2nd/10_Wiki/Topics/AI_and_ML/Axioms.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-axioms	Axioms	10_Wiki/Topics	verified	self	공리 axiom postulate ZFC Peano first principles AI constitution	none	A	0.93	applied	logic mathematics axiom foundations formal-system ai-alignment constitutional-ai first-principles		2026-05-10	pending	language applicable_to logic / mathematics Formal Verification AI Alignment Constitutional AI

Axioms

📌 한 줄 통찰

"매 의심 없는 지식 의 시작". 매 self-evident or 매 agreed. 매 모든 reasoning 의 ground. 매 wrong axiom = 매 정교한 logic 의 collapse. 매 modern AI 의 Constitutional AI = 매 axiom 의 hardcode.

📖 핵심

매 정의

• 매 logical reasoning 의 premise.
• 매 prove X (within the system).
• 매 derived 의 X.
• 매 system 의 closure 의 base.

매 type

1. Self-evident: 매 universal intuition. (예: A = A)
2. Defined / postulated: 매 system 의 convention. (예: 평행선 공리)
3. Empirical (科學): 매 experiment-based.
4. Practical / regulative: 매 useful 의 assume.

매 famous axiom system

Euclid (geometry, 300 BC)

1. 매 두 point 의 line.
2. 매 line 의 extend.
3. 매 center + radius 의 circle.
4. 매 right angle 의 equal.
5. 매 parallel postulate (controversial → 비유클리드).

Peano (natural numbers, 1889)

1. 0 ∈ ℕ.
2. n ∈ ℕ → S(n) ∈ ℕ.
3. S 의 injective.
4. 0 ∉ range(S).
5. Induction.

ZFC (set theory, modern)

• Extensionality, Pairing, Union, Power, Infinity, Replacement, Foundation, Choice.

Kolmogorov (probability)

1. P(A) ≥ 0.
2. P(Ω) = 1.
3. Countable additivity.

Field axioms (algebra)

• Associativity, commutativity, distributivity, identity, inverse.

매 limitation

Gödel (1931)

• 매 sufficient axiom system 의 매 incomplete (true 가 prove X) or 매 inconsistent.
• 매 self-reference 의 fundamental.

매 axiom 의 의 choice

• 매 different axiom → 매 different math.
• 매 Euclidean vs 매 non-Euclidean.
• 매 ZFC vs 매 ZF + AD.

매 First Principles Thinking

• 매 Aristotle / Descartes / Musk.
• 매 base 의 break.
• 매 reason from scratch.
• vs analogy / convention.

→ 매 reduce → 매 axiom → 매 rebuild.

매 AI 의 응용

Constitutional AI (Anthropic)

• 매 axiom 의 model 의 hardcode.
• 매 "be helpful, harmless, honest".
• 매 principle 의 critique + revise.

Asimov's Laws (fiction)

• 매 robot 의 3 law (+ 0).
• 매 simple 가, 매 conflict.

Formal verification

• 매 axiom + 매 inference rule → 매 theorem.
• 매 Lean / Coq.

Ontology

• 매 RDF / OWL.
• 매 axiom = 매 inference rule.

매 design 의 axiom (engineering)

• DRY: Don't Repeat Yourself.
• YAGNI: You Aren't Gonna Need It.
• KISS: Keep It Simple, Stupid.
• SOLID (5 axiom).
• 매 universal X 가, 매 useful guideline.

💻 패턴

Constitutional AI principles (Anthropic style)

constitution = [
    "Choose the response that is most helpful, honest, and harmless.",
    "Choose the response that is least likely to be perceived as harmful or offensive.",
    "Choose the response that is most truthful and avoids speculation.",
    "Choose the response that respects autonomy and dignity.",
    "Choose the response that protects privacy.",
]

def critique_and_revise(response, prompt):
    for principle in constitution:
        critique = llm.generate(f"""Given:
Prompt: {prompt}
Response: {response}
Principle: {principle}

Critique the response according to the principle.""")
        
        if critique.has_issue():
            response = llm.generate(f"""Revise the response to address: {critique}""")
    return response

Lean axiom

-- 매 Peano arithmetic 의 axiom
axiom Nat : Type
axiom zero : Nat
axiom succ : Nat → Nat
axiom succ_inj : ∀ a b, succ a = succ b → a = b
axiom zero_ne_succ : ∀ n, zero ≠ succ n
axiom induction : ∀ (P : Nat → Prop), 
  P zero → (∀ n, P n → P (succ n)) → ∀ n, P n

First principles (Musk style)

Problem: 매 batteries are too expensive.
Conventional reasoning: 매 lithium battery 의 $X / kWh.

First principles:
1. 매 battery = 매 cobalt + nickel + aluminum + carbon + polymer + steel.
2. 매 commodity price 의 sum = $80 / kWh.
3. 매 manufacturing markup = $X.
4. → 매 actual minimum.

Z3 with axioms

from z3 import *

# 매 axiom: 매 모든 person has a parent.
Person = DeclareSort('Person')
parent = Function('parent', Person, Person)

s = Solver()
x = Const('x', Person)
s.add(ForAll([x], Exists([parent(x)], True)))

# 매 query
y = Const('y', Person)
s.push()
s.add(parent(y) == y)  # 매 self-parent X?
print(s.check())  # sat / unsat
s.pop()

Formal property check

# 매 axiom: 매 transaction 의 atomicity
ATOMICITY = '∀ tx. completed(tx) ⟺ all_steps(tx) ∨ no_steps(tx)'

def verify_axiom(system, axiom):
    return formal_verifier.check(system, axiom)

Axiom-based fairness

# 매 axiom: 매 protected attribute 의 swap → prediction 의 same
def counterfactual_axiom_check(model, x, protected_idx):
    flipped = x.copy()
    flipped[protected_idx] = 1 - flipped[protected_idx]
    return model.predict(x) == model.predict(flipped)

🤔 결정 기준

상황	Approach
Math research	ZFC / type theory
AI alignment	Constitutional axioms
Formal verify	Lean / Coq
Engineering design	SOLID + KISS + YAGNI
Innovation	First principles
Fairness	Counterfactual axiom
Knowledge graph	Ontology axiom (OWL)

기본값: 매 axiom 의 explicit. 매 implicit assumption 의 surface.

🔗 Graph

• 부모: Logic · Mathematics · Foundations
• 변형: Postulate · Theorem · Definition
• 응용: Constitutional-AI · Formal-Verification · Ontology · First-Principles-Thinking
• 비판: Gödel-Incompleteness · Non-Euclidean · Asimov-Laws
• Adjacent: Lean-4 · Coq · ZFC · Peano · Z3

🤖 LLM 활용

언제: 매 reasoning 의 base 의 surface. 매 AI alignment 설계. 매 formal verify. 매 first principles 의 problem 의 break. 언제 X: 매 axiom 의 dogma 의 mistake. 매 specific 시 의 axiom 의 universal claim.

❌ 안티패턴

• Hidden axiom: 매 implicit 의 inconsistent.
• Too many axiom: 매 system 의 complex.
• Inconsistent axiom: 매 anything provable (ex falso).
• No falsifiability (empirical): 매 dogma.
• Asimov 의 simple 의 imitate: 매 conflict.
• Constitution 의 vague: 매 enforcement X.

🧪 검증 / 중복

• Verified (Euclid, Peano, ZFC, Anthropic Constitutional AI paper).
• 신뢰도 A.
• Related: Formal-Verification · Constitutional-AI · First-Principles-Thinking · Lean-4.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — system + Gödel + Constitutional AI + 매 Lean / Z3 / first principles