**Appendix C: Detailed Graph Data (Edges)** **C.6: Edges Originating from Logic Nodes** | Source Node ID | Target Node ID | Type | Rationale | Conf. | Dir. | |:------------------------- |:----------------------- |:------- |:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |:---- |:--- | | `FOL` | `ZFC` | `F-BAS` | First-Order Logic provides the formal language and deductive system within which the axioms of ZFC are stated and theorems are derived. | H | D | | `FOL` | `Math::FormalSystem` | `L-INST` | First-Order Logic is a primary example and foundation for many mathematical formal systems. | H | D | | `FOL` | `Logic::Soundness` | `L-ENT` | The standard proof systems for FOL are sound (only prove valid formulas). | H | D | | `FOL` | `Logic::Completeness` | `L-ENT` | FOL is semantically complete (all valid formulas are provable) according to Gödel’s Completeness Theorem. | H | D | | `FOL::Syntax` | `FOL` | `S-COMP` | The syntax (symbols, formation rules) is a defining component of FOL. | H | D | | `FOL::Semantics` | `FOL` | `S-COMP` | The semantics (rules for interpretation in models, truth definition) are a defining component of FOL. | H | D | | `FOL::ProofTheory` | `FOL` | `S-COMP` | The proof theory (logical axioms, inference rules) is a defining component of FOL. | H | D | | `FOL::ProofTheory` | `Logic::Soundness` | `L-ENT` | Soundness is a property *of* the proof theory relative to the semantics. | H | D | | `ZFC` | `Mathematics` | `F-BAS` | ZFC serves as the standard foundational axiomatic system for the vast majority of contemporary mathematics. | H | D | | `ZFC` | `Math::Continuum` | `S-FORM` | The real number continuum is typically constructed rigorously within ZFC using set-theoretic definitions (e.g., Dedekind cuts). | H | D | | `ZFC` | `Math::HilbertSpace` | `S-FORM` | The definition and properties of Hilbert spaces rely on concepts (sets, functions, limits) formally grounded in ZFC. | H | D | | `ZFC` | `Math::GoedelTheorems` | `F-CHL` | Gödel’s Incompleteness Theorems apply to ZFC (assuming its consistency), challenging its potential for syntactic completeness and demonstrating the unprovability of its own consistency from within | H | D | | `ZFC::Axioms` | `ZFC` | `S-COMP` | The specific axioms are the defining components of the ZFC system. | H | D | | `ZFC::AxiomOfChoice` | `ZFC::Axioms` | `S-COMP` | The Axiom of Choice is a distinct (and historically controversial) axiom within the ZFC axiom set. | H | D | | `ZFC::AxiomOfInfinity` | `ZFC::Axioms` | `S-COMP` | The Axiom of Infinity is crucial within ZFC for founding analysis and dealing with infinite sets. | H | D | | `ZFC::AxiomOfRegularity` | `ZFC::Axioms` | `S-COMP` | The Axiom of Regularity is a key axiom in ZFC ensuring sets are well-founded. | H | D | | `ZFC::SetConcept` | `ZFC` | `F-BAS` | The ontological commitment to pure sets is fundamental to the ZFC framework. | H | D | | `ZFC::CumulativeHierarchy` | `ZFC` | `F-BAS` | The cumulative hierarchy provides the standard intended model or picture for the universe of sets described by ZFC. | H | D | | `ZFC::CumulativeHierarchy` | `ZFC::AxiomOfRegularity` | `L-ENT` | The Axiom of Regularity formally implies the well-founded hierarchical structure. | H | D |