| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies canonical internal universes such as Gödel’s (L), fine-structure models, Jensen hierarchies, sharps, core models; includes definability-based constructions and minimal models satisfying ZFC. Excludes arbitrary non-definable sets and general forcing extensions unless used externally for comparison. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates at the definability and structural scale: ordinal-indexed levels (L_\alpha), fine-structure parameters, inner models extending (L), and minimal universes closed under definable operations. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Constructible sets, levels (L_\alpha), ordinals, parameters for definability, fine-structure sequences, sharps (e.g., (0^\sharp)), core models (K), directed systems of inner models. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Definability, minimality, absoluteness, coherence across levels, fine-structure ranks, canonical well-orderings, condensation, elementary substructure properties. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Levels of (L), admissible ordinals, fine-structural segments, premice, mice, sharps, core models, definability classes, iterable structures. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Stage index (\alpha), definability predicates, fine-structure parameters, Skolem functions, extender sequences (in advanced core models), coding functions. |
| | Parameterization | How variables encode and represent the system’s state. | System state encoded by definability parameters, ordinal height, fine-structure levels, internal Skolem hulls, and coding schemes within canonical inner models. |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Treat model constructions as perfectly iterable; assume definability behaves uniformly; idealize fine-structure hierarchy; assume coherence across levels; ignore large-cardinal-strength gaps unless explicitly included. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Break down in the presence of large cardinals beyond the scope of a given core model, in non-standard models of arithmetic, or in settings where definability fails to produce a well-behaved hierarchy. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Assumes classical ZFC; existence of Gödel’s constructible universe; well-foundedness; definability governing set existence; fine-structure coherence; condensation; absoluteness under certain embeddings. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes that canonical inner models meaningfully approximate the true universe; definability produces minimal models; fine structure mirrors global set-theoretic structure; internal well-orderability. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Requires coherent interaction among definability, condensation, fine structure, iteration strategy, and minimality; levels must fit together without contradiction. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires compatibility between rank hierarchies, definability classes, fine-structure segments, Skolem functions, and the embedding/iteration systems used in constructing inner models. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Definability patterns in (L), level-by-level construction (L_\alpha), condensation behavior, fine-structure signatures, presence/absence of sharps, elementary substructure patterns, canonical well-orderings. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limits of first-order definability; inability to detect non-constructible sets; failure to observe large-cardinal consequences beyond inner model strength; expressive limits of (L)’s definable hierarchy. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Ordinal height, fine-structure levels, definability complexity, admissibility levels, projecta, sharps indicators (e.g., existence of (0^\sharp)). |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Gödel operations, definability operators, fine-structure machinery, condensation tests, Skolem hulls, extender sequences (in advanced core models), elementary embeddings. |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Definitions of (L_\alpha), constructibility predicates, projecta, admissibility, premice, sharps, fine-structure parameters, elementary substructures, iterable core models. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Constructing (L) via Gödel operations; computing fine-structure parameters; generating Skolem hulls; checking condensation; analyzing sharps; iterating premice; verifying definability closures. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Producing initial segments (L_\alpha); generating fine-structure diagrams; performing inner-model reflection checks; building canonical embeddings; examining extender models. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Selecting specific (L_\alpha) levels; sampling definable subsets; evaluating projecta; examining types over small segments; sampling premice with varying extender strength. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Constructible hierarchies, fine-structure tables, projecta profiles, Skolem functions, condensation maps, extender sequences, sharps data. |
| | Resolution | The granularity or precision with which data is captured. | Fineness determined by granularity of definability hierarchy, depth of fine structure, precision in projecta calculations, and ordinal resolution. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Validating condensation; verifying fine-structure equations; confirming correct (L_\alpha) construction; checking iterability conditions; calibrating extender sequences; ensuring definability closure. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Miscomputed projecta, incorrect condensation results, non-iterable premice, misassigned fine-structure parameters, definability mistakes, or false identification of sharps. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Gödel operations generating (L); condensation lemma; fine-structure recurrence; coherence of definability across levels; canonical well-ordering; admissibility patterns; transitivity of all (L_\alpha). |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Ordinal hierarchy; projecta; Skolem hull invariants; definability ranks; admissible ordinals; core model invariants; iterability; structural minimality of (L). |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Mechanisms generating constructible sets (Gödel operations); fine-structure recursion; condensation processes; Skolem hull construction; extender-based iteration strategies in core models. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Level-by-level generation (L_\alpha); projectum descent; premouse iteration sequences; extender iteration trees; Skolem closure pathways; definability refinement sequences. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Constructibility, (L), (L_\alpha), fine structure, projecta, admissible ordinals, premice, mice, sharps ((0^\sharp)), condensation, iterability, core model (K). |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Admissible vs. inadmissible ordinals, standard vs. nonstandard segments, small vs. large mice, iterable vs. non-iterable structures, core model vs. extender models, definability tiers. |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Recursive definitions of (L_{\alpha+1} = \mathrm{Def}(L_\alpha)); limit stage equations (L_\lambda = \bigcup_{\beta<\lambda} L_\beta); fine-structure equations for projecta; condensation identities. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Gödel’s constructible universe (L); inner models closed under definability; fine-structure models; sharps models; core models; premice and mice; iterable extender models. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Pure (L_\alpha) segments; minimal fine-structure models; toy premice without extenders; simplified admissible hierarchies; idealized versions of (K) with reduced complexity. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Breakdown in presence of large cardinals incompatible with a given core model; failures of condensation; non-iterable premice; definability collapse; models where (0^\sharp) exists vs. does not. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Gödel’s constructible universe as foundational baseline; Jensen fine structure; core model theory; sharps and mice as organizing frameworks for large-cardinal approximations. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Links to recursion theory (admissible ordinals), descriptive set theory (projecta and scales), large cardinal theory (inner model approximations), forcing theory (comparison with outer models), and proof theory (constructible ordinals). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Manipulating definability parameters, altering Gödel operations, restricting or expanding fine-structure rules, testing iterability assumptions, constructing alternative inner models (e.g., premice with/without extenders). |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing condensation, definability patterns, Skolem hull behavior, sharps existence, and fine-structure regularities without modifying the underlying axioms. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing whether a structure satisfies condensation; checking iterability; evaluating fine-structure equations; determining whether large-cardinal-like features appear; testing minimality of inner models. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Reproducing constructibility stages (L_\alpha); replicating fine-structure calculations across different admissible levels; verifying iterability results in equivalent premice; recomputing projecta. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Logical analogues: analyzing frequency of admissibility levels, comparing projecta distributions, evaluating degree of definability in segments, identifying systematic patterns in extender sequences. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing inner models by fine-structure complexity, definability closure, strength of iteration strategies, presence/absence of sharps, and alignment with large cardinal axioms. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Identifying mistakes in fine-structure calculations, misassigned projecta, incorrect condensation claims, non-iterable premice, misconstructed extender sequences, or improper use of Gödel operations. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding selective attention to well-behaved segments; ensuring unbiased examination of premice with problematic extenders; preventing overreliance on canonical models; avoiding hidden assumptions about sharps. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Expert review of inner model constructions, fine-structure derivations, condensation proofs, iteration strategy correctness, and claims about sharps or core model completeness. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Revising fine-structure rules, modifying definitions of premice/mice, altering core model assumptions, adjusting iteration strategies, or incorporating additional large-cardinal-strength parameters. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of definability assumptions, Skolem functions used, fine-structure parameters, extender choices, iteration rules, and all constraints shaping the inner model. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of non-iterability, failures of condensation, misaligned fine structure, incorrect sharps claims; clear attribution of inner model methods; rigorous avoidance of overstated conclusions. |