Search Results (6)

The pursuit of self-evolving AI raises a critical question: when is autonomous self-improvement sustainable rather than degenerative? Drawing an analogy to von Neumann’s complexity threshold for self-reproducing automata, we argue that sustainable recursive self-improvement in large language models (LLMs) requires a functional analogue: introspection—the system’s capacity to simulate its own operations and target modifications. Grounded in Kleene’s Second Recursion Theorem, we construct such introspective self-improvement programs and prove their key properties: completeness of self-modification, necessity of the reflective architecture, undecidability of improvement in general, and equivalence with Schmidhuber’s Gödel machine under a rewrite-equivalence notion, which transfers the global optimality guarantee. An empirical review, organized around these functional criteria, suggests that current LLMs exhibit only quasi-introspection.The available evidence does not establish complete introspection in the formal sense developed here, while pointing to several candidate structural bottlenecks, including incomplete self-access, feedforward processing, and limited computational depth. We outline architectural paths toward the threshold and discuss the safety implications of crossing it. Full article

How can an LLM-based agent recognize the limits of its own self-knowledge and improve that self-knowledge over time? This paper proposes the metacognitive evolutionary system (MES), a functional framework for studying metacognitive evolution at the prompt level in large language model (LLM)-based agents. MES does not modify model weights; it evolves the prompt program around a fixed base model, keeping the system readable, auditable, and easier for humans to inspect. The framework introduces a recursive metacognitive tower, $M^n\left(P\right)=\mathrm{LLM}\left(⟨M^{n-1}\left(P\right)⟩\right)$, to model layered self-evaluation. The fixed-point behavior of this tower is interpreted as a Strange Loop, with a formal analogy to Gödelian incompleteness used to describe its epistemic limits. The system is built using five primitive functions: inference, grounding, awareness, adaptive self-replication, and population-level selection. A grounded fitness function guides prompt evolution across generations, evaluating uncertainty calibration, error detection, strategy adaptation, and epistemic boundedness. Through Quine-style prompt rewriting, MES studies how agents can revise their own prompt-level structure while remaining constrained by grounded evaluation. The paper presents the formal architecture, analyzes tower dynamics using Markov chains, discusses convergence results, identifies six application domains, and proposes QuineBench as an evaluation design. Numerical case studies serve as illustrative analytical examples rather than empirical experiments, and QuineBench is presented as a structured protocol for future validation, providing a theoretical foundation and a clear path toward empirical evaluation. Full article

1936 was a watershed year for computability. Debates among Gödel, Church and others over the correct analysis of the intuitive concept “human effectively computable”, an analysis at the heart of the Incompleteness Theorems, the Entscheidungsproblem, the question of what a finite computation is, and most urgently—for Gödel—the generality of the Incompleteness Theorems, were definitively set to rest with the appearance, in that year, of the Turing Machine. The question I explore here is, do the mathematical facts exhaust what is to be said about the thinking behind the “confluence of ideas in 1936”? I will argue for a cultural role in Gödel’s, and, by extension, the larger logical community’s absorption of Turing’s 1936 model. As scaffolding I employ a conceptual framework due to the critic Leo Marx of the technological sublime; I also make use of the distinction within the technological sublime due to Caroline Jones, between its iconic and performative modes—a distinction operating within the conceptual art of the 1960s, but serving the history of computability equally well. Full article

Despite his unreserved appreciation of Turing’s analysis for being a “precise and unquestionably adequate definition” of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a “philosophical error” in Turing’s argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing’s work, despite the error of its ways? Previous interpretative strategies by Feferman, Shagrir and others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing’s response to the “mathematical objection” based on Gödel’s incompleteness theorem and Gödel’s own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no “error” on Turing’s side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together. Full article

The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems Ξ and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: “it is consistent that…”. The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deductively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel’s theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant’s view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified. Full article

Here, we introduce a new class of computer which does not use any circuit or logic gate. In fact, no program needs to be written: it learns by itself and writes its own program to solve a problem. Gödel’s incompleteness argument is explored here to devise an engine where an astronomically large number of “if-then” arguments are allowed to grow by self-assembly, based on the basic set of arguments written in the system, thus, we explore the beyond Turing path of computing but following a fundamentally different route adopted in the last half-a-century old non-Turing adventures. Our hardware is a multilayered seed structure. If we open the largest seed, which is the final hardware, we find several computing seed structures inside, if we take any of them and open, there are several computing seeds inside. We design and synthesize the smallest seed, the entire multilayered architecture grows by itself. The electromagnetic resonance band of each seed looks similar, but the seeds of any layer shares a common region in its resonance band with inner and upper layer, hence a chain of resonance bands is formed (frequency fractal) connecting the smallest to the largest seed (hence the name invincible rhythm or Ajeya Chhandam in Sanskrit). The computer solves intractable pattern search (Clique) problem without searching, since the right pattern written in it spontaneously replies back to the questioner. To learn, the hardware filters any kind of sensory input image into several layers of images, each containing basic geometric polygons (fractal decomposition), and builds a network among all layers, multi-sensory images are connected in all possible ways to generate “if” and “then” argument. Several such arguments and decisions (phase transition from “if” to “then”) self-assemble and form the two giant columns of arguments and rules of phase transition. Any input question is converted into a pattern as noted above, and these two astronomically large columns project a solution. The driving principle of computing is synchronization and de-synchronization of network paths, the system drives towards highest density of coupled arguments for maximum matching. Memory is located at all layers of the hardware. Learning, computing occurs everywhere simultaneously. Since resonance chain connects all computing seeds, wireless processing is feasible without a screening effect. The computing power is increased by maximizing the density of resonance states and bandwidth of the resonance chain together. We discovered this remarkable computing while studying the human brain, so we present a new model of the human brain in terms of an experimentally determined resonance chain with bandwidth 10⁻¹⁵ Hz (complete brain with all sensors) to 10⁺¹⁵ Hz (DNA) along with its implementation using a pure organic synthesis of entire computer (brain jelly) in our lab, software prototype as proof of concept and finally a new fourth circuit element (Hinductor) based beyond Complementary metal-oxide semiconductor (CMOS) hardware is also presented. Full article