type: axiom
verification: machine_ready
dependencies: ["philosophy-formal.md"]
verification_points:
- axiom_statement
- five_fold_equivalence
- entropy_increase_necessity
- self_referential_dynamics
- discrete_continuous_equivalence
Axiom_A1 := ∀S : System . SelfReferentialComplete(S) → H(S_{t+1}) > H(S_t)
Symbols := {
S: System,
H: Function[System → ℝ₊], // 熵函数
t: Time ∈ ℕ,
SelfReferentialComplete: Property[System]
}
FiveFoldEquivalence := {
E1: ∀S . SRC(S) → (∀t . H(S_t) < H(S_{t+1})),
E2: ∀S . SRC(S) → TimeIrreversible(S),
E3: ∀S . SRC(S) → ObserverEmerges(S),
E4: ∀S . SRC(S) → StructuralAsymmetry(S_t, S_{t+1}),
E5: ∀S . SRC(S) → RecursiveUnfolding(S)
}
Theorem: ∀i,j ∈ {1,2,3,4,5} . E_i ⟺ E_j
ProofStructure := {
E1→E2: EntropyIncreaseImpliesIrreversibility,
E2→E3: IrreversibilityImpliesObserver,
E3→E4: ObserverImpliesAsymmetry,
E4→E5: AsymmetryImpliesRecursion,
E5→E1: RecursionImpliesEntropyIncrease
}
SRC_static(S) :=
SelfReferential(S) ∧
Complete(S) ∧
Consistent(S) ∧
NonTrivial(S)
SRC_dynamic(S) :=
SRC_static(S) ∧
∀t . SRC_static(S_t) ∧
∀t . Evolution(S_t) = S_{t+1}
H_desc(S) := log₂|Description(S)|
H_struct(S) := -∑_{s∈S} p(s)·log₂(p(s))
H_evol(S_t, S_{t+1}) := H_struct(S_{t+1}) - H_struct(S_t)
Proof_Necessity := {
Step1: SRC(S) → RequiresDescription(S),
Step2: RequiresDescription(S) → InformationAccumulation(S),
Step3: InformationAccumulation(S) → EntropyIncrease(S),
Step4: EntropyIncrease(S) → H(S_{t+1}) > H(S_t)
}
Contradiction_Proof := {
Assume: ∃S . SRC(S) ∧ H(S_{t+1}) ≤ H(S_t),
Derive: ¬CanDescribeSelf(S_{t+1}),
Conclude: ¬SRC(S),
Result: Contradiction
}
Discrete_Form := ∀n ∈ ℕ . H(S_n) < H(S_{n+1})
Continuous_Limit := lim_{Δt→0} (H(S_{t+Δt}) - H(S_t))/Δt > 0
Theorem: Discrete_Form ⟺ Continuous_Limit
Information(S) := {
Content: Description(S),
Measure: H(S),
Growth: ΔH(S) = H(S_{t+1}) - H(S_t)
}
Conservation_Growth := {
LocalConservation: ∀subsystem . ΔH_in + ΔH_out = 0,
GlobalGrowth: H_total(t+1) > H_total(t)
}
def verify_axiom_format():
axiom = "∀S : System . SelfReferentialComplete(S) → H(S_{t+1}) > H(S_t)"
return is_valid_formula(axiom)
def verify_five_fold_equivalence():
equivalences = [E1, E2, E3, E4, E5]
return all(prove_equivalence(ei, ej) for ei in equivalences for ej in equivalences)
def verify_entropy_necessity():
return prove_implication(SRC, entropy_increase)
def verify_dynamic_properties():
return all([
verify_evolution_exists(),
verify_src_preserved(),
verify_entropy_increases()
])
def verify_discrete_continuous():
return prove_limit_equivalence(discrete_form, continuous_form)