arxiv: 2512.10100 · v2 · submitted 2025-12-10 · 💻 cs.AI

Robust AI Security and Alignment: A Sisyphean Endeavor?

Apostol Vassilev This is my paper

Pith reviewed 2026-05-16 22:55 UTC · model grok-4.3

classification 💻 cs.AI

keywords AI securityAI alignmentGödel incompletenessinformation-theoretic limitsAI robustnessformal systems

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The pith

Extending Gödel's incompleteness theorem establishes fundamental information-theoretic limits on the robustness of AI security and alignment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models AI systems as formal axiomatic systems and applies an extension of Gödel's incompleteness theorem to prove that no such system can achieve complete consistency and completeness in its security and alignment properties. This creates unavoidable vulnerabilities and barriers that no amount of additional safeguards can fully eliminate. The work stresses that recognizing these limits is essential for responsible AI adoption and offers practical strategies to navigate the resulting challenges. It further extends the argument to show similar bounds on the cognitive reasoning capabilities of AI systems.

Core claim

By extending Gödel's incompleteness theorem to AI, the manuscript shows that AI systems modeled as formal systems cannot be both complete and consistent in their security and alignment, implying inherent information-theoretic limitations that prevent full robustness.

What carries the argument

An extension of Gödel's incompleteness theorems applied directly to AI systems treated as formal axiomatic systems, which carries the proof that security and alignment properties must contain undecidable or inconsistent elements.

If this is right

AI security measures will always leave some vulnerabilities unaddressed no matter how many layers are added.
Full alignment of AI with human intentions cannot be guaranteed through any formal specification.
Ongoing adaptation and monitoring strategies become necessary rather than one-time solutions.
Cognitive reasoning tasks in AI inherit the same incompleteness bounds as security properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Developers may need to prioritize detection and response mechanisms over attempts at perfect prevention.
Similar incompleteness arguments could apply to other domains like autonomous decision systems or verification tools.
Empirical tests could involve checking whether any deployed AI ever exhibits provably exhaustive coverage of its threat model.

Load-bearing premise

AI systems can be modeled as formal axiomatic systems in a manner that permits direct application of Gödel's incompleteness results to their security and alignment properties.

What would settle it

Construction of a specific AI system that is provably complete and consistent in all its security and alignment behaviors, with no undecidable propositions or hidden vulnerabilities.

Figures

Figures reproduced from arXiv: 2512.10100 by Apostol Vassilev.

**Figure 1.** Figure 1: Context window sizes for real LLM’s Π. This leaves room for defenders to harden their AI Systems, which is considered in the next section. 3.2. Considerations for real AI Systems with finite context windows The previous section established information-theoretic security and alignment limitations for ideal AI Systems based on LLM’s. This section considers real-life AI systems with finite context windows. Th… view at source ↗

read the original abstract

This manuscript establishes information-theoretic limitations for robustness of AI security and alignment by extending G\"odel's incompleteness theorem to AI. Knowing these limitations and preparing for the challenges they bring is critically important for the responsible adoption of the AI technology. Practical approaches to dealing with these challenges are provided as well. Broader implications for cognitive reasoning limitations of AI systems are also proven.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper recycles the Gödel-AI analogy for robustness limits without supplying the formal embedding or derivation that would make it a theorem.

read the letter

The main thing to know is that the paper asserts information-theoretic limits on AI security and alignment by extending Gödel's incompleteness theorem, yet it never builds the required mapping from a trained model to a consistent formal system that can express arithmetic and contain self-referential undecidable statements. The abstract and claims treat the extension as given, but the actual construction is absent. That leaves the central result as an informal restatement rather than a proof. On the positive side, the author does flag real practical difficulties in achieving robust behavior in deployed systems and sketches some high-level mitigation steps. Those reminders can be useful for teams handling high-stakes applications, even if they stay at the level of general advice. The broader discussion of cognitive limitations in AI reasoning follows the same pattern: it points to the issue without new technical content. The soft spot is exactly the modeling step the stress-test note flags. Neural networks are gradient-based approximators, not deductively closed axiom sets. Without explicit axioms, a consistency proof, and a clear correspondence between alignment properties and undecidable propositions, the Gödel extension does not go through. This gap has appeared in earlier philosophy-of-AI-safety pieces, so the work does not add a fresh formal result. Readers who want philosophical framing of why perfect robustness may be impossible will find the discussion familiar. Anyone looking for rigorous bounds, reproducible derivations, or testable predictions will not. I would not bring this to a reading group. I would not cite it. It does not merit peer review in its current form because the load-bearing claim rests on an unmade reduction rather than on delivered evidence.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to establish information-theoretic limitations for the robustness of AI security and alignment by extending Gödel's incompleteness theorem to AI systems. It asserts that this extension proves fundamental limits, provides practical approaches to dealing with these challenges, and proves broader implications for cognitive reasoning limitations of AI systems.

Significance. If the extension of Gödel's incompleteness theorem to AI systems were rigorously formalized with explicit mappings from learned models to consistent axiomatic systems and proofs preserving the theorem's conditions, it would represent a significant contribution by identifying inherent limits on perfect robustness in AI security and alignment. This could usefully inform expectations and strategy in the field. The inclusion of practical approaches would add applied value if directly tied to the theoretical result.

major comments (2)

Abstract: the assertion that the manuscript 'establishes' limitations by extending Gödel's incompleteness theorem supplies no derivation steps, formal definitions of how an AI system is modeled as a consistent axiomatic theory capable of expressing arithmetic, or explicit mappings showing that 'robust security' or 'alignment' corresponds to an undecidable Gödel sentence.
Main text on the extension: the central modeling step—that AI decision procedures or alignment objectives can be represented as a formal system S to which Gödel's theorem applies directly—is not constructed; no axioms are defined, no consistency of S is shown, and no reduction from gradient-based approximators to a deductively closed theory is supplied, leaving the claimed information-theoretic limit as an unverified analogy.

minor comments (1)

Clarify in the abstract and introduction whether the 'proofs' of broader implications are formal derivations or informal arguments, to avoid overstatement of results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and precise feedback. We agree that the manuscript's central claim requires a more explicit formalization of the extension from Gödel's theorem and will revise accordingly to strengthen the rigor of the argument.

read point-by-point responses

Referee: Abstract: the assertion that the manuscript 'establishes' limitations by extending Gödel's incompleteness theorem supplies no derivation steps, formal definitions of how an AI system is modeled as a consistent axiomatic theory capable of expressing arithmetic, or explicit mappings showing that 'robust security' or 'alignment' corresponds to an undecidable Gödel sentence.

Authors: We accept this criticism. The abstract was written for brevity and therefore omitted the requested details. In the revision we will expand the abstract to include a concise outline of the modeling steps, the definition of the formal system, and the explicit correspondence between robustness properties and undecidable sentences. revision: yes
Referee: Main text on the extension: the central modeling step—that AI decision procedures or alignment objectives can be represented as a formal system S to which Gödel's theorem applies directly—is not constructed; no axioms are defined, no consistency of S is shown, and no reduction from gradient-based approximators to a deductively closed theory is supplied, leaving the claimed information-theoretic limit as an unverified analogy.

Authors: We agree that the current presentation leaves the mapping at a conceptual level. The revised manuscript will add a dedicated subsection that (i) defines the axiomatic system S with explicit axioms for AI decision procedures, (ii) proves consistency of S under the stated assumptions, and (iii) supplies a reduction showing how gradient-based approximators can be embedded into a deductively closed theory, thereby converting the information-theoretic claim from analogy to a formal corollary. revision: yes

Circularity Check

1 steps flagged

Gödel extension to AI security reduces to unproven modeling assumption by definition

specific steps

self definitional [Abstract]
"This manuscript establishes information-theoretic limitations for robustness of AI security and alignment by extending Godel's incompleteness theorem to AI."

The claimed limitations are established precisely by the act of extending the theorem, which requires defining AI decision procedures as consistent formal systems S containing a Gödel sentence for alignment properties. This definition is not derived from external properties of neural networks but is instead the mechanism that produces the incompleteness result, rendering the limitation tautological to the modeling choice.

full rationale

The paper's core result claims information-theoretic limits on AI robustness by extending Gödel incompleteness, but this extension is achieved solely by positing that AI systems are formal axiomatic systems capable of self-reference. No independent construction, consistency proof, or parameter-free reduction from gradient-based models to Peano arithmetic is supplied; the undecidability of alignment properties is therefore true by the choice of analogy rather than derived. This matches the self-definitional pattern exactly, with the modeling step serving as both premise and conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on an unproven domain assumption that AI systems are sufficiently formal to inherit Gödel-style incompleteness directly; no free parameters or invented entities are visible in the abstract.

axioms (1)

domain assumption AI systems can be modeled as formal axiomatic systems to which Gödel's incompleteness theorem applies directly for security and alignment properties.
Invoked when extending the theorem to AI robustness limits.

pith-pipeline@v0.9.0 · 5338 in / 1159 out tokens · 33912 ms · 2026-05-16T22:55:23.266863+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2: For any checker C_Π(T_Π, p) there exist a truth T_Π such that C_Π(T_Π, p) ≠ 1, ∀p.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

All types of AI systems rely on computation for reasoning.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 2 internal anchors

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