ethicsRealization
This definition constructs the ethical realization as a LogicRealization instance whose carrier is the natural numbers counting morally meaningful improvement steps. Researchers embedding ethical structure into the universal forcing chain would cite it when linking moral improvement counts to the arithmetic orbit. The definition proceeds by direct field assignment, with reflexivity on toNat and induction lemmas from LogicNat discharging the required equalities.
claimThe ethical realization is the LogicRealization structure with carrier the natural numbers (as counts of morally meaningful improvement steps), cost function ethicsCost, zero element 0, successor the standard successor on naturals, orbit the inductive type LogicNat, interpretation map sending each orbit element to its iteration count, and axioms including identity, non-contradiction, excluded middle, composition, action invariance, and nontriviality witnessed by the element 1.
background
LogicRealization supplies the interface for realizing logical structures inside arithmetic carriers, with fields for carrier, cost, orbit, interpretation, and the standard Peano and logic axioms. MoralImprovementStep is the abbreviation for the natural numbers, serving here as the count of ethical improvement steps. The upstream LogicNat inductive type models the orbit under repeated application of the generator, with constructors identity and step; its derived theorems include succ_injective (Peano P2) and induction, which mirror the multiplicative orbit closed under the generator.
proof idea
The definition directly populates the LogicRealization record: Carrier to MoralImprovementStep, Cost to Nat, compare to ethicsCost, Orbit to LogicNat, interpret to ethicsInterpret, and the remaining fields by direct assignment. interpret_step holds by rfl on the definition of toNat. orbit_no_confusion applies LogicNat.zero_ne_succ, orbit_step_injective applies LogicNat.succ_injective, orbit_induction applies LogicNat.induction, and nontriviality is witnessed by the element 1 with a simp on ethicsCost.
why it matters in Recognition Science
This definition supplies the ethical carrier for the Universal Forcing chain and feeds directly into ethics_arith_equiv_nat, which establishes the equivalence between the arithmetic Peano carrier of ethicsRealization and LogicNat. It realizes the ethical improvement count within the T0-T8 forcing sequence, where the orbit structure mirrors the phi-ladder and eight-tick octave. The module keeps the ethics domain lightweight, leaving open the full integration with kinship systems or cellular automata rules.
scope and limits
- Does not reconstruct the full domain theory of ethics.
- Does not prove any specific ethical axioms beyond the LogicRealization interface.
- Does not connect to physical constants, mass formulas, or the phi-ladder.
- Does not address nontrivial kinship systems or cellular automata locality.
Lean usage
example : (arithmeticOf ethicsRealization).peano.carrier ≃ LogicNat := ethics_arith_equiv_natformal statement (Lean)
38def ethicsRealization : LogicRealization where
39 Carrier := MoralImprovementStep
proof body
Definition body.
40 Cost := Nat
41 zeroCost := inferInstance
42 compare := ethicsCost
43 zero := 0
44 step := Nat.succ
45 Orbit := LogicNat
46 orbitZero := LogicNat.zero
47 orbitStep := LogicNat.succ
48 interpret := ethicsInterpret
49 interpret_zero := by rfl
50 interpret_step := by
51 intro n
52 show LogicNat.toNat (LogicNat.succ n) = Nat.succ (LogicNat.toNat n)
53 rfl
54 orbit_no_confusion := by intro n h; exact LogicNat.zero_ne_succ n h
55 orbit_step_injective := LogicNat.succ_injective
56 orbit_induction := by
57 intro P h0 hs n
58 exact LogicNat.induction (motive := P) h0 hs n
59 orbitEquivLogicNat := Equiv.refl LogicNat
60 orbitEquiv_zero := rfl
61 orbitEquiv_step := by intro n; rfl
62 identity := ethicsCost_self
63 nonContradiction := ethicsCost_symm
64 excludedMiddle := True
65 composition := True
66 actionInvariant := True
67 nontrivial := by
68 refine ⟨1, ?_⟩
69 simp [ethicsCost]
70