Gödel Agents: Self-Extending Systems

• Gödel Agents are self-referential computational systems that combine routine Turing computation with noncomputable oracle decisions for theory extension.
• They integrate deterministic processing with recursive self-improvement, enabling agents to manage paradoxes and ambiguous evidence.
• Empirical models, such as the Darwin and Huxley-Gödel Machines, illustrate practical self-modification, improved performance, and controlled safety measures.

Gödel Agents are self-referential or self-extending computational systems that operationalize Gödelian principles in theory extension, recursive self-improvement, symbol-handling, or graded epistemic reasoning. The concept originates from Gödel’s incompleteness phenomena and has evolved into concrete agent architectures with empirical and formal instantiations, notably in AI and logic, where agents are capable of self-modification, continual belief revision, or managing ambiguous and paradoxical information.

1. Theoretical Foundations: Incompleteness, Agency, and Self-Reference

Gödel’s incompleteness theorems establish that, for sufficiently expressive formal systems (e.g. Peano Arithmetic), there exist true statements unprovable within the system. The implication is that any extension of knowledge or explanation ultimately depends on agency—the insertion of new axioms, guesses, or explanatory choices not determined by the system’s pre-existing rules. In arithmetic, this is visible in the need to decide independent sentences; in physics, Myers & Madjid show that, for any given body of quantum evidence, the set of compatible explanations (density operators and POVMs producing the same statistics) is uncountably infinite, requiring an agent to choose among alternatives in a noncomputable fashion (Myers et al., 2018, Myers et al., 2018).

Formally, a Gödel Agent is thus characterized by the interplay between:

• Routine computation ("a-machine," Turing 1936): Deterministic symbol manipulation.
• Noncomputable guesswork ("oracle" or "choice" actions): Decisions not algorithmically determined, invoked at critical junctures for theory extension or interpretation.

This dual structure models scientists extending axioms, machines self-improving, and physical symbol-handling agents bridging the gap between evidence and explanation.

2. Formal Models and Logical Machinery for Gödel Agents

a. Theory-Extending Learners and Bot Belief Dynamics

A Gödel Agent can be formalized as a sequence of self-extending belief states or recursively enumerable theories, $T_0 \subseteq T_1 \subseteq \dots$, where each $T_{n+1}=T_n\cup\{\varphi_n\}$ and $\varphi_n$ is the least not-yet-provable but "true" formula at step $n$. The process continues transfinitely, producing limit theories that, while always incomplete, become pragmatically "complete" in the sense of capturing all observations encountered so far (Pavlovic et al., 2023). The fixpoint construction ensures that there exists a belief assignment consistent with any computable update policy.

• Testability vs. Unfalsifiability: Gödel Agents may reach self-fulfilling endpoints (belief fixpoints) that explain all observable data and thus cannot be empirically falsified—representing both the power and peril of Gödelian reasoning.
• Convergence: Under enumerability and fair update rules, the belief state sequence stabilizes at a countable ordinal stage, achieving weak completeness for the agent's lived experience.

b. Symbol-Handling Agents in Physics

Myers & Madjid develop a c-machine variant for symbol-handling agents: Turing machines with an oracle tape and a clock tape. The oracle tape introduces noncomputable information (guesses) at "choice states," while the clock tape provides an intrinsic temporal and symbolic record. A central result is the uncountability of quantum explanations for any finite evidence set, meaning that an agent’s theory choice is not determined by logic or data, but by agency itself—encoded as oracle interactions (Myers et al., 2018).

This framework also models communication and synchronization (e.g., atomic clock networks) via recorded symbol exchange, deducing all spatiotemporal structure from the frequencies and echo counts in agent tapes.

c. Paraconsistent Agents and Contradiction-Driven Recursion

The Gödel Mirror calculus provides a minimal mechanized architecture in which symbolic paradoxes are treated as control signals for recursive structural evolution, not as points of logical explosion. Here, a Gödel Agent maintains a state in the term algebra of the calculus, with deterministic reduction rules encoding paradox → encapsulate → reenter → node. Encountering a contradiction (such as "liar"-style self-reference) triggers controlled handling rather than collapse, enabling unbounded agent growth without global inconsistency (Chan, 16 Sep 2025).

3. Agent Architectures for Self-Improvement and Empirical Gödelization

a. Classical Gödel Machine and Modern Instantiations

The original Gödel machine architecture, due to Schmidhuber, is a universal problem solver that rewrites its own code whenever it finds a formal proof (within its own axiomatic system) that doing so increases expected utility. The requirements are:

• Axiom system $A$: Includes code, utility function $U$, and environment rules.
• Proof searcher: Systematic enumeration of proofs in $A$.
• Switching mechanism: Executes a self-rewrite upon proof of net utility gain.

Optimality is formally guaranteed, but intractable due to proof search complexity (Wang et al., 24 Oct 2025).

b. LLM-Driven Gödel Agents

Gödel Agents with LLM backbones replace formal proof search with LLM-driven heuristic reasoning. The LLM simultaneously decides which portions of the codebase to alter and synthesizes new code. Empirical validation substitutes for proof: proposed modifications are adopted only if they empirically increase utility (on held-out benchmarks). The overall update is

$(T_{t+1},\,I_{t+1}) \;=\; I_t(T_t,\;I_t,\;g,\;r_t)$

where $I_t$ represents the self-modification routine, which is itself modifiable (Yin et al., 2024).

c. Empirical Self-Improvement Loops: Darwin Gödel Machine

The Darwin Gödel Machine (DGM) maintains an archive of self-modifying agents (coding agents with FMs) and iteratively explores modifications through empirical benchmarking. The mutation operator is FM-driven: given code and logs, the FM proposes patches. Acceptance is granted only with empirical utility improvement ($\hat U(p') > \hat U(p)$), relaxing the formal proof obligation (Zhang et al., 29 May 2025).

Safety is enforced via sandboxing, traceability, and human oversight. The DGM achieves significant gains: on SWE-bench, pass@1 improves from 20% (initial) to 50%; on Polyglot, from 14.2% to 30.7%. The design enables open-ended, parallel exploration unavailable to purely human-designed agents.

d. Lineage-Aware Optimization: Huxley-Gödel Machine

The Huxley-Gödel Machine (HGM) advances the DGM by addressing metaproductivity–performance mismatch: immediate agent performance is not a reliable indicator of self-improvement potential. HGM introduces clade metaproductivity (CMP)—the expected maximal utility achievable in an agent’s future lineage. Under mild assumptions, access to a perfect CMP oracle recovers true Gödel machine optimality. In practice, HGM employs Bayesian estimation of CMP and tree-based bandit exploration. Empirically, HGM outperforms DGM and baseline agents, with substantial speedup and achievement of human-level coding benchmarks (Wang et al., 24 Oct 2025).

Agent	SWE-bench	Polyglot
DGM Best $U(p^*)$	50.0%	30.7%
HGM Best	56.7%–61.4%	30.5%

4. Gödel Agents in Graded Epistemic Logics

Multi-agent epistemic Gödel logic provides a formalism for agents with graded beliefs, enabling reasoning about plausibility on the interval [0,1]. Syntax is based on Gödel operations (conjunction, implication), involutive negation, and S5-style modalities for each agent. Two semantics are used: [0,1]-valued Kripke frames and finite-token eF-models (for the finite model property). Tableaux systems enable effective proof search, with the logic PSPACE-complete for multi-agent settings and coNP-complete for single-agent cases (Bílková et al., 6 Oct 2025).

B_a φ expresses the degree to which agent a “knows” φ:

• B_a φ(w)=1: agent a knows φ certainly at w.
• B_a φ(w)=0: agent a knows φ is false at w.
• Intermediate values yield graded epistemic plausibility.

This formalization supports modeling of bounded rationality and collective graded knowledge, extending to graded fuzzy ontologies and distributed agent settings.

5. Empirical Results, Benchmarks, and Safety

Gödel Agent frameworks applied to LLM coding and reasoning tasks achieve state-of-the-art or surpass it on diverse benchmarks (DROP, MGSM, MMLU, GPQA, SWE-bench, Polyglot). Notably, Gödel Agent and DGM frameworks converge to improved performance and generalize well across architectures and datasets (Yin et al., 2024, Zhang et al., 29 May 2025, Wang et al., 24 Oct 2025). Key empirical findings include:

• Continuous self-improvement via recursive code rewriting.
• Cost-efficient convergence (e.g., 30 iterations ≈$15 for Gödel Agent vs. ≈$300 for baseline meta-agent).
• Robustness to temporary performance drops and rare catastrophic failures, often mitigated by error handling and rollback.
• Safety enforced via sandboxing, traceability, human oversight, and proposals for immutable “safety cores.”

6. Limits, Metatheory, and Open Questions

Gödel Agents, while operationalizing recursive self-improvement and self-extension, face limitations:

• Loss of Formal Guarantee: Empirical validation replaces the classical requirement for formal proof of improvement, sacrificing guaranteed optimality for practical tractability.
• Blind Spots: Self-fulfilling fixed points (“lock-in” of unfalsifiable worldviews) may arise, resistant to empirical disconfirmation unless explicit meta-level revision is engineered (Pavlovic et al., 2023).
• Implicit Agency: All contact between evidence and theory (mathematics or physics) depends fundamentally on noncomputable, agent-driven choices (Myers et al., 2018, Myers et al., 2018).
• Paraconsistency: Recent formal systems (Gödel Mirror) allow contradiction to trigger controlled structural evolution, enabling agents to persist and expand without logical collapse in face of inconsistency (Chan, 16 Sep 2025).

Open directions include more principled estimation of lineage productivity, extensions to non-coding domains and real-time environments, integration with explicit proof search, and formal safety guarantees for recursive self-modification.

7. Summary Table: Gödel Agent Paradigms

Paradigm	Core Mechanism	Agent Functionality	Key Reference
Classical Gödel Machine	Proof-based self-mod	Provably optimal self-improvement	(Wang et al., 24 Oct 2025)
LLM Gödel Agent	Heuristic LLM search	Empirical recursive self-change	(Yin et al., 2024)
Darwin Gödel Machine	Archive+empiricism	Open-ended, empirical evolution	(Zhang et al., 29 May 2025)
Huxley-Gödel Machine	Lineage-based metric	Clade-maximal, bandit-guided	(Wang et al., 24 Oct 2025)
Symbol-Handling/Oracle	Oracle-in-c-machine	Guesswork bridging evidence/theory	(Myers et al., 2018, Myers et al., 2018)
Paraconsistent Loop	Controlled contradiction	Consistency via encapsulation	(Chan, 16 Sep 2025)
Epistemic Gödel Logic	Graded beliefs, S5	Multiagent plausibility reasoning	(Bílková et al., 6 Oct 2025)

• (Myers et al., 2018) Incompleteness theorem for physics
• (Myers et al., 2018) Agency and the physics of numbers
• (Pavlovic et al., 2023) From Gödel's Incompleteness Theorem to the completeness of bot beliefs
• (Yin et al., 2024) Gödel Agent: A Self-Referential Agent Framework for Recursive Self-Improvement
• (Zhang et al., 29 May 2025) Darwin Gödel Machine: Open-Ended Evolution of Self-Improving Agents
• (Chan, 16 Sep 2025) Gödel Mirror: A Formal System For Contradiction-Driven Recursion
• (Bílková et al., 6 Oct 2025) Tableaux for epistemic Gödel logic
• (Wang et al., 24 Oct 2025) Huxley-Gödel Machine: Human-Level Coding Agent Development by an Approximation of the Optimal Self-Improving Machine