Element Ontology

FLOW ALGEBRA
TITLE: Element Ontology
VERSION: 1.0
DOMAIN: Discrete Mathematics / Set Theory / Element
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/element-ontology-2.html
STATUS: Domain-based formalisation; source page contents not independently verified.
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Represent elements as mathematical objects and formalise their membership, identity, typing, and relationships to sets without conflating membership with subset inclusion.

2. TYPED ENTITIES
Element: object
Set: collection of distinct objects
Universe: ambient set or domain of discourse
OrderedPair: ordered tuple (first: object, second: object)
MembershipStatement: proposition (element: object, set: Set)
EqualityStatement: proposition (left: object, right: object)
Predicate: function(object) -> Boolean
Cardinality: nonnegative integer or cardinal number
Proof: derivation of a proposition
Context: logical assumptions and scope

3. CORE RELATIONSHIPS
MEMBER_OF(x: Element, A: Set) -> Boolean [x ∈ A]
NOT_MEMBER_OF(x: Element, A: Set) -> Boolean [x ∉ A]
EQUALS(x: Element, y: Element) -> Boolean [x = y]
SUBSET_OF(A: Set, B: Set) -> Boolean [A ⊆ B]
SATISFIES(x: Element, P: Predicate) -> Boolean [P(x)]
BELONGS_TO_UNIVERSE(x: Element, U: Universe) -> Boolean [x ∈ U]

4. OPERATIONS
MEMBERSHIP_TEST(x, A) := x ∈ A
SINGLETON(x) := {x}
PAIR_SET(x, y) := {x, y}
SET_BUILDER(U, P) := {x ∈ U | P(x)}
EXTENSIONAL_EQUALITY(A, B) := ∀x (x ∈ A ↔ x ∈ B)

5. FORMAL INFERENCE RULES
R1: x ∈ A ∧ A ⊆ B => x ∈ B
R2: x = y ∧ x ∈ A => y ∈ A
R3: x ∈ A ∩ B => x ∈ A ∧ x ∈ B
R4: x ∈ A ∪ B => x ∈ A ∨ x ∈ B
R5: x ∈ A \ B => x ∈ A ∧ x ∉ B
R6: x ∈ {y} ↔ x = y
R7: x ∈ {z ∈ U | P(z)} ↔ (x ∈ U ∧ P(x))
R8: A = B ↔ ∀x (x ∈ A ↔ x ∈ B)
R9: (x ∈ A) does not imply (x ⊆ A); membership and inclusion are distinct predicates.
R10: x ∈ ∅ is false for every x.

6. VALIDATION AND CONSTRAINTS
- Every membership statement must specify an element, set, and logical context.
- Membership is not automatically transitive: x ∈ A and A ∈ B do not generally imply x ∈ B.
- A set may itself be an element of another set; its role is context-dependent.
- Extensional equality is equality of sets by their members, not equality of their names.
- Avoid unrestricted comprehension; use bounded set-builder expressions or a specified axiomatic set theory.
- Distinguish mathematical certainty from empirical or heuristic confidence.

7. CONFIDENCE
DEDUCTIVE_CONFIDENCE := 1.0 only when premises and inference are formally verified.
HEURISTIC_CONFIDENCE := [0,1] for extracted or proposed mappings; not a proof probability.
UNKNOWN := unresolved proposition, never silently coerced to false.

8. SEMANTIC_EQUIVALENCE
"element of", "member of", "belongs to" => MEMBER_OF when the mathematical context matches.
"is contained in" => ambiguous; resolve whether membership or subset inclusion is intended.
"identical to", "equal to" => EQUALS when referring to mathematical equality.

9. PROVENANCE
Record source URL, source version, extraction date (if verified), axiomatic framework, proof dependencies, and author/reviewer for each assertion.
Source verification status: unverified domain-based formalisation.

10. FLOW_HASH
FLOW_HASH := SHA256(canonical_utf8(normalised_flow_without_FLOW_HASH))
Canonicalisation requires stable field ordering, whitespace rules, Unicode normalisation, and explicit versioning.

11. UPDATE_MODEL
On a source or ontology change:
  (a) compare canonical definitions and axioms;
  (b) classify added, removed, and changed relationships;
  (c) revalidate dependent proofs and inference rules;
  (d) preserve prior versions and provenance;
  (e) recompute FLOW_HASH and publish the revision.

12. AI_CONSUMPTION
Input: typed mathematical statement + context + source/proof metadata.
Process: parse -> type-check -> disambiguate MEMBER_OF vs SUBSET_OF -> apply sound rules -> validate -> attach provenance.
Output: proposition, status {proved, disproved, unknown}, proof trace, assumptions, confidence metadata, and flow version.
Safety: do not claim theoremhood from lexical similarity or unverified retrieval alone.

END FLOW ALGEBRA