type: definition
verification: machine_ready
dependencies: ["A1-formal.md"]
verification_points:
- self_referential
- completeness
- consistency
- non_triviality
SelfReferentialComplete(S : System) : Prop ≡
SelfReferential(S) ∧
Complete(S) ∧
Consistent(S) ∧
NonTrivial(S)
SelfReferential(S : System) : Prop ≡
∃f : Function[System → System] . S = f(S)
Complete(S : System) : Prop ≡
∀x : Element . x ∈ S →
∃y : Element . y ∈ S ∧
∃g : Function[System → System] . x = g(y)
Consistent(S : System) : Prop ≡
¬∃x : Element . (x ∈ S ∧ ¬x ∈ S)
NonTrivial(S : System) : Prop ≡
|S| > 1
Notation: SRC(S) := SelfReferentialComplete(S)
Notation: S := S := SelfReferential(S)
Irreducible(S) :=
SRC(S) → ¬∃(S₁, S₂) . S = S₁ ∪ S₂ ∧ ¬SRC(S₁) ∧ ¬SRC(S₂)
Closed(S) :=
SRC(S) → ∀op : Operation . op(S) ⊆ S
Recursive(S) :=
SRC(S) → ∃desc : Description . desc ∈ S ∧ Describes(desc, S)
Dynamic(S) :=
SRC(S) → ∃t : Time . S_t ≠ S_{t+1}
def verify_self_referential(system):
return exists_function_f_such_that_S_equals_f_S(system)
def verify_completeness(system):
return all_elements_have_internal_origin(system)
def verify_consistency(system):
return no_contradictory_elements(system)
def verify_non_triviality(system):
return len(system.elements) > 1