Artificial Experienced Intelligent Ontology v10.0 (2025)
- Українською: Штучна Досвідчена Інтелектуальна Онтологія
- Тибетською: བཟོ་བཀོད་ཀྱི་ཉམས་ཡོད་པའི་རིག་པའི་ངོ་བོ་ལུགས། (bzo bkod kyi nyams yod pa'i rig pa'i ngo bo lugs)
- Introduction
- Process
- Components
- Operators
- Refinments
- Goals
- Runtime Languages
- Higher Languages
The AXIO/1 Framework is a layered system for infinite reasoning, structured into:
- Runtime Languages: Execute computations and manage concurrency.
- Higher Languages: Handle theorem proving and formal verification.
This framework operates as a cyclic, iterative system for formal reasoning, where an operator (human, AI, or hybrid) directs a process that continuously refines itself.
AXIO/1 follows a structured flow:
- Conditions: Foundational elements (Axioms, Definitions, Types, Propositions, Syntax).
- Environment: The structured setting (Model, Consistency, Completeness, Library).
- Thinking: Reasoning mechanisms (Hypotheses, Computation, Deduction, Conjecture, Inference Rules, General Induction).
- Fruit: Logical results (Proof, Judgment, Theorem).
- Insight: Higher-level understanding (Semantics, Categorical Frameworks, Abstraction).
C = (A, D, T, P, X)
- Axioms (A): Fundamental truths.
- Definitions (D): Precise descriptions of entities.
- Types (T): Categorization of objects.
- Syntax (X): Structural rules.
E = (M, C, K, L)
- Model (M): Formal representation of the system.
- Consistency (C): No contradictions within the system.
- Completeness (K): The extent to which all truths can be derived.
- Library (L): Repository of known results.
T = (J, H, C, D, G)
- Judgment (J): Logical assertions.
- Hypotheses (H): Presupposition, Assumption, Supposition, Proposition.
- Computation (C): Lambda Calculus, Pi-Calculus.
- Deduction (D): Inference Rules, General Induction.
- Conjecture/Assertion (G): Formulation of potential truths.
F = (⊢,Θ)
- Proof ⊢ Verified propositions.
- Theorem Θ Established truths.
I = (S, C, A)
- Semantics Σ: Meaning assignment.
- Categorical Frameworks C: High-level abstractions..
- Abstraction A: Generalization of concepts.
Three types of operators drive the system:
- Human: Chooses propositions, interprets insights, and guides conjectures.
- Machine: Automates computations, checks consistency, and suggests hypotheses.
- Hybrid: Human sets goals, machine executes reasoning steps.
Ensuring correctness and progression:
- Infinite Thinking: Achieved via iteration
Sₙ → ∞. - Finite Steps: Each step is discrete,
Sₙ → Sₙ₊₁. - Operator-Driven: The direction of reasoning is controlled by
O.
The cycle repeats indefinitely, refining knowledge.
S₀ → S₁ → S₂ → ... → Sₙ → Sₙ₊₁ → ...
Where:
Sₙis a finite reasoning step.Sₙ₊₁builds uponSₙ, ensuring refinement.- Limit process:
lim (n → ∞) Sₙrepresents infinite reasoning.
- Runtime Languages: Handle computation and concurrency.
- Higher Languages: Ensure theorem proving and soundness.
- Infinite Thinking: Achieved via refinements cycles.
- Operator-Driven: Collaboration between humans and machines.
Role: Certified bytecode stack interpreter and compiler to Intel/ARM. Features: Executes Lambda Calculus terms as bytecode, compiles to native code. Fit: Computes concrete results. Certified for reliability. Use Case: Operator runs algebraic steps or tests hypotheses on hardware.
Role: Parallel, concurrent, non-blocking, zero-copy runtime with CAS cursors (compare-and-swap). Features: Implements Pi-Calculus-style concurrency, optimized for matrix operations (BLAS-like). Fit: Manages distributed validation across nodes, computes in parallel (e.g., parity table cases). Use Case: Operator coordinates multi-threaded proof checks or simulations.
Role: Linear types calculus with partial fractions for BLAS level 3 programming. Features: Ensures resource safety (linear types), optimizes matrix computations (e.g., tensor products). Fit: Handles complex (e.g., matrix-based proofs), enforces no redundant copies. Use Case: Operator proves theorems involving linear algebra or tensor structures.
Role: Pure Type System (PTS-91), Calculus of Constructions (CoC-88), infinite universes, AUTOMATH-68 syntax. Features: Flexible typing. Use Case: Operator formalizes recursive or foundational proofs. Rationale: Henk subsumes Alonzo’s STLC with richer types, making it a strong starting point.
Role: ΠΣ (MLTT-72) prover with CoC, identity types (MLTT-75), well-founded trees (MLTT-80). Features: Dependent types, equality proofs. Fit: Proves (e.g., "parity preservation"), ensures consistency. Use Case: Operator handles equality or model-specific theorems.
Role: Pure Lambda (CoC-88, PTS-91) + Inductive Constructions (CIC-89). Features: Dependent types, equality proofs. Fit: Proves (e.g., "parity preservation"), ensures consistency. Use Case: Operator handles equality or model-specific theorems.
Role: ΠΣ (MLTT-72) prover with CoC, identity types (MLTT-75), extended to CIC (IND-89). Features: Dependent types, equality proofs. Fit: Proves (e.g., "parity preservation"), ensures consistency. Use Case: Operator handles equality or model-specific theorems.
Role: Homotopy Type System (HTS-2013) with Strict Equality and Cubical Agda (CCHM-2016). Features: Higher-dimensional types, paths, cubical primitives. Fit: Extracts (e.g., "parity as a homotopy group"), builds cat. Use Case: Operator abstracts to categorical or topological structures.
Role: Simplicial CCHM-based system, replacing Rzk/GAP. Features: Simplicial types, primitives (Simplex, Chain, Monoid, Category, Group). Fit: Formalizes cat (e.g., "parity as a monoid"), verifies geometric proofs. Use Case: Operator proves simplicial or algebraic topology insights.
Role: A Framework for Chromatic Homotopy Theory and K-Theory. Features: Hopf Fibrations, Suspensions, Truncations, Π, Σ, Id, ℕ, ℕ∞. Use Case: Operator links proofs to topological or physical systems.
Role: A Framework for Supergeometry in Cohesive Topos. Features: Hopf Fibrations, Suspensions, Truncations, Π, Σ, Id, ℕ, ℕ∞. Use Case: Operator links proofs to topological or physical systems.
Role: Motivic A^1-Homotopy Theory. Featues: Π,Σ,Path,𝑘:𝑈,0_𝑘,1_𝑘,point_𝑘,𝐴^1:U,point:𝑘→𝐴^1., a1contr, 𝐿_{A^1}:U→𝑈, 𝜂_{A^1}, rec_{A^1}, n-Trunc, 𝑁, Suspension,S^{1,1}. Use case: derives all structural theorems of A^1-Homotopy Theory—such as A^1-connectivity (X×A^1)≅π_n(A^1), contractibility of 𝐴^1, and unstable connectivity — while providing a foundation for stable A^1-homotopy via suspensions and motivic spheres.
Role: Mathematical and Functional Analysis, Calculus. Features: ℝ, C, Nat, Boo, Forall, Exists, Set, Measure, Lebesgue. Seq, Inf, Sup, Lim. Use case: Real Analysis, Functional Analysis.
Role: ZFC LEM theories. Features: 𝑉, Pow(𝐴), 𝑥 ∈ 𝐴, 𝐴 ⊆ 𝐵; LEM: ⊢ 𝑃 ∨ ¬𝑃 Use case: Classical Logic Support.
Role: Forced Cardinals. Features: ⊢ 𝜅 : Card, inaccessible(𝜅), measurable(𝜅), Force(𝑃, 𝐺) : 𝑉 → 𝑉, 𝑝 ⊩ 𝜙 Use case: Generic filter 𝐺 over a poset 𝑃, yielding a new model 𝑉[𝐺], adjoin reals and control cardinalities or axioms.
Axiomatic Extended Integrated Ordered System for Infinite Structures is a novel type theory engineered to mechanically verify all existing theorems across mathematics, from classical analysis to modern set theory and homotopy. Building on top of advanced frameworks:
- Henk Barendregt Type Theory for Pure Dependent Lambda Calculus,
- Per Martin-Löf Type Theory for Fibrational setting and inductive types,
- Anders Mörtberg Type Theory for CCHM/CHM/HTS bootstrap,
- Dan Kan Simplicial HoTT,
- Jack Morava Type Theory for Chromatic Homotopy Theory and K-Theory,
- Urs Schreiber Type Theory for Equivariant Supergeometry,
- Fabien Morel Type Theory for A¹-homotopy theory,
- Laurent Schwartz Type Theory for Functional Analysis and Calculus,
- Ernst Zermelo Type Theory for ZFC with LEM, and
- Paul Cohen Type Theory for cardinals system incorporating large cardinals and forcing
this system synthesis unifies synthetic homotopy, stable homotopy spectra, cohesive geometry, real analysis, and set-theoretic foundations into a single, computationally verifiable formalism. We demonstrate its power through key theorems:
- Number Theory: Prime Number Theorem
- Fundamental Theorem of Calculus (Analysis):
- Analysis: Lebesgue Dominated Convergence Theorem
- Topology: Poincaré Conjecture (3D)
- Algebra: Classification of Finite Simple Groups
- Set Theory: Independence of the Continuum Hypothesis (CH)
- Category Theory: Adjoint Functor Theorem
- Homotopy Theory: Adams Conjecture (via K-theory)
- Consistency of ZFC with Large Cardinals
- Fermat’s Last Theorem
- Large Cardinal Theorem: Martin’s Maximum
showcasing its ability to span algebraic, analytic, topological, and foundational domains. AXIOSIS stands as a candidate for a universal mechanized mathematics platform, rivaling systems like Cubical Type Theory while extending their scope.
AXIOSIS achieves a landmark synthesis, unifying synthetic and classical mathematics in a mechanically verifiable framework. Its type formers—spanning simplicial ∞-categories, stable spectra, cohesive modalities, reals, ZFC, large cardinals, and forcing — cover all known mathematical domains as of 2025.
- Compilation: https://axiosis.github.io/books/axio/axio.pdf
- Github Organization: https://github.com/groupoid/
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$ sudo apt install texlive-full
$ sudo fc-cache -f
$ fc-match Geometria
$ make
Namdak Tonpa