Incompleteness theorems via Turing category (2412.14084v1)

Abstract: We give a reframing of Godel's first and second incompleteness theorems that applies even to some undefinable theories of arithmetic. The usual Hilbert-Bernays provability conditions and the diagonal lemma are replaced by a more direct diagonalization argument, from first principles, based in category theory and in a sense analogous to Cantor's original argument. To this end, we categorify the theory G\"odel encodings, which might be of independent interest. In our setup, the G\"odel sentence is computable explicitly by construction even for $\Sigma ^{0} _{2}$ theories (likely extending to $\Sigma ^{0} _{n}$). In an appendix, we study the relationship of our reframed second incompleteness theorem with arguments of Penrose.

• The paper introduces a novel categorification of Gödel encodings that constructs computable Gödel sentences for Σ⁰₂ theories without traditional provability conditions.
• The work employs Turing categories to define stable computability and decision maps, culminating in a stable analogue of Turing’s halting problem.
• The reframing of incompleteness theorems offers fresh insights into the limitations of formal systems and opens pathways for further exploration in mathematical logic and theoretical computer science.

A Reframing of Gödel's Incompleteness Theorems Through Category Theory

The paper "Incompleteness theorems via Turing category" by Yasha Savelyev presents a novel approach to Gödel's incompleteness theorems by leveraging the framework of category theory, particularly Turing categories, to extend these theorems to certain undefinable theories of arithmetic. This work represents a shift from traditional provability conditions towards more foundational, direct diagonalization arguments akin to Cantor's original methods. The use of category theory and this particular form of diagonalization serves to broaden the applicability of incompleteness theorems, providing insights into theories that are not definable within the typical structures dictated by Gödel's original formulations.

The primary contribution of this research lies in the categorification of Gödel encodings, which, as noted by the author, might possess independent interest beyond the immediate results obtained. The formal setup proposed allows for the construction of Gödel sentences that are explicitly computable by design, even for $\Sigma^0_2$ theories, and potentially extendable to $\Sigma^0_n$ theories. This work reinterprets the classic incompleteness results without requiring the conditions of Hilbert-Bernays provability or the diagonal lemma, which are traditionally cumbersome to apply outside of definable contexts.

Key Results

1. Stable Computability and Decision Maps: The paper introduces the concept of stable computability in the environment of category theory, which provides a framework for discussing the computability of sets and the decision processes involved. This involves a pioneering use of Turing categories to manage the logical constructs necessary for the main proofs.
2. Speculative Theories: The notion of speculative theories is introduced to explore the intersection of arithmetic theories and abstract languages, enabling the establishment of new forms of consistency. The operation $Spec$, as defined in the paper, transforms stably c.e. theories into speculative ones with retained consistency, illustrating how abstraction in logic can provide robust frameworks for understanding definability in complex systems.
3. The Stable Halting Problem: A stable analogue of Turing's halting problem is formulated, extending the classical problem to accommodate stable conditions. This formulation demonstrates the universal undecidability properties within the field of category theory, thereby reinforcing Gödel's original assertions about the limitations of formal systems.
4. Incompleteness Theorems for Stable Theories: By reframing the incompleteness theorems, the paper shows that even strong consistency within a theory cannot resolve the presence of undecidable propositions when interpreted through these abstract categorical constructs. The construction of the Gödel sentence here is tightly linked with the stable properties of Turing machines, ultimately resulting in a refined understanding of formal limitations in mathematical theories.

Implications and Future Directions

The implications of this research extend beyond merely reestablishing Gödel's results in a new context. By exploring the reframing of classical logical theorems in terms of category theory, new paths are opened for understanding complex mathematical and physical systems that reside outside classical computation and definability. Future developments may explore the ramifications of these methods in other branches of theoretical computer science and mathematical logic, potentially offering novel insights into unresolved theoretical questions across a variety of domains.

Further exploration into how these categorical frameworks relate to modern physics, perhaps in connection with theories of computation and consciousness as suggested in the appendix discussing Roger Penrose's ideas, might bring additional clarity to long-standing debates on the nature of computability in our universe.

In conclusion, this work provides a significant contribution to the field of mathematical logic, introducing categorical perspectives that could redefine our understanding of formal systems and their limitations. The methodological advancements presented invite further theoretical exploration that may significantly impact both mathematics and its applications in broader scientific contexts.