1. Introduction: Gödel as a boundary object in the history of AI

Gödel’s incompleteness theorems belong, strictly speaking, to mathematical logic: they establish limits on what can be proved within certain kinds of formal axiomatic systems. Yet from the mid-twentieth century onward, “Gödel” acquired a second life in debates about artificial intelligence. It appears in arguments about whether minds are mechanizable, whether intelligence is reducible to formal reasoning, whether machines can be creative, and—more recently—whether complex AI systems can be made provably safe or reliably self-improving. The historical puzzle is not simply why a theorem about formal arithmetic is cited in discussions of AI. It is why Gödel is cited recurrently and in different ways—often as if incompleteness could decide questions that the theorem itself does not, by itself, settle.

This paper advances a historical-philosophical thesis: in AI discourse, incompleteness functions as a boundary object—a stable result whose meaning and argumentative role remain plastic as it travels across communities (logicians, philosophers of mind, AI researchers, machine-learning engineers) and across shifting research regimes.^[1] In the late GOFAI reception, incompleteness is mobilized as a putative decider of the mind–machine question: anti-mechanist arguments treat Gödel as a principled barrier to formalization, while mechanist replies insist that the Gödel move targets only fixed systems and can be mechanized or iterated. What matters for the history and the philosophy of science is that the debate repeatedly turns not on the theorem alone but on auxiliary premises about consistency, soundness, meta-level reasoning, and the appropriate unit of comparison—premises that prevent decisiveness.^[2]

In contemporary Machine Learning (ML), by contrast, “Gödel incompleteness” is often invoked less as a claim about provability than as a limit marker within performance-driven methodological disputes—an instructive case of reception drift in which “Gödel” signals principled limitation even when the operative constraints concern feature representations and evaluation practices.^[3] And in recent work on self-modifying coding agents, “Gödel” returns as a design horizon: provably beneficial self-improvement is framed as an ideal, but it is relaxed into benchmark-based empirical validation when proof is treated as infeasible in practice.^[4] Taken together, these episodes suggest that Gödel’s afterlife in AI is not best understood as a single “Gödelian argument,” but as a family of deployments that shift with AI’s changing epistemic norms.

The payoff of this approach is a disciplined alternative to two familiar but unsatisfactory narratives. One treats incompleteness as if it straightforwardly refuted strong AI; the other treats Gödel’s invocation in AI as mostly rhetorical noise. Against both, I argue that Gödel’s AI afterlife is historically intelligible once we track how different AI regimes renegotiate what counts as epistemic warrant. The recurrent “return to Gödel” is best understood as a marker of closure anxiety: whether intelligence, explanation, or safety can be fully captured within a single fixed formal or representational scheme. That anxiety appears in different guises—formal provability in the symbolic era, evaluation and representational closure in ML, and provable guarantees for autonomous self-modification in recent agent research. In each case, Gödel’s authority is recruited to articulate where a community senses a principled limit, even when the relevant limit is not incompleteness in the narrow technical sense.

Methodologically, the paper distinguishes three levels of analysis that are too often conflated in the literature: (1) theorem-level content (what incompleteness entails and under what preconditions), (2) argument-level deployment (how incompleteness is used in claims about minds and machines), and (3) rhetorical boundary work (how “Gödel” signals depth or principled constraint, including explicitly analogical uses). This layered method prevents overextension of Gödel’s result while allowing the reception history to do explanatory work.

The paper proceeds as follows. Section 2 provides scope control: a minimal, accurate account of incompleteness and the key distinctions needed to avoid category mistakes. Sections 3 and 4 examine the established trajectory of AI discourse (from Turing to Lucas and subsequent standard responses), and demonstrate how, following the symbolic era, shifts in Gödelian language serve distinct purposes—transitioning from assertions about the boundaries of mechanised cognition to delineating limits within performance frameworks and shaping design standards for provability-restricted self-improvement. Section 4 further substantiates this shift through a small set of practical cases, clarifying the impact of Gödelian overextension in ML-adjacent research on evaluation culture, claims of optimality, and the status of proof as an ideal of warrant. Section 5 examines how Gödel’s incompleteness has evolved from a strict logical boundary in AI to a guiding ideal for proof-backed self-improvement in contemporary machine learning.

2. What incompleteness does—and does not—license in AI contexts

A recurring problem in AI debates is a slide from Gödel’s technical theorem to expansive claims about intelligence. This section provides the minimal technical and conceptual background needed to prevent that slide. The aim is not to teach mathematical logic, but to delimit the inferential reach of “Gödel” in AI discussions, and to make explicit where additional philosophical premises are doing the argumentative work.

Gödel’s incompleteness theorems apply to formal axiomatic systems with specific properties—systems that are effectively specified, sufficiently expressive to encode elementary arithmetic, and consistent (or at least not trivially inconsistent).^[5] Under these conditions, there exist arithmetical statements not provable within the system (first incompleteness), and the system cannot, using only its own resources, prove its own consistency (second incompleteness). The core point is not that “truth always exceeds proof” in all contexts, but that within sufficiently expressive formal systems, provability has intrinsic limits.^[6]

Two clarifications matter for AI. First, incompleteness is a result about provability in a specified formal system, not a general result about what machines can do. The theorem does not say that machines cannot compute certain functions, learn certain patterns, or solve certain tasks. It says that if one identifies reasoning with proof in a fixed system of the relevant kind, then there are arithmetical truths the system cannot prove. Second, incompleteness is conditional. The truth of the relevant Gödel sentence (in the intended interpretation) depends on assumptions such as consistency or soundness; many AI arguments quietly treat “unprovable” as if it entailed “false,” or treat “human can see it” as if humans had unproblematic access to meta-level truth. Those steps require additional premises about mathematical knowledge and about what an agent is entitled to infer from consistency-style assumptions.^[7]

A common argumentative move is to infer from incompleteness a limitation on “machines” in general. That inference is not licensed without a further premise connecting intelligence to proof in a fixed axiomatic system. Even in symbolic AI contexts, where formal reasoning is central, it remains contestable whether an intelligent agent is best modeled as a single fixed system rather than as a revisable procedure, a hierarchy of systems, or an adaptive ensemble. The same point can be stated more sharply: incompleteness blocks the claim that any one fixed formal system captures all arithmetical truth; it does not block the possibility of mechanized procedures that revise, extend, or vary their formal resources over time.^[8]

This caution is even more important for contemporary ML where systems are not naturally represented as theorem-provers. In such contexts, the relevance of Gödel is often analogical: “Gödel” operates as a name for principled limitation, but the operative constraints concern representational choices, optimization dynamics, data regimes, or evaluation protocols. Distinguishing theorem-level entailment from analogical limit-marking is therefore part of the scope control that prevents category mistakes.

Another frequent conflation is between Gödel’s incompleteness (limits on provability within formal systems) and Turing’s undecidability results (limits on algorithmic decision procedures). The two are historically related, but not interchangeable. Incompleteness concerns what cannot be proved inside a sufficiently strong proof system; undecidability concerns what no general algorithm can decide in all cases. An argument that depends on undecidability is not, strictly speaking, an incompleteness argument even if it gestures at “Gödel,” and an argument that depends on proof limitation does not automatically yield a claim about computability.^[9]

The canonical anti-mechanist template runs as follows: represent a machine’s mathematical reasoning by a formal system; construct the corresponding Gödel sentence; conclude that the machine cannot prove it but a human can “see” it is true; infer that the human mind is not equivalent to any machine. The template’s persuasive force rests on two hidden steps: (1) fixing the machine as a single formal system, and (2) granting the human a reliable meta-level privilege.

Once these steps are explicit, it becomes clear why the template yields controversy rather than decision. If the machine is allowed revision, extension, or meta-level routines, “fixity” becomes unstable; if the human’s meta-level access is treated as fallible or itself formalizable, the privilege becomes contestable. This is why careful reconstructions conclude that Gödel is deployed “pro and contra AI” without decisively settling the issue.^[10] It is also why the literature repeatedly emphasizes that Gödel-style arguments hinge less on incompleteness alone than on substantive views about what “understanding” amounts to and what kinds of meta-inference are legitimate.

These distinctions do more than prevent technical mistakes; they are central to the paper’s historiographical argument. Once theorem-level content is separated from argument-level deployment, one can see why Gödel persists in AI debates: incompleteness supports boundary work under shifting epistemic norms. In the symbolic era, Gödel’s authority is recruited to articulate a limit on formalization. In ML contexts, “Gödel” may function as a limit marker within debates organized around evaluation and representational closure.^[11] In self-improvement research, “Gödel” can name an ideal of proof-backed self-modification that is explicitly displaced by benchmark-driven validation when proof is treated as infeasible in practice.^[12]

3. Gödel in early AI debates: from Turing’s caution to the Gödelian template

The canonical reception sequence that ties Gödel to AI debates runs from Turing’s early framing of “mathematical objections,” through Lucas’s anti-mechanist crystallization, to a set of standard replies that deflate decisiveness by challenging the comparison protocol and by questioning the alleged non-mechanical status of the meta-step. Reconstructing this sequence reveals that incompleteness functions as boundary work: it is used to demarcate what intelligence must be (mathematical insight or understanding) and what machines allegedly cannot capture (meta-level truth or reflection).

Turing’s 1950 discussion of machine intelligence is often remembered for the imitation game, but it also contains a methodological point that becomes central in later Gödel-related exchanges. In the section on “mathematical objections,” Turing treats Gödel-style limitations as objections to particular formalized procedures rather than as decisive limits on machinery as such: a Gödel-type result bears on what follows from adopting a given fixed formal scheme, not on what any possible machine could do under any architecture or learning history.^[13] The difference is one of quantification and fixity: defeating “this fixed machine” does not defeat “any possible machine,” particularly if a machine can be revised, extended, or supplied with additional procedures.

This point does not dissolve the relevance of limit theorems; it relocates it. Incompleteness becomes evidence that no single formalization is final, not evidence that mechanization is impossible in principle. This posture is already the seed of later controversy. Gödel becomes relevant to AI by an identification—explicit or implicit—between intelligence and formalizable reasoning. But once that identification is made, incompleteness becomes a pressure point for philosophical claims about what “reasoning” must amount to. Turing’s caution suggests a stable moral: if one wants to infer from incompleteness to a boundary on AI, one must make explicit the bridge premise connecting intelligence to proof in a fixed formal system. Without that premise, the inference is not licensed.

Lucas’s 1961 paper is widely recognized as the moment when incompleteness becomes a ready-made anti-mechanist template.^[14] The template treats a machine’s mathematical reasoning as captured by a formal system; the Gödelian construction then yields a sentence the system cannot prove; and the human is said to recognize the sentence’s truth (often conditional on the system’s consistency or soundness). The conclusion is that the human mind is not equivalent to any machine.

Two features of this move matter for our narrative. First, it depends on auxiliary premises about consistency, soundness, and the human’s access to truth at the meta-level; the theorem alone does not confer the epistemic privilege that the argument requires.^[15] Second, the argument’s rhetorical force lies in its apparent generality. It does not target a particular engineering architecture; it targets the very project of formalizing intelligence. “Gödel” becomes a boundary marker: a theorem about the limits of formal proof is recruited to demarcate a presumed non-mechanical residue in human understanding.

The canonical replies, already prominent in the early 1960s, challenge Lucas’s advantage by exposing how the comparison is staged. Lucas fixes the machine as a single formal system and then permits the human to move—to construct the Gödel sentence, to reason at the meta-level, and to extend beyond the system. But if the relevant Gödelian move is systematic—if it is a repeatable construction—there is no obvious reason why a machine cannot be built to perform it as well, or why a machine cannot be strengthened by adding the relevant sentence as a new axiom, by adding a meta-level routine, or by being replaced with a more powerful system. This “extension/race” line is compactly expressed in Good’s discussion of “human and machine logic,” which treats Lucas-style conclusions as resting on an asymmetry produced by holding the machine fixed rather than allowing systematic extension.^[16] Smart’s earlier intervention makes the same point in a different idiom: Gödel and related limit theorems block the completeness of fixed formal systems, but they do not straightforwardly yield anti-mechanist conclusions about what machines can be.^[17]

Once framed this way, the Lucasian victory becomes a “race” rather than a principled asymmetry. For any fixed system, a stronger system can be defined; for any strengthened machine, a new Gödelian sentence can be constructed. The debate turns on what is held fixed and what is allowed to change. This is an important historical pivot. Incompleteness no longer appears as a refutation of mechanization; it appears as a theorem that guarantees open-endedness of formal extension. The boundary claim (“machines cannot…”) is displaced by a methodological claim (“no fixed formalization is complete”).

A second family of replies targets the claim that the human meta-step is essentially non-mechanical. Whiteley’s immediate reply to Lucas is exemplary here: he explicitly rejects the idea that the mechanist issue can be “settled” by a Gödel-based argumentative maneuver, stressing that the inference from incompleteness to anti-mechanism is not straightforwardly warranted.^[18] If the Gödelian construction can be specified and executed reliably, then it looks like the kind of operation that can be implemented as a procedure. In this line of response, the human’s “seeing” the Gödel sentence’s truth is redescribed as a combination of formal reasoning plus substantive assumptions about consistency, soundness, and meaning -assumptions whose legitimacy and scope are themselves open to dispute.^[19]

Jongeneel and Koppelaar’s reconstruction serves as a valuable reference point for analyses of history and philosophy of science. Their central claim is that Gödel is deployed “pro and contra AI” without yielding a decisive outcome because the debate repeatedly shifts between theorem-level claims and philosophical premises about what humans can legitimately claim to know and how machines should be represented.^[20] In other words, incompleteness does real work in the controversy, but it cannot do the decisive work that the anti-mechanist template assigns to it without importing contested epistemological assumptions.

The Turing–Lucas–reply sequence establishes the initial function of “Gödel” in AI discourse: it is used to police claims about mechanizing intelligence by presenting a principled limit with the authority of mathematics. At the same time, the sequence explains why incompleteness proves so durable as a boundary object. It invites strong conclusions while repeatedly forcing attention to the auxiliary assumptions those conclusions require. The theorem is stable; the philosophical attachments are unstable. That combination—stability of mathematical result, plasticity of use—is what enables Gödel to travel into later AI regimes, where the boundary work it performs will no longer primarily concern “mind versus machine,” but rather the limits of guarantees, evaluation, and justified confidence in complex systems.

4. After the symbolic era: from Gödel’s challenge to Strong AI to limits of guarantees and evaluation

If the canonical reception spine makes incompleteness central to the mind–machine controversy, the post-symbolic era reveals a significant shift: Gödel’s name persists, but its function changes. In the symbolic setting, incompleteness is mobilized to argue about what intelligence is and whether minds are mechanizable.^[21] In ML-centered settings, “Gödel” is more often used to articulate closure anxieties within performance-driven regimes—anxiety about whether optimization within a fixed representational scheme can deliver unbounded improvement, and anxiety about what can be warranted when systems become too complex for proof-like assurance. In other words, the same name begins to mark two different worries: closure of formaljustification in the mechanist debate and closure of representational/evaluative warrant in contemporary ML practice.

The shift from symbolic AI to contemporary ML is not merely a change of tools; it is a change in epistemic norms. In symbolic AI, formal representation and proof-like ideals of correctness could be treated as central, which helps explain why Gödel was an attractive resource for boundary arguments.^[22] In contemporary ML, the dominant currency is performance under specified evaluation protocols: benchmarks, robustness tests, and comparative empirical results. This transformation does not eliminate interest in proof. Rather, proof becomes an aspirational form of warrant invoked precisely where complexity and opacity make guarantees desirable but difficult to obtain. “Gödel” can then function as a signifier for the gap between what one would like to guarantee and what one can practically validate.

This shift is crucial for the paper’s historical claim. Once AI’s epistemic culture moves toward benchmark-centered validation, Gödel’s incompleteness no longer enters primarily as a theorem about provability within formal systems. It enters as a widely recognizable emblem of principled limitation, capable of being redeployed—sometimes loosely, sometimes carefully—to discipline overclaims about “optimality,” “exhaustiveness,” or “proof-level assurance” in settings where what counts as warrant is largely empirical.

Muscoloni and Cannistraci provide an instructive symptom text. Their paper is primarily a methodological and empirical critique of strong “near optimality” claims for stacking and meta-learning in network link prediction. They argue that adding or combining “good rules” does not necessarily yield improvement and emphasize that what is taken as progress depends critically on evaluation methodology. In this context, they invoke Gödel incompleteness to express the idea that stacking cannot extract more than what is already present in the chosen features and their interactions.^[23]

For present purposes, the importance of this invocation is historiographical. It exemplifies reception drift: Gödel incompleteness operates as a culturally available label for principled limitation inside a performance- and evaluation-driven discourse. The operative constraint is not provability in an axiomatic system; it is representational closure—what a method can extract given a fixed feature space and combination scheme. Whether or not the analogy is philosophically exact, its use reveals what is being policed: inflated claims of methodological superiority, ungrounded “near optimality” narratives, and the tendency to treat performance gains as if they were theoretically guaranteed.

A few practical cases help clarify the impact of forced Gödelian interpretations in ML-adjacent research. One clear case appears in the link-prediction literature. Ghasemian and collaborators present stacked models as yielding “nearly optimal” link prediction across a large and diverse corpus of networks, thereby framing ensemble performance in unusually strong terms.^[24] Muscoloni and Cannistraci respond by arguing that such claims depend heavily on evaluation methodology and by denying that stacking “good” predictors guarantees cumulative improvement; in this context, Gödelian language functions as a way to mark the closure limits of a fixed representational and combinatory scheme rather than as a strict theorem-level application of incompleteness to ML systems.^[25] A second case concerns current research on mathematical reasoning and theorem discovery. Bengio and Malkin explicitly invoke Gödel’s incompleteness theorem in reflecting on the open-endedness of mathematical discovery and on the need for AI systems capable not merely of proving given statements but of generating interesting conjectures and revisable theoretical structures.^[26] Here again, the impact of the Gödelian reference is not to impose a direct formal limit on ML architectures as such, but to lend conceptual force to a research agenda organized around incompleteness, revisability, and the insufficiency of closed representational schemes. Recent preprint literature extends this pattern further by invoking Gödelian incompleteness to frame LLM hallucination as structurally unavoidable rather than merely contingent or remediable.^[27]

Taken together, these cases show that the impact of forced Gödelian interpretation in ML is not primarily technical in the narrow logical sense. Rather, it is epistemic and methodological: Gödelian language is used to discipline overclaiming, to redescribe local representational or evaluative limits as principled ones, and to preserve the authority of impossibility-style reasoning in domains governed largely by empirical validation.

Framed this way, the ML episode does not weaken the paper’s thesis by drifting away from logic; it strengthens it. It shows how Gödel’s authority can migrate from theorem-level entailment to a more general form of epistemic boundary work: “Gödel” marks the point at which a community suspects that a given representational and evaluative scheme cannot, on its own terms, guarantee the kinds of completeness or optimality claims being asserted.^[28]

A second contemporary register returns to Gödel in a way that is closer to its original association with self-reference: the Gödel Machine tradition and its recent reconfigurations in self-improving agents. Schmidhuber’s original proposal frames the Gödel machine as a self-referential system that searches for self-modifications and adopts them only when it can prove that the modification is beneficial according to a specified utility criterion.^[29] This is a maximalist expression of proof-as-warrant: self-improvement is legitimate only if it can be justified in a formal way that the system itself can verify.

Zhang and collaborators explicitly place their Darwin Gödel Machine in this lineage, but they also make the shift in epistemic currency explicit. They treat proof as infeasible in practice and substitute benchmark-driven empirical validation of code changes.^[30] Here “Gödel” names a design aspiration—provably beneficial self-modification—while the operative warrant is empirical performance. The key point is not that proof disappears, but that proof is displaced from the status of gatekeeper to the status of an idealized horizon. Precisely where proof would be most valuable—systems that rewrite themselves—the research regime relies on performance evidence, archive-based search, and open-ended variation.

Read together, these post-symbolic uses clarify the paper’s central historical-philosophical claim. Gödel’s incompleteness has mattered in AI less as a stable technical constraint than as a recurrent resource for articulating closure anxieties under changing norms of justification. The symbolic-era debate tried to turn Gödel into a decisive boundary on intelligence and foundered on auxiliary premises.^[31] Contemporary ML and self-improvement research repurpose Gödelian language to mark limits of representational closure, limits of evaluation-based claims, and limits of provability-driven assurance. The continuity across regimes is not that Gödel consistently refutes something. It is that Gödel’s name repeatedly appears where a community senses that no single fixed system—formal, representational, or evaluative—can guarantee completeness, finality, or fully secure warrant.

5. From incompleteness to “limits of guarantees”: the return of proof as a design ideal

One might expect that the shift to empirical ML regimes would marginalize proof-centered ideals entirely. Yet “Gödel” returns in an unexpected form: not as a mind–machine argument but as a design ambition for self-referential systems that can improve themselves. This return is instructive precisely because it places proof and formal guarantee back on the table—only to show why, under contemporary conditions, proof is repeatedly displaced by performance-based validation.

The relevant lineage is the Gödel machine tradition, which frames self-improvement as legitimate only when a proposed self-modification can be proved (relative to a formalized objective and background assumptions) to be beneficial.^[32] In this setting, “Gödel” does not name incompleteness as a theorem about arithmetic; it names an ideal of proof-backed self-modification anchored in self-reference and internal justification. The aspiration is maximal: the system ought to be able to justify its own self-changes within a formal framework that it can itself check.

Zhang and collaborators explicitly frame their project as an attempt to realize this “Gödel machine” ambition while also acknowledging why its proof condition becomes operationally prohibitive. They treat proof as infeasible in practice for most candidate modifications and therefore replace proof with empirical evidence from coding benchmarks. Their Darwin Gödel Machine (DGM) iteratively self-modifies its own code and retains changes that improve benchmark performance, maintaining an archive of variants in an open-ended, evolution-inspired exploration scheme.^[33] Here, “Gödel” names a design horizon—provably beneficial self-improvement—even as the operative research method is empirical selection under evaluation.

For an historico-philosophical account of Gödel’s role in AI, this is a crucial contemporary episode precisely because it makes the reception drift explicit. The historical point is not that Gödel’s 1931 incompleteness result directly constrains DGM. It is that “Gödel” becomes a name for an ideal of internal justification—proof-backed self-modification—that becomes newly salient in a context where systems are increasingly complex, autonomous, and difficult to audit. The very reasons that make proof attractive as a norm of warrant—opacity, scale, and autonomy—also make proof difficult to operationalize within a working research pipeline.

The philosophical point is equally sharp: the DGM framework dramatizes the modern form of Gödelian anxiety as an anxiety about guarantees. In a world of large, opaque systems, we would like proof-like assurances that modifications are beneficial and safe. But the system is too complex, the modification space too large, and the relevant desiderata too context-dependent for proof to be a routine gatekeeper. As a result, the epistemic currency shifts: DGM replaces proof with benchmark validation and treats improvement as what survives empirical selection under evaluation. One can read this episode as an instance of a broader pattern in contemporary AI: formal guarantees remain desirable as ideals, but practical research regimes increasingly rely on empirical proxies when guarantees are infeasible.

This episode also sharpens the paper’s central comparative claim about AI regimes. In GOFAI-era debates, Gödel was used to argue about what minds are and whether intelligence outruns mechanism.^[34] In contemporary self-improvement research, “Gödel” is used to argue about what justification would ideally look like for systems that rewrite themselves—and about why that ideal is hard to satisfy.^[35] The question shifts from “can machines think?” to “what would it take to warrant claims of safe improvement?” The label “Gödel” thus continues to perform boundary work, but the boundary is no longer human versus machine; it is proof versus performance, guarantee versus empirical proxy.

6. Conclusion: what the shifting uses of Gödel reveal about changing AI epistemologies

The history traced here supports a simple but consequential moral: Gödel’s incompleteness has mattered in AI less as a stable technical constraint than as a recurrent argumentative resource whose function shifts with AI’s epistemic regime. In the GOFAI reception, incompleteness is repeatedly mobilized as if it could decide the mind–machine question, yet close reconstruction shows that its force depends on auxiliary assumptions about consistency, meta-level reasoning, and the fixity of formal systems—assumptions that undermine the hope of a decisive conclusion. In contemporary ML discourse, Gödel’s name often functions as a marker of principled limitation inside debates that are actually organized around feature representations, model classes, and evaluation methodology—an instructive case of reception drift that nonetheless reveals what kinds of claims are being policed. In recent work on self-modifying agents, “Gödel” reappears as a design ideal of provable improvement, only to be displaced by benchmark-driven empirical validation under the explicit claim that proof is infeasible in practice.

What incompleteness reveals—through its shifting reception—is the changing structure of justification in AI. Proof-centered ideals have not disappeared; they return precisely where autonomy, opacity, and self-modification make guarantees most desirable. Yet the practical research regime often substitutes empirical validation for proof. Gödel’s name, in this setting, becomes a way of naming the gap between what we would like to guarantee and what we can actually warrant.

In the context of the history and the philosophy of science, the broader implication is that foundational theorems can have scientific lives that are not reducible to their original content. Incompleteness becomes a boundary object because it travels: it can remain stable as a theorem while being reinterpreted as a warning, a weapon, a trope, or a design ideal.

Tracking that travel is not an antiquarian exercise. It provides a historically grounded way to understand contemporary AI’s tensions between formal assurance and empirical performance, between interpretability and opacity, and between proof and benchmark. In that sense, Gödel’s incompleteness continues to matter—not because it settles what intelligence is, but because it keeps reappearing at the points where AI communities renegotiate what counts as a justified claim.

[1] The notion of boundary object is drawn from S. L. Star, J. R. Griesemer, Institutional Ecology, “Translations” and Boundary Objects: Amateurs and Professionals in Berkeley’s Museum of Vertebrate Zoology, 1907-1939, in «Social Studies of Science», 19, 3, 1989, pp. 387-420.

[2] C. J. B. Jongeneel, H. Koppelaar, Gödel Pro and Contra AI: Dismissal of the Case, in «Engineering Applications of Artificial Intelligence», 12, 1999, pp. 655-659.

[3] A. Muscoloni, C. V. Cannistraci, “Stealing Fire or Stacking Knowledge” by Machine Intelligence to Model Link Prediction in Complex Networks, in «iScience», 26, 105697, 2023, pp. 1-15.

[4] J. Zhang, S. Hu, C. Lu, R. Lange, J. Clune, Darwin Gödel Machine: Open-Ended Evolution of Self-Improving Agents, in «arXiv», 2025, https://doi.org/10.48550/arXiv.2505.22954.

[5] The literature on Gödel’s incompleteness theorems is too vast to summarize here. For an exhaustive overview and bibliography see P. Raatikainen, Gödel’s Incompleteness Theorems, in The Stanford Encyclopedia of Philosophy (Spring 2026 Edition), edited by E. N. Zalta, U. Nodelman, forthcoming, https://plato.stanford.edu/archives/spr2026/entries/goedel-incompleteness/

[6] P. Raatikainen, On the Philosophical Relevance of Gödel’s Incompleteness Theorems, in «Revue Internationale de Philosophie», 59, 2005, pp. 513-534.

[7] P. Benacerraf, God, the Devil, and Gödel, in «The Monist», 51, 1, 1967, pp. 9-32.

[8] J. J. C. Smart, Gödel’s Theorem, Church’s Theorem, and Mechanism, in «Synthese», 13, 1, 1961, pp. 105-110.

[9] P. Raatikainen, On the Philosophical Relevance of Gödel’s Incompleteness Theorems, cit.

[10] C. J. B. Jongeneel, H. Koppelaar, op. cit.

[11] A. Muscoloni, C. V. Cannistraci, op. cit.

[12] J. Zhang, S. Hu, C. Lu, R. Lange, J. Clune, op. cit.

[13] A. M. Turing, Computing Machinery and Intelligence, in «Mind», 59, 236, 1950, pp. 433-460; G. Piccinini, Alan Turing and the Mathematical Objection, in «Synthese», 136, 1, 2003, pp. 23-48.

[14] J. R. Lucas, Minds, Machines and Gödel, in «Philosophy», 36, 137, 1961, pp. 112-127.

[15] P. Benacerraf, op. cit.; P. Raatikainen, On the Philosophical Relevance of Gödel’s Incompleteness Theorems, cit.

[16] I. J. Good, Human and Machine Logic, in «The British Journal for the Philosophy of Science», 18, 2, 1967, pp. 144-147.

[17] J. J. C. Smart, op. cit.

[18] C. H. Whiteley, Minds, Machines and Gödel: A Reply to Mr Lucas, in «Philosophy», 37, 141, 1962, pp. 153-157.

[19] P. Benacerraf, op. cit.; P. Raatikainen, On the Philosophical Relevance of Gödel’s Incompleteness Theorems, cit.

[20] C. J. B. Jongeneel, H. Koppelaar, op. cit.

[21] A. M. Turing, op. cit.; J. R. Lucas, op. cit.

[22] J. R. Lucas, op. cit.; C. H. Whiteley, op. cit.

[23] A. Muscoloni, C. V. Cannistraci, op. cit., p. 2: «The result of this analysis agrees with the Gödel incompleteness: stacking might be nearly optimal but incomplete, by stacking you cannot squeeze out more than what is in your features and their interaction».

[24] A. Ghasemian, H. Hosseinmardi, A. Galstyan, E. M. Airoldi, A. Clauset, Stacking Models for Nearly Optimal Link Prediction in Complex Networks, in «Proceedings of the National Academy of Sciences», 117, 38, 2020, pp. 23393-23400.

[25] A. Muscoloni, C. V. Cannistraci, op. cit.

[26] Y. Bengio, N. Malkin, Machine Learning and Information Theory Concepts towards an AI Mathematician, in «Bulletin of the American Mathematical Society», 61, 3, 2024, pp. 457-469.

[27] S. Banerjee, A. Agarwal, S. Singla, LLMs Will Always Hallucinate, and We Need to Live With This, in «arXiv», 2024.

[28] S. L. Star, J. R. Griesemer, op. cit.

[29] J. Schmidhuber, Gödel Machines: Self-Referential Universal Problem Solvers Making Provably Optimal Self-Improvements, in «arXiv», 2003.

[30] J. Zhang, S. Hu, C. Lu, R. Lange, J. Clune, op. cit.

[31] J. R. Lucas, op. cit.; C. H. Whiteley, op. cit.; I. J. Good, op. cit.; P. Benacerraf, op. cit.

[32] J. Schmidhuber, op. cit.

[33] J. Zhang, S. Hu, C. Lu, R. Lange, J. Clune, op. cit.

[34] J. R. Lucas, op. cit.; C. H. Whiteley, op. cit.

[35] J. Schmidhuber, op. cit.; J. Zhang, S. Hu, C. Lu, R. Lange, J. Clune, op. cit.