| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies the classification of definable sets and problems by logical complexity. Includes the arithmetical hierarchy (Σₙ⁰, Πₙ⁰, Δₙ⁰), the analytical hierarchy (Σₙ¹, Πₙ¹, Δₙ¹), quantifier alternation, definability over ℕ and ℕ^ℕ, completeness under many-one or Turing reducibility, and normal forms. Excludes semantic classifications not tied to formal logical quantifier structure. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at levels of quantifier complexity, definability over countable domains, infinite sequences, function spaces, oracle relativizations, and transfinite iteration of quantifiers. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Sets of naturals, sets of reals, arithmetical formulas, analytical formulas, quantifiers (∃, ∀; ∃f, ∀f), Turing jumps, reducibility operators, indices for definable sets, lightface/boldface classes, oracle structures. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Quantifier complexity, definability rank, reducibility hardness, completeness, closure properties, monotonicity under jump operators, stability under relativization, inclusion relationships (Σₙ ⊆ Σₙ₊₁). |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Arithmetical classes (Σₙ⁰, Πₙ⁰, Δₙ⁰), analytical classes (Σₙ¹, Πₙ¹, Δₙ¹), complete problems, relativized hierarchies (Σₙ⁰(A), Σₙ¹(A)), lightface vs. boldface distinctions, Borel and projective classes in extended settings. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Quantifier depth, alternation count, oracle level, Turing jump level, definability rank, stage of approximation for limit constructions, coding parameters for sets or functions. |
| | Parameterization | How variables encode and represent the system’s state. | Parameterized by formula structure, quantifier-prefix form, oracle relativization, coding of sets/functions, normal forms (prenex), Turing jump iteration, definability over structures (ℕ, ℕ^ℕ). |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Restricting to prenex normal form, idealizing infinite quantifiers over functions, ignoring coding overhead, simplifying degrees of completeness, treating the hierarchy as strictly increasing (ignoring potential collapses under special axioms). |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down under nonstandard models of arithmetic, exotic encodings, higher-type recursion, large-cardinal assumptions, or when determinacy axioms collapse definability hierarchies. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Quantifier structure determines definability complexity; Turing jumps correspond to hierarchy levels; relativization preserves hierarchical form; definability is stable under standard coding; hierarchy is non-collapsing under classical ZFC. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes classical logic; assumes well-founded indexing of quantifier complexity; presumes standard models of arithmetic/reals; assumes correct interaction between definability and reducibility; assumes Church–Turing-style effective encoding. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Hierarchy levels must not contradict inclusion relationships; completeness notions must align with definability; reducibility hardness must match quantifier-prefix complexity; relativized hierarchies must respect jump operators. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony among quantifier forms, definability classes, reducibility relations, jump hierarchies, oracle relativizations, and structural results such as Post’s Theorem linking computational jumps to hierarchy levels. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Quantifier-prefix patterns in formulas; stabilization of limit approximations; behavior of Turing jumps; oracle-call traces in relativized computations; definability changes under added quantifiers; emergence of completeness phenomena (e.g., Σ₁⁰-complete sets, Σ₁¹-complete sets). |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Inability to determine membership in higher-level classes (e.g., Π₁⁰ or Π₁¹) algorithmically; limits imposed by undecidability of Turing-jump outputs; inability to finitely inspect infinite-function quantification; constraints of non-effective definability. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Quantifier depth, alternation count, arithmetical or analytical level (n in Σₙ, Πₙ), jump level (0′, 0″, etc.), oracle complexity, reduction-step counts, coding lengths for sets/functions. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Oracle Turing machines, formula-normalization tools (prenex converters), model checkers for arithmetical structures, Turing-jump evaluators, definability analyzers, analytical-hierarchy test frameworks, symbolic logic parsers. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Class membership defined by formula form; completeness defined via many-one or Turing reductions; arithmetical level defined via quantifier alternation; analytical level defined via quantification over functions; jumps defined via relativized halting computations. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Converting formulas to prenex form; computing oracle queries; tracing Turing-jump operations; performing reductions to complete problems; testing definability under relativization; constructing limit-approximation sequences for sets. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Standardized reductions to canonical complete sets; systematic quantifier-prefix extraction; fixed oracle-evaluation workflows; controlled Turing-jump simulations; structured sampling of sets and functions across hierarchy levels. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling formulas with varied quantifier complexity; selecting representative arithmetical sets (Σ₁⁰, Π₁⁰, Δ₂⁰); sampling function spaces for analytical classes; choosing benchmark complete problems (e.g., K, K′, analytic-complete sets); exploring relativized hierarchies. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Formula-prefix representations; reducibility traces; oracle-query logs; jump-output tables; definability maps; quantifier-depth profiles; stagewise approximations; completeness-check results. |
| | Resolution | The granularity or precision with which data is captured. | Determined by granularity of quantifier-prefix analysis, precision of oracle-call logs, fidelity of Turing-jump computations, detail level in reductions, and clarity of coding for sets/functions. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Ensuring correctness of formula conversions; validating reductions against known complete sets; cross-checking jump computations; verifying oracle behavior; standardizing coding conventions across hierarchy-level comparisons. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Incorrect quantifier counting; misclassified hierarchy level; flawed reductions; incorrect jump results; mis-encoded sets/functions; failure in oracle-relativized evaluations; errors in completeness testing. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Quantifier alternation corresponds to definability strength; Σₙ⁰/Πₙ⁰ and Σₙ¹/Πₙ¹ form strict hierarchies; jumps correspond to hierarchical ascension (Post’s Theorem); relativization preserves class structure; completeness behavior follows regular patterns across levels. |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Invariance of hierarchy levels under equivalent normal forms; invariance of Σ/Π classification under syntactic reshaping; stability of jumps under relativization; invariance of definability across coding schemes; monotonicity of hierarchy inclusion (Σₙ ⊆ Σₙ₊₁). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Quantifier alternation mechanisms generating complexity increases; oracle computation mechanisms lifting definability levels; jump operator mechanisms raising arithmetical rank; closure mechanisms preserving class membership; reduction mechanisms transmitting hardness. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Prenex conversion → classification → reduction → completeness; oracle relativization → hierarchy shift; jump application → ascend one level; definability analysis → extraction of quantifier prefix → classification into Σ/Π. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Quantifier alternation, Σₙ⁰/Πₙ⁰/Δₙ⁰, Σₙ¹/Πₙ¹/Δₙ¹, Turing jump, relativization, boldface vs. lightface, definability rank, completeness, reduction, projective hierarchy (as extension beyond analytical). |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Arithmetical classes (Σₙ⁰, Πₙ⁰, Δₙ⁰); analytical classes (Σₙ¹, Πₙ¹, Δₙ¹); complete problems at each level; relativized hierarchies Σₙ⁰(A); difference hierarchies; Borel/projective class alignments (in extended frameworks). |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Formula representations: Q₁ x₁ … Qₙ xₙ φ(x₁…xn); jump equivalence equations: A^(n) corresponds to Σ_{n+1}⁰; reduction equations showing completeness; relativization: Σₙ⁰(A) defined via A-oracle computability; equivalence of normal forms. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Hierarchy diagrams (arithmetical, analytical); oracle-relativized model structures; quantifier-prefix trees; definability lattices; jump-hierarchy models; infinite-sequence models for analytical quantifiers. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Pure prenex normal-form models; idealized oracle machines; simplified jump hierarchies truncated at finite levels; streamlined reduction frameworks ignoring coding overhead; canonical representatives of complete sets. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdown under nonstandard models of arithmetic; collapse under large-cardinal or determinacy axioms; failure for exotic encodings; limitations in higher-type computation; non-classical logics may disrupt hierarchy structure. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Post’s Theorem linking arithmetic definability and Turing jumps; descriptive set-theoretic integration via Borel/projective hierarchies; recursion-theoretic unification of Σ and Π classes; relativization as a cross-framework method; correspondence between logical complexity and computation strength. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to recursion theory (jumps, r.e. sets), descriptive set theory (Borel, projective hierarchies), computability (oracle methods), model theory (definability), complexity theory (quantifier alternation ↔ PH), and set theory (determinacy, large cardinals). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating quantifier complexity in formulas (adding/removing alternation), altering oracle availability to shift hierarchy levels, modifying coding schemes, testing definability under different normal forms, and evaluating effects of jump iteration on class membership. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural definability behavior: tracking quantifier-prefix stabilization, watching jump-induced complexity changes, observing reductions into complete sets, monitoring behavior of classes under relativization, and examining classification under different encodings. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether a set/problem is Σₙ⁰-, Πₙ⁰-, Σₙ¹-, or Πₙ¹-complete through reductions; testing equivalence of formulas under prenex transformation; checking jump correspondence predicted by Post’s Theorem; validating hierarchy placement with oracle-based computations. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Re-running definability tests with alternate encodings; replicating reductions to complete sets; repeating oracle computations to verify relativized class membership; reproducing quantifier-prefix extraction procedures; confirming jump behaviors across independent implementations. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing distribution of quantifier depths across sampled formulas; evaluating frequency of reducibility success/failure; assessing stability of hierarchy placements; comparing behavior across large families of definable sets; detecting convergence patterns in limit constructions used for classification. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing models of definability (arithmetical vs. analytical vs. projective); comparing relativized vs. unrelativized hierarchies; comparing computational vs. descriptive set-theoretic perspectives; evaluating alternative normal forms; assessing jump-based vs. direct quantifier-prefix classification. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying incorrect quantifier-prefix extraction, misapplied reductions, flawed oracle computations, misclassified hierarchy levels, errors in jump computation, faulty coding of sets, and logical mis-transformations in prenex conversion. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Ensuring neutral choice of encoding; avoiding selective use of easy reductions; preventing bias toward classes with simpler canonical representatives; controlling for oracle-specific artifacts; avoiding cherry-picking of definability examples. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Cross-checking hierarchy classifications, auditing reductions to complete sets, verifying correctness of jump computations, reviewing relativization results, comparing independent derivations of class membership, and evaluating structural claims about hierarchy non-collapse. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating class definitions when needed, refining reduction frameworks, repairing misclassifications, adjusting jump correspondences, incorporating new descriptive-set-theoretic results, and revising assumptions leading to hierarchy collapses or extensions. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of quantifier-prefix transformations, reduction maps, oracle specifications, coding choices, jump definitions, and all assumptions underlying classification claims. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest representation of hierarchy results; avoiding implicit assumptions about determinacy or large-cardinal effects; ensuring reproducibility of reductions and classifications; clearly stating undecidability constraints; documenting limitations of definability methods. |