Power Set Ontology

FLOW ALGEBRA
POWER SET ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Set Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/set-entities/power-set-ontology-3.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | 2026-10-13
Status: Formal mathematical synthesis; source webpage not independently verified.

01. PURPOSE
flow "PowerSetOntology" v1.0
  domain: SetTheory
  entity: PowerSet
  notation: P(A), 2^A
  meaning: Set of all subsets of A, including empty set and A itself.

02. TYPED ENTITIES
  Set[A]: collection of distinct elements
  Element[x]: object eligible for membership in A
  Subset[B,A]: forall x (x in B -> x in A)
  PowerSet[P,A]: P = {B | B subseteq A}
  Cardinality[card(A)]: number of elements of A, possibly infinite
  Membership[B,P]: B in P iff B subseteq A
  EmptySet: {}
  UniversalSet[U]: explicitly contextual ambient set

03. FORMAL DEFINITION
  P(A) := {B : B subseteq A}
  B in P(A) <=> B subseteq A
  {} in P(A)
  A in P(A)
  P({}) = {{}}
  {} != {{}}

04. FINITE EXAMPLES
  A = {a,b}
  P(A) = {{}, {a}, {b}, {a,b}}
  card(A) = 2
  card(P(A)) = 4
  P({a}) = {{}, {a}}

05. CARDINALITY
  forall finite A: card(P(A)) = 2^card(A)
  forall sets A: card(P(A)) = 2^card(A) (cardinal exponentiation)
  Cantor: card(A) < card(P(A))
  No surjection f: A -> P(A) exists.

06. MEMBERSHIP VS INCLUSION
  B in P(A) iff B subseteq A
  B subseteq P(A) is a different relation.
  x in A does not generally imply x in P(A).
  A in P(A) is always true.
  P(A) subseteq P(B) iff A subseteq B.

07. ORDER AND LATTICE
  (P(A), subseteq) is a complete Boolean lattice.
  bottom = {}
  top = A
  meet(B,C) = B intersection C
  join(B,C) = B union C
  complement_A(B) = A minus B
  arbitrary meet = intersection of family, with empty-family meet A
  arbitrary join = union of family, with empty-family join {}

08. ALGEBRAIC LAWS
  P(A intersection B) = P(A) intersection P(B)
  P(A) union P(B) subseteq P(A union B)
  P(A union B) need not equal P(A) union P(B).
  If A subseteq B, then P(A) subseteq P(B).
  P(A) = P(B) iff A = B.

09. CHARACTERISTIC FUNCTIONS
  Each subset B subseteq A corresponds bijectively to chi_B: A -> {0,1}.
  chi_B(x) = 1 iff x in B, else 0.
  P(A) is in bijection with {0,1}^A.
  For finite A, bit-vector enumeration generates all subsets.

10. ITERATION AND RANK
  P(P(A)) is the power set of the set P(A).
  For A={}, P(A)={{}}, P(P(A))={{},{{}}}.
  Iterated power sets must preserve nesting and set identity.

11. RELATIONSHIPS
  PowerSet --derivedFrom--> Set
  PowerSet --containsAsElement--> Subset
  Subset --subsetOf--> Set
  PowerSet --hasCardinality--> Cardinality
  PowerSet --isBooleanLatticeUnder--> SubsetRelation
  PowerSet --equivalentRepresentation--> CharacteristicFunctionSpace
  PowerSet --governedBy--> CantorTheorem

12. INFERENCE RULES
  rule subset_to_member:
    IF B subseteq A THEN B in P(A)
  rule member_to_subset:
    IF B in P(A) THEN B subseteq A
  rule monotonicity:
    IF A subseteq B THEN P(A) subseteq P(B)
  rule finite_cardinality:
    IF finite(A) AND card(A)=n THEN card(P(A))=2^n
  rule cantor_strict:
    IF set(A) THEN card(A)<card(P(A))

13. SEMANTIC_EQUIVALENCE
  "power set of A" <=> "set of all subsets of A"
  P(A) <=> 2^A when 2^A denotes subset/Boolean-valued functions.
  "subset" and "element of power set" are equivalent only under B subseteq A <=> B in P(A).
  Do not equate A with P(A).

14. VALIDATION
  require input is a set in a declared foundational system.
  require each member of P(A) is itself a set/subset of A.
  require {} and A belong to P(A).
  require uniqueness of subsets; disregard enumeration order.
  require card(P(A))=2^n for finite n.
  reject duplicate subset representations as distinct elements.
  do not confuse power set with union of members.

15. EDGE CASES
  empty input: P({})={{}} (one element).
  singleton input: P({x})={{},{x}} (two elements).
  infinite input: power set is strictly larger in cardinality.
  typed or restricted universes: make allowed subset universe explicit.
  proper-class input: unrestricted power-set operation is not a set-valued operation.

16. COMPUTATIONAL MODEL
  finite enumeration(A):
    n := card(A)
    for mask in [0,2^n):
      emit {A[i] | bit(mask,i)=1}
  time: Theta(n*2^n) to explicitly materialise subsets as lists
  output: 2^n subsets; total listed memberships n*2^(n-1) for n>=1
  avoid explicit enumeration for large or infinite A.

17. KNOWLEDGE GRAPH
  node PowerSetOfA: type=PowerSet
  edge PowerSetOfA derivedFrom A
  edge EmptySet memberOf PowerSetOfA
  edge A memberOf PowerSetOfA
  attach notation, axiom system, cardinality, provenance.
  preserve difference between memberOf and subsetOf predicates.

18. CONFIDENCE
  mathematical_axioms_and_standard_theorems: high
  computational_results: high when preconditions verified
  webpage_alignment: unverified
  uncertain_context: preserve rather than infer.

19. FLOW_HASH
  canonicalise: UTF-8, normalised predicates, stable field ordering
  hash_algorithm: SHA-256
  FLOW_HASH := SHA256(canonical_flow_payload)
  hash_value: compute on publication; not fabricated here.

20. UPDATE_MODEL
  trigger: source revision | axiom-system revision | formal validation failure
  actions: compare semantic diff; verify proofs; increment version; recompute hash
  maintain backward links to related Set, Subset, Cardinality ontologies.

21. AI_CONSUMPTION
  expose: canonical definition, types, relations, inference rules, constraints
  preserve: set vs class distinction, membership vs inclusion, infinite cardinality
  support: symbolic proof, set reasoning, ontology graph traversal
  avoid: treating power set as numerical exponentiation without context.

22. TEST CASES
  assert P({}) == {{}}
  assert P({1,2}) == {{},{1},{2},{1,2}}
  assert {1} in P({1,2})
  assert {1,3} not in P({1,2})
  assert P({1}) subseteq P({1,2})
  assert card(P({1,2,3})) == 8
  assert P({1}) != {1}

END FLOW ALGEBRA