We have earlier shown by construction that a proposition can have a welldefined nonzero probability, even if it is justified by an infinite probabilistic regress. We thought this to be an adequate rebuttal of foundationalist claims that probabilistic regresses must lead either to an indeterminate, or to a determinate but zero probability. In a comment, Frederik Herzberg has argued that our counterexamples are of a special kind, being what he calls ‘solvable’. In the present reaction we investigate what Herzberg means (...) by solvability. We discuss the advantages and disadvantages of making solvability a sine qua non, and we ventilate our misgivings about Herzberg’s suggestion that the notion of solvability might help the foundationalist. (shrink)

Initial Judgment of Solvability (iJOS) is a metacognitive judgment that reflects solvers’ first impression as to whether a problem is solvable. We hypothesized that iJOS is inferred by combining prior expectations about the entire task with heuristic cues derived from each problem’s elements. In two experiments participants first provided quick iJOSs for all problems, then attempted to solve them. We manipulated expectations by changing the proportion of solvable problems conveyed to participants, 33%, 50%, or 66%, while the true proportion (...) was the 50% for all. In Experiment 1 we used the non-verbal Raven’s matrices and examined nameability as the element-based heuristic cue. Unsolvable matrices were generated by switching locations of elements in original Raven’s matrices. In Experiment 2 we used the verbal Compound Remote Associate (CRA) problems and examined word’s frequency in the language as the element-based heuristic cue. Unsolvable CRAs were random word triads from the same word pool. The results were consistent in suggesting that quick iJOS integrates prior expectations and experience-based heuristic cues. Notably, iJOS was predictive for the subsequent solving attempt only for Raven’s matrices. (shrink)

Background: Classical computability and complexity theory analyze how costly a computation is in time or space, assuming inputs are already encoded as finite strings. The knowledge-representation tradition recognizes that representation choices constrain inference—most notably through the expressiveness–tractability tradeoff—but focuses on the complexity of querying a fixed representation, not on whether certain representational resources are necessary for a task to be expressible at all. Meanwhile, deep learning systems exhibit systematic failures on compositional and multi-object reasoning, symbolic systems face the classical symbol-grounding (...) problem, and neuro-symbolic architectures lack a formal account of which representational resources are required for which tasks. -/- Objectives: I aim to develop a computability-theoretic framework that classifies computational tasks by their minimal representational requirements—an axis orthogonal to time–space complexity—thereby providing formal criteria for determining when connectionist, symbolic, or neuro-symbolic architectures are representationally adequate, or provably inadequate, for a given task. -/- Methods: I introduce the Hierarchical Representational Machine (HRM), which stratifies representational resources into four layers: Structured Input Space (L1), grounded in Type-2 Theory of Effectivity; Equivalence and Measurement (L2), formalized via computable quotient spaces and representational measurement theory; Formal Symbolic Representation (L3), providing compositional syntax with effective derivation; and Admissibility Constraints (L4), supplying decidable meta-level filters for safety, legality, and verification. For each layer, I define necessity schemas that identify structural conditions on tasks—invariance, unbounded compositionality, and hard admissibility—under which that layer is provably required. I establish separation results via diagonal and undecidability arguments, and introduce the meta-problem HRM-Classify with the minimal requirement operator Req(T). -/- Results: The main results are: (i) three necessity schemas identifying when specific layers are unavoidable; (ii) separation theorems establishing strict gaps between layer-restricted solvability classes—{L1} ⊊ {L1,L2}, {L3} ⊊ {L1,L2,L3}, and {L1,L2,L3} ⊊ {L1,L2,L3,L4} under external-standard semantics—demonstrating that these layers correspond to genuine computability-theoretic distinctions even for constant-output tasks; and (iii) a solvability classification via Req(T) with worked examples spanning object recognition, program verification, and autonomous driving that illustrate the full layer hierarchy. -/- Conclusions: HRM provides a formal vocabulary for diagnosing representational insufficiency in AI systems: distribution-shift brittleness is identified as missing invariance structure, systematic compositional failures as insufficient symbolic resources, and constraint violations as absent admissibility gates. The framework serves as a principled pre-architectural analysis tool that determines minimal representational commitments before algorithm selection. Because HRM addresses representational feasibility rather than computational efficiency, it complements existing frameworks such as PAC/VC theory, knowledge compilation, and category-theoretic approaches, and is applicable wherever the question "what must be representable?" precedes "how hard is inference?". (shrink)

This paper studies how the introduction of non-additive probabilities (capacities) affects the solvability of strategic games.

Theorem 0.1LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an abstract elementary class with amalgamation and no maximal models. Letλ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is categorical inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, then the model of cardinalityλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}is Galois-saturated.This answers a question asked independently by Baldwin and (...) Shelah. We deduce several corollaries: K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document} has a unique limit model in each cardinal below λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document} is weakly tame below λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, and the thresholds of several existing categoricity transfers can be improved.We also prove a downward transfer of solvability :Corollary 0.2LetK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}be an AEC with amalgamation and no maximal models. Letλ>μ>LS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda> \mu > {LS}$$\end{document}. IfK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is solvable inλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, thenK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {K}$$\end{document}is solvable inμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. (shrink)

The consistent tanh expansion method is successfully applied to the coupled integrable dispersionless system. A nonauto-Bäcklund transformation theorem includes two fields f and v 1 is obtained by using the CTE method. One obtains the consistent condition in the nonauto-BT theorem by means of the relation between the fields f and v 1. The CID system possesses the CTE solvability property by some detailed analysis. Many interactions between one soliton and multiple resonant solitons, and between one soliton and cnoidal (...) waves are generated by using the nonauto-BT theorem. The types of bright and gray two front waves are shown by some figures. In the meanwhile, the nonlocal symmetry is obtained by the truncated Painlevé method and the Möbious invariant form. The initial value problem and an auto-BT are constructed by the localization procedure. (shrink)

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a (...) definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian)p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T]. At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition. (shrink)

The topic of this article is Hilbert’s axiom of solvability, that is, his conviction of the solvability of every mathematical problem by means of a finite number of operations. The question of solvability is commonly identified with the decision problem. Given this identification, there is not the slightest doubt that Hilbert’s conviction was falsified by Gödel’s proof and by the negative results for the decision problem. On the other hand, Gödel’s theorems do offer a solution, albeit a (...) negative one, in the form of an impossibility proof. In this sense, Hilbert’s optimism may still be justified. Here I argue that Gödel’s theorems opened the door to proof theory and to the remarkably successful development of generalized as well as relativized realizations of Hilbert’s program. Thus, the fall of absolute certainty came hand in hand with the rise of partially secure and reliable foundations of mathematical knowledge. Not all was lost and much was gained. (shrink)

Consciousness is traditionally examined through the lens of its content—qualia, representation, neural correlates—or its computational mechanisms. This paper proposes a fundamental shift toward a **structural theory of awareness**, arguing that the coherent self is not an emergent epiphenomenon but the necessary signature of a cognitive system operating within a precise **solvability regime of constraint**. We synthesize the unsolvability threshold in Galois theory, the capacity limit of working memory, and the precision-weighting mechanisms of predictive processing into a single, unified mathematical (...) framework: the **Solvability Threshold Theorem of Conscious Constraint**. Within this model, probability is reconceptualized not as epistemic uncertainty but as the ontological architecture of experience, a generative density that shapes the very possibility of state-space trajectories. We further demonstrate that the most profound cognitive event is a **Galois-type transformation** of this probability measure itself—a phase transition in the symmetry group governing the plausible. A novel deduction from the pigeonhole principle, applied to the expanded state-space of reflexive awareness, reveals the structural inevitability of the mind-world duality. Ultimately, the self is formalized as a solvable orbit in a constrained symmetry group, and its transcendence is framed as a controlled navigation to the meta-cognitive "Witness State" that exists at the very edge of this solvable horizon. (shrink)

This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of continuity (full, separate, Wold, weak Wold, Archimedean, mixture) with those of solvability (restricted, unrestricted) under the completeness and transitivity of a binary relation. Theorem 2 uses the primitive notion of a separately continuous function to answer the question when an analogous property on a relation is fully continuous. Theorem (...) 3 provides a portmanteau theorem on the equivalence between restricted solvability and various notions of continuity under weak monotonicity. Finally, Theorem 4 presents a variant of Theorem 3 that follows Theorem 1 in dispensing with the dimensionality requirement and in providing partial equivalences between solvability and continuity notions. These theorems are motivated for their potential use in representation theorems. (shrink)

In this paper we establish the - and -transfer principles for finitely decidable locally finite varieties, where a class of structures is finitely decidable if the first order theory of its finite members is recursive. The transfer principles deal with the local structure of finite algebras and have strong global consequences.

The main objective of this paper is the following two results. There exists a computable bi-orderable group that does not have a computable bi-ordering; there exists a bi-orderable, two-generated computably presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$. [a]nswers a question posed by Downey and Kurtz; answers a question posed by Bludov and Glass in Kourovka Notebook.One of the technical tools used to obtain the main results (...) is a computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest. (shrink)

The importance of the Stability Problem in neurocomputing is discussed, as well as the need for the study of infinite networks. Stability must be the key ingredient in the solution of a problem by a neural network without external intervention. Infinite discrete networks seem to be the proper objects of study for a theory of neural computability which aims at characterizing problems solvable, in principle, by a neural network. Precise definitions of such problems and their solutions are given. Some consequences (...) are explored, in particular, the neural unsolvability of the Stability Problem for neural networks. (shrink)

We prove that in a locally finite variety that has definable principal congruences , solvable congruences are nilpotent, and strongly solvable congruences are strongly abelian. As a corollary of the arguments we obtain that in a congruence modular variety with DPC, every solvable algebra can be decomposed as a direct product of nilpotent algebras of prime power size.

The affirmative answer to the title question is justified in two ways: logical and empirical. The logical justification is due to Gödel’s discovery that in any axiomatic formalized theory, having at least the expressive power of PA, at any stage of development there must appear unsolvable problems. However, some of them become solvable in a further development of the theory in question, owing to subsequent investigations. These lead to new concepts, expressed with additional axioms or rules. Owing to the so-amplified (...) axiomatic basis, new routine procedures like algorithms, can be reached. Those, in turn, help to gain new insights which lead to still more powerful axioms, and consequently again to ampler algorithmic resources. Thus scientific progress proceeds to an ever higher scope of solvability. The existence of such feedback cycles –in a formal way rendered with Turing’s systems of logic based on ordinal – gets empirically supported by the history of mathematics and other exact sciences. An instructive instance of such a process is found in the history of the number zero. Without that insight due to some ancient Hindu mathematicians there could not arise such an axiomatic theory as PA. It defines the algorithms of arithmetical operations, which in turn help intuitions; those, inturn, give rise to algorithmic routines, not available in any of the previous phases of the process in question. While the logical substantiation of the point of this essay is a well-established result of logico-semantic inquiries, its empiricalclaim, based on historical evidences, remains open for discussion. Hence the author’s intention to address philosophers and historians of science, and to encourage their critical responses. (shrink)

The mind–body problem is analyzed in a physicalist perspective. By combining the concepts of emergence and algorithmic information theory in a thought experiment, employing a basic nonlinear process, it is shown that epistemologically emergent properties may develop in a physical system. Turning to the significantly more complex neural network of the brain it is subsequently argued that consciousness is epistemologically emergent. Thus reductionist understanding of consciousness appears not possible; the mind–body problem does not have a reductionist solution. The ontologically emergent (...) character of consciousness is then identified from a combinatorial analysis relating to universal limits set by quantum mechanics, implying that consciousness is fundamentally irreducible to low-level phenomena. (shrink)

Global Cumulative Treatment Analysis is a novel clinical research model combining expert knowledge, and treatment coordination based upon global information-gain, to treat every patient optimally while efficiently searching the vast space that is the realm of cancer research.

We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a model where the equation x 2 1 + x 2 2 + x 2 3 + x 2 4 = a has a solution. The main lemma states that there is no polynomial f(x,y) with coefficients in a (nonstandard) DOR M such that $|f(x,y)| for every (x,y) ∈ C, where C is (...) the curve defined on the real closure of M by C: x 2 + y 2 = a and $a > 0$ is a nonstandard element of M. (shrink)

In the introduction the paper presents, based on the work of Michael Devitt, the conflicting ontological positions of Realism and Constructivism. The former insists on the independence of the nature of the world from our conceptual apparatus, language or scientific theories, whereas the latter affirms its dependence. The central part of the paper is concerned with showing that the Realism/Constructivism dispute is unsolvable by way of a thought experiment followed by refutation of the arguments of key constructivists and realists. The (...) views of Hilary Putnam are also briefly assessed and rejected. In conclusion it is argued, partly with recourse to Carnap’s arguments, that the dispute cannot be resolved, that it is a kind of Kantian antinomy, and that being a realist or a constructivist is therefore a matter of decision. In the course of the article a view is also expressed with regard to the nature of philosophy. (shrink)

A differential equation with a known solution is transformed by changing both its dependent and independent variables, and the resulting nonlinear differential equation is then compared with the Schrödinger equation. The method is demonstrated using the confluent hypergeometric differential equation and the solutions to hydrogen, SHO and l=0 Morse potential problems are obtained.

In modern societies collective identity is both an empty signifier and a sacred center: even as its existence is taken for granted, what is or should be is subject to a host of different and often conflicting interpretations. However, the narratives and representations of collective identity are in no way undermined by these public debates; these signifiers are seen rather as a problem that is in principle amenable to solution, as something that ought to be (re)solved. In fact, the empty (...) signifiers of collective identity are constructed as solvable secrets precisely and primarily in public speech, open debate and perpetual critique. This article identifies the public and private modes of dealing with empty signifiers – through collective traumatic repressions, private resentments, public discourses adhering to argumentation ethics, and individual fabulations. (shrink)

Let G be a group definable in a monster model $\germ{C}$ of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U þ (G) < ∞. We prove that if G acts definably on a definable set of U þ -rank 1, then, under some general assumption about this action, there is an infinite field interpretable in $\germ{C}$ . We conclude that if G is not solvable-by-finite and it acts faithfully and definably on (...) a definable set of U þ -rank 1, then there is an infinite field interpretable in $\germ{C}$ . As an immediate consequence, we get that if G has a definable subgroup H such that U þ (G) = U þ (H) + 1 and $G/\bigcap _{g\in G}H^{g}$ is not solvable-by-finite, then an infinite field interpretable in $\germ{C}$ also exists. (shrink)