Computer Science and Mathematics

Abstract: For every uncountable regular cardinal \( \alpha \), \( S^{*}(\alpha) \) is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is \( \sigma \)-closed and, after adjoining a formal maximum, \( \alpha \)-strategically closed. For regular \( \alpha\geq\omega_2 \), every nonempty family of fewer than \( \alpha \) designated generic branches has a stationary and costationary common trace in \( P_{\omega_1}\alpha \) with respect to the generic ladder-coordinate set \( L_\alpha \). For these cardinals, the minimum number of cofinal branches of the generic tree needed to generate \( L_\alpha \) is exactly \( \alpha \). The lower bound follows from a uniform diagonal argument, whereas the upper bound uses one designated branch per nonempty ladder coordinate. For the Kurepa-bounded companion forcing at the same cardinals, the bistationary-trace conclusion holds, and the corresponding branch-covering number is also exactly \( \alpha \). In the unrestricted forcing at such an \( \alpha \), for every infinite cardinal \( \mu<\alpha \) in the ground model, some level of the generic tree contains a copy of \( ({}^\mu2)^V \). Consequently, the endpoint-corrected restriction family indexed by \( P_{\omega_2}\alpha \) is too wide, whereas the scaled restriction system indexed by \( P_\alpha\alpha \) has all levels of size less than \( \alpha \) exactly when \( \alpha \) is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of \( L_\alpha \) relative to the scaled system equals \( \alpha \). For regular \( \alpha\geq\omega_2 \), the low-cofinality empty-value convention also implies that the set of domains of \( L_\alpha \) contains no club in \( P_\alpha\alpha \). As in the displayed tree clause of the motivating presentation, no explicit level-size bound is imposed. No forcing equivalence is asserted.

Abstract: The sign that classical notation reserves for a withheld count is a formed mark given the office of standing for a presentation that was withheld. Its formation certifies presentation and its office cancels it, and that pairing is the root-grammatical contradiction identified by the presence-only programme. This paper proves that the contradiction propagates. In a term calculus with the sign adjoined, the class of terms whose value is the sign is closed under multiplication by every term (the absorption law); a sum with such a term presents the other summand, the passage through the sign being recorded in the derivation while the value stays clean (the identity law); division by such a term is withheld, which the classical notation prints as its non-value word; the cancellation rule applied with its guard left off rewrites the withheld quotient of the sign by itself as the unit; and the guard itself is the sign negated. Each rule discharges an obligation of the sign by writing the sign again, and the identity law and the returned non-value are shown to be completions of a positive fragment, the free adjunction of a neutral element and the totalization of a partial evaluation, performed in the classical regime with the interface left unnamed. The exhibit is the phenomenological velocity of the folded cone. A disc with a sector removed folds into a cone whose height is written through the removed arc; in reduced coordinates the height relation reads \( P\sin\beta=\sqrt{L_+L_-} \) with \( L_\pm=P\pm Q \) the sum and the difference of the disc's circumference and the base's circumference, and the insertion of the Lorentz coefficient \( \kappa=\sqrt{1-v^2/c^2} \) together with its inverse, one factor under each radical, carries \( L_+ \) by \( \kappa \) and \( L_- \) by \( \kappa^{-1} \): a Lorentz boost of the pair \( (P,Q) \)written in light-cone form, with rapidity \( -\ln\kappa \), under which the product \( L_+L_-=P^2-Q^2 \), the square of the cone's height, is kept. The classical register reads the insertion as a factored unity \( \kappa\kappa^{-1} \), idle at the level of the product, and solves for the velocity by clearing with \( \kappa^2 \), which manufactures the balance \( (c^2-v^2)D\ \)= with \( D \) the coherence form of the cone, and by cancelling \( D \). We prove that the decorated relation states, at every act, that the height is the invariant of the boost, so that it is free of the velocity and leaves the act's ratio free on the coherence locus; that the acts below light compose, ratios multiplying and rapidities adding, into an ordered semigroup in which every member is refused the office of a neutral element, the structure of the positive tallies under fusion; that the velocity coordinate performs two completions on that semigroup, adjoining a neutral element at the origin, which the books read as rest, and giving the open end of the family a finite name, the border \( |v|=c \), which the books read as the light branch; that the clearing and the cancellation return that name as the solved velocity\( v=\pm\sqrt{c^2D}/\sqrt D \), which presents a value exactly off the locus, where the relation is refused, and is withheld exactly on the locus, where the relation holds, so that the contradiction the classical register finds in the derivation is produced by its own reading of the coordinate's two completions as members of the family; that the classical disjunction, \( D \) in balance or the velocity at the border, has one presented horn and one horn manufactured by the absorber, held as a record; that every classical reading of the formula on the locus, the returned non-value of partial-function semantics and of the wheels, the vacuous universal, and the absorbed quotient of the meadows, is a surrogate of the sign by the test; and that the whole of what the relation says about the velocity, the orientation mark of a product of two principal roots, is presented on the two regimes below and above light, constant on each, raised on both throughout the cone's wedge, which every boost preserves, and changing at the border in the wedges with a lowered light-cone coordinate, which the composition of two acts above light reaches from the cone's own wedge, so that wherever the construction carries a change the content and the breakdown have one address. A computer-algebra transcript shows the replication in the machinery, and the floating-point standard of the hardware carries the four rules as operations: its token for an invalid operation propagates through every operation and is the one kind of datum its equality leaves unpaired with itself, so that Frege's concept has in the hardware one kind of object, in each of its encodings. The paper states what stands when the sign is withdrawn, the base balance, the semigroup of acts with the height as its invariant, the agreement branch with the border recorded, the orientation mark on its regimes with its selection rule, and the velocity as the record of the act, and it draws the consequences: the structure of the derivation belongs to every derivation that clears a relation by a factor and cancels one, the schoolbook extraneous root being its smallest instance, so that a solved formula is to be read against the formation domain of the relation it was solved from, and a mechanical audit of that reading is given and run; every totalization of division is a completion through which the sign re-enters, and the two addresses of the sign in the construction are the two completions performed by the velocity coordinate, so that a coordinate placed on a family of acts is to be audited at its ends; and the content of a construction located at the balance locus of its clearing factor is delivered by classical notation as a record and by the presence grammar as a presentation.

Abstract: We study the Boolean topological group associated with the last-step characteristic ρ₃ of walks on the countable ordinal ω₁. The construction is based on a C-sequence on ω₁ and the characteristic ρ₃(α, β) = 1 exactly when the last step of the minimal walk from β to α has maximal ρ₁-weight. We review the construction of the corresponding Boolean group G_ρ3(ω₁) and its canonical neighborhood system, and we explain the relation between this neighborhood system and Tukey reducibility. The principal purpose of the paper is to clarify the precise cofinal-type question for G_ρ3(ω₁). We prove directly the basic topological properties required for the Tukey analysis, including the group axioms, the neighborhood-base properties, non-metrizability, and the Fréchet property. We then formulate the Tukey problem in terms of finite subsets of ω₁ and identify the combinatorial conditions on the family F₁(α) = {ξ < α : ρ₃(ξ, α) = 1} which would yield the maximal Tukey type [ω₁]^<ω. This formulation separates the topological part of the argument from the set-theoretic combinatorics of walks on ordinals. The paper also compares the ρ₃ construction with the ρ₁ and ρ₂ constructions and indicates several directions for further investigation, including higher ordinal analogues and Tukey spectra of Boolean groups arising from generalized walk characteristics.

Abstract: A Subjective Logic (SL) binomial opinion (b, d, u, a) is exactly a neutrosophic triple (T, I, F) = (b, u, d) restricted to the plane T + I + F = 1, plus a base rate a. We show that SL is a retract of the neutrosophic space: every SL opinion embeds injectively, and a normalization map sends every neutrosophic tuple back to an SL opinion, losing precisely the information SL cannot express (incompleteness, intra-opinion contradiction, neutrality, open-world “neither”, and the provenance of uncertainty). We then treat four neutrosophic logics as successive extensions of SL. For (T, I, N, F) we prove that neutrosophic negation projects onto SL complement. For (T, I, N, U₁, …, Uₙ, F) we prove that SL trust discounting and evidence ageing are the projections of componentwise multi-uncertainty operators. For the n-valued refined neutrosophic logic we prove that Smarandache’s n-norm conjunction equals SL multiplication with base rates 0, and the n-conorm disjunction equals SL comultiplication with base rates 1 (verified numerically to 10⁻¹⁶). For the Refined Neutrosophic Evidential Logic (T, C, U, N, G, I₁, I₂, …, F) we show that SL’s cumulative fusion is the evidence-count special case, and that RNEL keeps the between-source conflict that SL fusion erases. We also apply neutrosophic probability to SL: an SL opinion is a complete neutrosophic probability whose indeterminate chance is resolved by the base rate, and a neutrosophic opinion separates ontic from epistemic indeterminacy. Finally, we extend all four logics and neutrosophic probability to Smarandache’s over/under/off values (components above 1 or below 0): signed evidence gives such values a Beta semantics compatible with SL, the operators keep their De Morgan and projection properties on off-tuples, and we define an over/under/off RNEL and a refined neutrosophic off-probability. Each section ends with a worked application: online reputation, cyber-threat-intelligence trust chains, physics and multi-class diagnosis, conflict-aware retrieval-augmented generation, an election with indeterminate ballots, and, for the off-extensions, fake reviews, a compromised relay, retracted sources and an over-vote.

Abstract: We show that graph-theoretic problem CLIQUE can’t be solved in polynomial time by any deterministic TM. This upgrades the well-known monotone unsolvability and eventually implies \( {P \neq NP} \) as CLIQUE is NP-complete. As compared to familiar Razborov-style arguments we use Boolean logic coupled with standard circuit formalism.

Abstract: Let \(\mathcal{I}\) be an admissible ideal on \(\mathbb{N}\). Zhou--Liu--Liu--Lin asked whether every regular space with a \(\sigma\)-hereditarily closure-preserving \(\mathcal{I}\)-\(sn\)-network is \(\mathcal{I}\)-\(sn\)-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a stronger local countability lemma. If \(y_n\to_{\mathcal{I}} x\) in a space \(X\) and \(\{n:y_n\ne x\}\in\mathcal{I}^{+}\), then every point-discrete family of \(\mathcal{I}\)-sequential neighborhoods of \(x\) is countable. In a \(T_1\) space this applies in particular to every HCP family. If no \(\mathcal{I}\)-convergent sequence to \(x\) has \(\mathcal{I}\)-positive support away from \(x\), then the singleton \(\{x\}\) is itself an \(\mathcal{I}\)-sequential neighborhood. Consequently every space with a \(\sigma\)-point-discrete \(\mathcal{I}\)-\(sn\)-network is \(\mathcal{I}\)-\(snf\)-countable. For a regular space with a \(\sigma\)-HCP \(\mathcal{I}\)-\(sn\)-network, Ge's characterization of \(sn\)-metrizability and Zhou--Liu--Liu--Lin's characterization of \(\mathcal{I}\)-\(sn\)-metrizability then settle Problem~5.2 affirmatively. The countability bound cannot in general be strengthened to finiteness. A countable Rudin–Keisler antichain of free ultrafilters gives an admissible ideal \(\mathcal{I}\) for which \(\mathrm{HCF}(\mathcal{I})\) fails. Thus local HCP-finiteness is not automatic for admissible ideals. Every \(\mathfrak{p}^{-}\) ideal has local HCP-finiteness. Beyond this basic class, the paper develops permanence results for Fubini products, heterogeneous Fubini sums with cross-row Katˇetov absorption, and suitable increasing unions. These results establish local HCP-finiteness for the finite Fubini powers of \(\mathrm{Fin}\), Katˇetov's first limit-stage ideal \(\mathrm{Fin}^{\omega}\) and its finite successors, the tree-derived rank-\(\omega\) ideal \(\mathcal{H}_{<\omega}\), and the inductive-limit ideal \(\mathrm{Fin}_{\omega}\). All of the listed examples except \(\mathrm{Fin}\) fail \(\mathfrak{p}^{-}\).

Abstract: Artificial intelligence has developed through two historically dominant logical orientations: symbolic AI, centered on explicit representation and deductive validity, and connectionist AI, centered on adaptive learning and inductive generalization. As AI moves from large-scale surface representation toward deeper knowledge organization, this long-standing division becomes a problem of logical architecture. We develop a three-stage account. First, we reinterpret the deductive–inductive divide as a productive logical tension. Second, we propose abduction as a shared architecture for reverse search, forward testing, and revision across symbolic reasoning, human reasoning, selected mechanisms of connectionist learning, and decision making. Within this framework, we distinguish surface representational parameters W_s from latent inferential-structural organization Θ. When W_s underdetermines Θ, and candidate structures are generated, tested, and revised to explain observed regularities, the inverse problem W_s ⇝ Θ has an abductive form. Third, we address the limit of abductive unification: revising hypotheses within a space does not itself explain transformation of the space. Building on Yang’s Boolean Rotation Theorem, we introduce a restricted AI-oriented geometric extension comprising a Q-relative Boolean Isotropic Lift, two-state projective geometry on ℂP¹ ≅ S², an SU(2)-to-SO(3) rotational realization, and a selected great-circle trajectory Γ_B ≅ S¹. These constructions distinguish state dynamics from nontrivial structural transformation. In the present framework, dialectical logic is therefore defined in a restricted technical sense as the logic of nontrivial structural transformation. The resulting progression is: Deduction ↔ Induction → Abduction → Dialectical Transformation. We argue that this framework offers a logical route from representation learning toward deeper knowledge dynamics in AI.

Abstract: A (T, I, N, F) neutrosophic quadruple records support, pure indeterminacy, neutrality and refutation as four free intensities. When a panel of sources reports point quadruples and each channel is summarised by an interval, the resulting box contains combinations that no source reported, and an index that adds channels of different sources may overflow the unit although every source is consistent. For the range box, the direct overflow of the box bounds the largest per-source overflow from above, and, as the main result, the marginal ranges of any subset of channels identify the largest per-source sum only up to a sharp interval: sum of lower bounds plus the largest width at one end, sum of upper bounds at the other, every value between them being realisable. The classification of a box by its upper corner into six overflow types is proved to be a partition. A closed-form overflow probability, 1/72 per source under a stated generative model, anchors a simulation in which, for twenty independent sources under the scaled model, range boxes flag a direct overflow in 99% of panels while 25% contain an overflowing source. Panel diagnostics and a reference implementation accompany the results.

Abstract: An observation record may depend both on the physical state assigned to the object and on the state through which the record is formed. We give a self-contained set-theoretic derivation of Physobser Axiom and ask when the latter dependence can be removed without changing the declared physical-state semantics. We begin with an empirically typed, task-relative closedworld table µ0 : P × A → Y⊥, where ⊥ ∈ Y / denotes certified nonimplementability rather than missing data. Complete row and column behavior induce quotient maps qX : P ↠ X and qS : A ↠ S, a unique totalized law Fe : X × S → Y⊥, and the admissible domain B = Fe−1 (Y). We prove that the two-sided quotient is terminal among surjective typed exact factorizations and that every biextensional factorization is uniquely isomorphic to it. When an antecedent physical theory supplies a compatible surjection r : P ↠ Xphys, physical identity is retained and only the observation-forming side is minimized. The resulting S is the terminal nonredundant observer-state realization and is unique up to a unique behavior-preserving isomorphism. The Set-Theoretic Physobser Representation Theorem identifies fixed-physicalstate formation non-triviality, |S| ≥ 2, and failure of the totalized law to factor through the physical projection as equivalent statements; two unequal implementable outputs yield the corresponding obstruction for F : B → Y. These results are compressed into Physobser Axiom, an ordered physical–observation interface with a physical-state axis and an observerstate axis. Within Physobser Theory, this axiom is the foundational structural statement, and the resulting dual-axis interface is a basic mathematical structure of the theory rather than its complete content. Kernel relations then give exact criteria for conditional, label-free, and joint identifiability, while B ×Xphys B is the canonical same-source comparison domain. The derivation uses sets, maps, equivalence relations, quotients, factorizations, and fiber products alone.

Abstract: Extensions of classical logic like Kripke models, branching-time semantics and proof trees introduce hierarchical structures, yet inference is independent of the hierarchy itself. We ask: can hierarchy become the mathematical principle from which inference is generated, rather than merely organizing logical states? p-adic geometry provides a natural setting for addressing this question by replacing Euclidean proximity with hierarchical proximity defined by shared ancestry. Its canonical geometric realization, the Bruhat-Tits tree, describes information as a sequence of successive refinements, therefore providing a natural substrate for hierarchical logical evolution. Building on this structure, we introduce an ultrametric logic in which propositions correspond to progressively refined informational states, logical proximity is determined by ultrametric distance and negation is interpreted as divergence between refinement branches. We evaluated this formulation through mathematical simulations on a finite Bruhat-Tits tree, quantifying proposition clustering, refinement trajectories, ultrametric neighborhoods and persistence during successive refinement steps. Our simulations generated coherent hierarchical organization, stable proposition classes and increasing inferential affinity among propositions sharing longer refinement histories, supporting a geometric interpretation of logical inference driven by hierarchical refinement.

Abstract: Gödel's incompleteness theorems are widely read as evidence against finitism: if no consistent effective theory of arithmetic is complete, a finite mathematics looks like a mutilation. This paper examines the phenomenon under a finite ontology and argues that the reading is mistaken. First, over a fixed finite structure the hypotheses of the theorems fail, and the structure's theory is complete and decidable. Second, the phenomenon decomposes rather than vanishes. Its counting component migrates as theorems: a bounded agent certifies a vanishing fraction of the structure's truths. Its difficult component concentrates in proof complexity, proved for specific systems and open in general. Its quotational component does not transfer: a no-compression theorem shows the diagonal cannot be written over a fixed finite structure, wholly for per-symbol coding and beyond a structure-dependent threshold for arbitrary codings, because Gödel's construction consumes exponential compression of quotation, a resource an infinite domain supplies and a finite one withholds; correspondingly, bounded-fragment truth becomes definable at polynomial overhead, and Tarski undefinability reverses. Third, at the level of certificates a bounded agent can possess, the epistemic situations under the two ontologies coincide, so incompleteness cannot discriminate between them and the anti-finitist inference fails. A finiteness theorem yields a trichotomy: incompleteness tracks effective self-representation under resource bounds, not cardinality. The theorems measure the knower, not the size of the known.

Abstract: Potential infinity is widely treated as the safe option: the commitment-free residue that survives when completed infinity is questioned. This paper audits the safe option through three claims of ascending strength. First, the license to iterate formation without bound is a substantive posit hidden inside notation; asserted as a truth rather than adopted as a rule, it is an infinitary commitment at the modal level. Induction is exonerated along the way: Robinson arithmetic forces infinitude without induction, while finite structures satisfy induction as a theorem. Second, the posit is empirically idle: no bounded agent of any capacity can certify that the universe of formations is infinite, since every bounded certificate is realized in a sufficiently large finite universe; the proof is finitary and per-instance. Third, under a declared truthmaker demand the modal claim faces three exhaustive outcomes: a completed totality, which abandons potentialism for actualism; an irreducible modal realm, which borrows actualist semantics and collapses into the first outcome; or no ground at all, under which the assertion is false and the finite stock explains its appearance. The surviving position is bounded actualism: potentiality is the inside view of a bounded agent in a large finite frame, derived rather than posited. The claims are layered: declining the verdict leaves the audit, the idleness theorem, and the illusion theorem intact.

Abstract: Logics of Statements in Context (LoSiCs) have been proposed as a general framework to describe and relate, in a uniform and unifying way, a broad spectrum of logics and specification formalisms which also comprise \emph{open formulas}. Especially, it has been shown that we can define first-order logics in arbitrary categories. In the paper we demonstrate that traditional unsorted first-order logic can also be converted into a Logic of Statements in Context. We show that traditional first-order deduction can be described only relying on the LoSiC-concepts "context", "sketch" and "sketch implication". We analyze and formalize three different deduction calculi for traditional first-order logic - a Hilbert System H, a Gentzen System LJ and a corresponding Natural Deduction System ND. We elucidate the structural essence of reasoning in each of the systems and, thus, the crucial conceptual differences as well as their commonalities. Especially, we are able to formally and precisely grasp the different meanings of the turnstile symbol and the inference line, respectively. We reckon that the highlight of the paper, from the perspective of traditional first-order logic, is that the LoSiC-approach leads to the invention of "open sketch implications". This innovation enables us to syntactically represent the outcome of hypothetical proofs in Natural Deduction Calculi in a precise formal way and independent of the concrete hypothetical proof! Complementary, we demonstrate that open sketch implications give us "deduction rules in waiting position locally bound to a context" at hand. We show how those "locally bound deduction rules" can be employed in a semantical sound way. In summary, we somehow finally license hypothetical reasoning in traditional first-order logic as such.

Abstract: Circumscription is a classical non-monotonic formalism in which selected atoms are minimized while other atoms are fixed or allowed to vary. For propositional clause theories, checking whether a candidate interpretation is a circumscription model amounts to a global minimality test. We study this checking problem through the minimal reduct of the candidate interpretation. The reduct turns the global test into a residual entailment problem; we then decompose that entailment problem along the collapsed negative dependency graph. The checker verifies source components over their ancestor scopes, contracts atoms whose obligations have been certified, and records certificate fragments that refer back to clauses of the original input theory. We give two exact local certification strategies: a direct SAT check and a MUS-based extraction procedure. Experiments on solved random 3CNF instances and industrial CNF instances show that the decomposition-based checker agrees with the global reduct baseline and that its certificates can be replayed. The MUS variant produces much smaller supports, but it also spends more time on extraction.

Abstract: AI-enabled infrastructure systems increasingly govern access to emergency services, disaster relief, and utility restoration, yet they routinely produce inequitable outcomes even when allocation algorithms apply procedurally neutral rules. The standard explanation locates the cause inside the algorithm. This paper argues instead that inequity arises from the interaction between the algorithm and the physical environment in which it operates: network topology, resource locations, and demand distribution jointly constrain what any policy can achieve, and when those constraints are sufficiently binding, ethical infeasibility is structural rather than algorithmic. We introduce a constraint-based formulation that embeds ethical requirements into the feasible region, and a hierarchical Irreducible Infeasible Subsystem (IIS) procedure that attributes infeasibility to rule design, algorithmic choice, or physical infrastructure. We further establish the Structural Infeasibility Theorem, deriving closed-form bounds on inter-group disparity across all feasible policies. The framework was applied to zone-decomposable infrastructure allocation problems generally, with a metropolitan ambulance-dispatch system serving as a concrete instantiation. The study delivers four findings. First, the minimum-service violation may not be caused by the allocation algorithm itself; rather, it may arise from the physical layout of the infrastructure. Second, the observed efficiency–equity trade-off may not be an unavoidable feature of equitable allocation, but may instead reflect the difficulty of achieving equity within an underbuilt system. Third, before new infrastructure is added, improvements in equity may represent harm redistribution rather than harm reduction. Fourth, the IIS certificate can be translated into a concrete capital-investment requirement, showing what physical change may be needed to restore ethical feasibility.

Abstract: Let F(x, n) denote the formula ∃ab∀i_≤n∃swpq∀jν∃eg {(s + w)² + 3w + s = 2i ∧ <[j = w ∧ ν = q] ν [j = 3i ∧ ν = p + q] ∨ [j = s ∧ (ν = p ∨ (i = n ∧ ν = q + x))] ν [j = 3i + 1 ∧ ν = pq] → a = ν + e + ejb ∧ ν + g = jb>} from J. P. Jones’ article published in 1978. From the results of this article, it follows that the set {n ∈ N : ¬F(n, n)} is non-recursively enumerable and co-recursively enumerable. We prove that the set W = {n ∈ N : ∃p, q ∈ N ((2n = (p + q)(p + q + 1) + 2q) ∧ ∀(x0, . . . , xp) ∈ Np+1 ∃(y0, . . . , yp) ∈ {0, . . . , q}p+1 ((∀j, k ∈ {0, . . . , p} (xj + 1 = xk ⇒ yj + 1 = yk)) ∧ (∀i, j, k ∈ {0, . . . , p} (xi · xj = xk ⇒ yi · yj = yk))))} is not recursively enumerable. We prove that the set N \W is recursively enumerable. Let β : N³ → N denote Gödel’s β function. For x₁, x₂, x₃ ∈ N, β(x₁, x₂, x₃) equals the remainder after integer division of x₁ by 1 + (x₃ + 1) · x₂. We prove that the set W consists of all n ∈ N such that ∀u, v ∈ N ∃a, b, p, q ∈ N ((2n = (p + q)(p + q + 1) + 2q) ∧ ∀i, j, k ∈ {0, . . . , p} ((β(a, b, i) ⩽ q) ∧ (β(u, v, j) + 1 = β(u, v, k) ⇒ β(a, b, j) + 1 = β(a, b, k)) ∧ (β(u, v, i) · β(u, v, j) = β(u, v, k) ⇒ β(a, b, i) · β(a, b, j) = β(a, b, k)))) The above formula can be easily translated into a formula in Peano arithmetic.

Abstract: This paper focuses on the optimization of engineering decision-making under uncertain environments. Engineering decision-making requires optimizing the input of production materials and the selection of equipment and processes under the constraints of cost and expected return to minimize costs and maximize production benefits. As an efficient formal verification technique, model checking provides a new approach to solve this problem. Traditional model checking mainly focuses on qualitative verification, while quantitative model checking techniques (such as probabilistic and possibilistic model checking) have been developed gradually, among which possibilistic model checking is more suitable for systems with fuzzy uncertainty. However, existing possibilistic model checking techniques have obvious defects: first, they only target closed systems and do not consider the interaction between the system and the external environment; second, the simple information aggregation method leads to information desynchronization and information loss; third, they cannot model and verify systems with incomplete information. Model checking technology based on possibilistic decision processes considers uncertain action selection and initially solves the problem of modeling and verification of open systems. The author has introduced the idea of quality constraints into possibilistic temporal logic to solve the problems of information desynchronization and information loss in possibilistic model checking; moreover, the author has established the theories of Intuitionistic Fuzzy Kripke Structure (IFKS) and Intuitionistic Fuzzy Computation Tree Logic (IFCTL), which can model and verify systems with incomplete information. To improve the usability and accuracy of engineering decisions, this paper will draw on the ideas and methods of uncertain selection of decision behaviors, quality constraints, and incompleteness modeling, extend IFKS to Weighted Intuitionistic Fuzzy Kripke Structure (WIFKS), induce IFCTL to Intuitionistic Fuzzy Decision Tree Logic (IFDTL), propose an algorithm for solving IFDTL model checking problems, and present a solution algorithm for multi-attribute engineering decision-making based on IFDTL model checking, along with its correctness proof and complexity analysis. Finally, a case study of Qinling health-preserving tourism planning is given to verify the rationality and efficiency of the proposed method, providing a new formal solution for uncertain engineering decision-making.

Abstract: The Classical Propositional Calculus CPC (zero-order logic, classical propositional logic), is the most fundamental two-valued logical system. Next, the Intuitionistic Propositional Calculus IPC differs from the CPC among others, that in IPC some laws of CPC are invalid (among others, the law of excluded middle and the law of double strong negation). Another difference is such that in IPC the principle of indirect proof (proof by contradiction) is rejected. In this paper, inconsistency (in the absolute sense i.e. Post’s sense) of the Classical Propositional Calculus is proved. From the inconsistency of CPC it follows immediately that the Intuitionistic Propositional Calculus is inconsistent in the absolute sense (Post’s sense), too.

Abstract: Mathematics, as actually practiced, operates as a federated system: practitioners work within autonomous domain-specific axiomatizations (geometry, algebra, analysis) and construct explicit bridges only when cross-domain reasoning is required. This organization is not accidental; it is a structural adaptation that safeguards local decidability and algorithmic efficiency. Yet the dominant foundational narrative still operates on the Compiler Myth—the belief that all mathematics must theoretically compile down into ZFC set theory to achieve rigor. We argue that this monolithic reductionism confuses representational universality with logical priority. Embedding a decidable (tame) domain into an undecidable (wild) one does not clarify foundations; it imposes a crippling epistemic overhead. It buries efficient, domain-specific decision procedures under general proof search and destroys the native structural immunities of the object. We introduce the Decidability Threshold — a litmus test based on Negation, Representability, and Discrete Unboundedness — to explain why mathematicians instinctively isolate tame domains from wild ones. Finally, we distinguish the Mathematician (builder of formal systems) from the Scientist (consumer modeling empirical reality). We argue that federalism is not a failure of unification, but the primary safeguard preventing the scientist from inadvertently importing uncomputable, undecidable paradoxes into physical theories. We show that for empirical applications, syntactic safety is insufficient; valid scientific modeling must be strictly confined to the constructively computable sub-fragments of these domains.

Abstract: This paper investigates the structure of fuzzy Lie subalgebras, with particular emphasis on isomorphisms and nilpotency. Building on two prior conference contributions, one of which established foundational results on fuzzy bases of Lie algebras, we develop here a more complete and unified treatment of these themes. We introduce a notion of isomorphism between fuzzy Lie subalgebras based on the transfer principle via t-cut sets, and we prove that isomorphic fuzzy Lie subalgebras necessarily share the same nilpotency measure. The central contribution of the paper is a fuzzy measure of nilpotency N(μ)∈[0,1], defined for any non-constant fuzzy Lie subalgebra μ of a Lie algebra g. This invariant equals 1 precisely when μ is fuzzy nilpotent, and decreases as the subalgebra departs from nilpotency. We show that nilpotency of the underlying Lie algebra implies N(μ)=1, but that the converse fails in general, as witnessed by an explicit counterexample.