isolates a structure sufficient for self-reference and is grounded in finite-support recursion.

Paper Structure

This paper contains 34 sections, 26 theorems, 67 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Zeckendorf Arithmetic
  3. Fibonacci Basis and Growth Laws
  4. Typed Bounds: Length, Rank, and Delimiters
  5. Bounded Computation of Supports
  6. Carryless Pairing
  7. Finite Carriers and Typed Bands
  8. Definition of Typed Carryless Pairing
  9. Bounded Evaluation
  10. Typed Unpairing
  11. Algorithm
  12. Stability Properties
  13. Typed $k$-Ary Pairing
  14. Syntax Encoding
  15. Symbol Tags and Typed Payloads
  16. ...and 19 more sections

Key Result

Lemma 1.1

For each $x$, the carriers $\tau_x$ and $\varepsilon_x$ are finite and determined entirely by $r(x)$. Their bounds are fixed before evaluating $\pi_{\mathrm{CL}}^\tau(x,y)$, so no quantification over $\mathbb{N}$ is required during computation.

Figures (3)

  • Figure 1: Additive Zeckendorf Composition avoids positional carries and is invertible in $O(\log n)$ time by separating even and odd Fibonacci indices.
  • Figure 2: Gödel Numbering encodes syntactic structures into arithmetic form, while the inverse mapping exhibits superpolynomial complexity, reflecting the intrinsic compression of the encoding.
  • Figure 3: Bit-Interleaving generates the characteristic Z-Order, providing a semantic ordering of multidimensional data for efficient indexing.

Theorems & Definitions (54)

  • Lemma 1.1: Typed Finiteness
  • Definition 2.1: Fibonacci Sequence
  • Lemma 2.2: Fibonacci Growth
  • Theorem 2.3: Zeckendorf Decomposition
  • Definition 2.4: Zeckendorf Support
  • Remark
  • Definition 2.5: Zeckendorf Length, Rank, and Delimiter
  • Lemma 2.6: Monotonicity of Rank and Delimiter
  • Lemma 2.7: Greedy Decomposition by Bounded Recursion
  • Remark
  • ...and 44 more