C11-2 理论不完备性形式化规范

依赖组件

from typing import Set, Dict, List, Tuple, Optional, Callable
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
from C11_1 import Theory, Formula, Proof, ReflectionOperator
from C10_1 import GödelEncoder, FormalSystem
from base import No11Number

核心数据结构

1. 可证性谓词

@dataclass
class ProvabilityPredicate:
    """可证性谓词 Prov_T(x)"""
    theory: Theory
    symbol: Symbol  # 表示可证性的关系符号
    
    def apply(self, formula_code: No11Number) -> Formula:
        """构造 Prov_T(⌜φ⌝)"""
        code_term = ConstantTerm(f"F_{formula_code.value}")
        return AtomicFormula(self.symbol, (code_term,))
    
    def is_provable(self, formula: Formula) -> bool:
        """检查公式是否可证明"""
        proof = self.theory.prove(formula)
        return proof is not None
    
    def verify_reflection_principle(self) -> bool:
        """验证反射原理：T ⊢ φ ⇒ T ⊢ Prov_T(⌜φ⌝)"""
        for theorem in self.theory.theorems:
            code = self.theory.encoder.encode_formula(theorem)
            prov_formula = self.apply(code)
            if prov_formula not in self.theory.theorems:
                return False
        return True

2. 对角化机制

@dataclass
class DiagonalizationOperator:
    """对角化算子"""
    encoder: GödelEncoder
    
    def diagonalize(self, formula_schema: Callable[[Term], Formula]) -> Formula:
        """
        对角化：给定 φ(x)，返回 φ(⌜φ(x)⌝)
        """
        # 创建表示formula_schema的项
        schema_code = self._encode_schema(formula_schema)
        schema_term = ConstantTerm(f"S_{schema_code.value}")
        
        # 应用到自身编码
        diagonal = formula_schema(schema_term)
        
        return diagonal
    
    def _encode_schema(self, schema: Callable) -> No11Number:
        """编码公式模式"""
        # 使用函数的字符串表示作为编码基础
        schema_str = str(schema.__code__.co_code)
        code_value = sum(ord(c) for c in schema_str) % 10000
        return No11Number(code_value)
    
    def construct_fixed_point(self, operator: Callable[[Formula], Formula]) -> Formula:
        """
        构造不动点：找到 φ 使得 φ ↔ F(φ)
        """
        def schema(x: Term) -> Formula:
            # 构造 F(decode(x))
            decoded = self._decode_term(x)
            return operator(decoded)
        
        return self.diagonalize(schema)
    
    def _decode_term(self, term: Term) -> Formula:
        """解码项为公式（示意性实现）"""
        # 这里需要完整的解码机制
        if isinstance(term, ConstantTerm):
            # 从编码恢复公式
            return AtomicFormula(
                Symbol("Decoded", SymbolType.RELATION, 0),
                ()
            )
        return AtomicFormula(Symbol("Unknown", SymbolType.RELATION, 0), ())

3. Gödel句

@dataclass
class GodelSentence:
    """Gödel句 G ↔ ¬Prov_T(⌜G⌝)"""
    theory: Theory
    formula: Formula
    encoding: No11Number
    
    @classmethod
    def construct(cls, theory: Theory) -> 'GodelSentence':
        """构造理论的Gödel句"""
        prov_pred = ProvabilityPredicate(
            theory,
            theory.language.symbols.get("Prov", 
                Symbol("Prov", SymbolType.RELATION, 1))
        )
        
        diag_op = DiagonalizationOperator(theory.encoder)
        
        # G(x) := ¬Prov(x)
        def G_schema(x: Term) -> Formula:
            prov_formula = AtomicFormula(prov_pred.symbol, (x,))
            return NotFormula(prov_formula)
        
        # 对角化得到 G ↔ ¬Prov(⌜G⌝)
        godel_formula = diag_op.diagonalize(G_schema)
        godel_encoding = theory.encoder.encode_formula(godel_formula)
        
        return cls(theory, godel_formula, godel_encoding)
    
    def verify_self_reference(self) -> bool:
        """验证自引用性质"""
        # G应该等价于¬Prov(⌜G⌝)
        prov_g = AtomicFormula(
            self.theory.language.symbols["Prov"],
            (ConstantTerm(f"F_{self.encoding.value}"),)
        )
        neg_prov_g = NotFormula(prov_g)
        
        # 检查等价性（这里简化为结构相等）
        return self._structurally_equivalent(self.formula, neg_prov_g)
    
    def _structurally_equivalent(self, f1: Formula, f2: Formula) -> bool:
        """检查两个公式是否结构等价"""
        # 完整实现需要考虑逻辑等价
        return type(f1) == type(f2)

4. 不完备性分析器

@dataclass
class IncompletenessAnalyzer:
    """不完备性分析器"""
    theory: Theory
    
    def find_undecidable_sentences(self, max_search: int = 100) -> List[Formula]:
        """寻找不可判定的语句"""
        undecidable = []
        
        # 生成候选公式
        for formula in self._generate_formulas(max_search):
            if self.is_undecidable(formula):
                undecidable.append(formula)
        
        return undecidable
    
    def is_undecidable(self, formula: Formula) -> bool:
        """判断公式是否不可判定"""
        # 尝试证明公式
        proof_pos = self.theory.prove(formula)
        
        # 尝试证明否定
        neg_formula = NotFormula(formula)
        proof_neg = self.theory.prove(neg_formula)
        
        # 都无法证明则不可判定
        return proof_pos is None and proof_neg is None
    
    def verify_first_incompleteness(self) -> bool:
        """验证第一不完备性定理"""
        godel = GodelSentence.construct(self.theory)
        
        # G应该不可证明
        if self.theory.prove(godel.formula) is not None:
            return False
        
        # ¬G也应该不可证明（如果理论一致）
        neg_godel = NotFormula(godel.formula)
        if self.theory.prove(neg_godel) is not None:
            return False
        
        return True
    
    def verify_second_incompleteness(self) -> bool:
        """验证第二不完备性定理"""
        # 构造一致性陈述 Con(T) := ¬Prov_T(⊥)
        bottom = AtomicFormula(
            Symbol("⊥", SymbolType.RELATION, 0),
            ()
        )
        bottom_code = self.theory.encoder.encode_formula(bottom)
        
        prov_bottom = AtomicFormula(
            self.theory.language.symbols["Prov"],
            (ConstantTerm(f"F_{bottom_code.value}"),)
        )
        
        consistency = NotFormula(prov_bottom)
        
        # 一致性陈述应该不可证明
        consistency_proof = self.theory.prove(consistency)
        return consistency_proof is None
    
    def _generate_formulas(self, count: int) -> List[Formula]:
        """生成测试用公式"""
        formulas = []
        
        # 基本原子公式
        for i in range(count // 4):
            pred = Symbol(f"P{i}", SymbolType.RELATION, 0)
            formulas.append(AtomicFormula(pred, ()))
        
        # 否定
        for i in range(count // 4):
            base = formulas[i] if i < len(formulas) else AtomicFormula(
                Symbol("Q", SymbolType.RELATION, 0), ()
            )
            formulas.append(NotFormula(base))
        
        # 合取
        for i in range(count // 4):
            left = formulas[i % len(formulas)] if formulas else AtomicFormula(
                Symbol("R", SymbolType.RELATION, 0), ()
            )
            right = formulas[(i + 1) % len(formulas)] if formulas else AtomicFormula(
                Symbol("S", SymbolType.RELATION, 0), ()
            )
            formulas.append(AndFormula(left, right))
        
        # 量词
        for i in range(count // 4):
            body = formulas[i % len(formulas)] if formulas else AtomicFormula(
                Symbol("T", SymbolType.RELATION, 1),
                (VariableTerm("x"),)
            )
            formulas.append(ForAllFormula("x", body))
        
        return formulas[:count]

5. 熵计算器

@dataclass
class EntropyCalculator:
    """理论熵计算器"""
    
    def compute_entropy(self, theory: Theory, sample_size: int = 1000) -> float:
        """
        计算理论的熵（不可判定陈述的比例）
        """
        analyzer = IncompletenessAnalyzer(theory)
        undecidable_count = 0
        
        # 采样公式空间
        formulas = analyzer._generate_formulas(sample_size)
        
        for formula in formulas:
            if analyzer.is_undecidable(formula):
                undecidable_count += 1
        
        # 熵定义为不可判定陈述的比例
        entropy = undecidable_count / sample_size
        return entropy
    
    def compute_entropy_growth(self, theory_tower: List[Theory]) -> List[float]:
        """计算理论塔的熵增长"""
        entropies = []
        
        for theory in theory_tower:
            entropy = self.compute_entropy(theory)
            entropies.append(entropy)
        
        return entropies
    
    def verify_entropy_increase(self, t1: Theory, t2: Theory) -> bool:
        """验证熵增原理"""
        entropy1 = self.compute_entropy(t1)
        entropy2 = self.compute_entropy(t2)
        
        # 反射后的理论应该有更高的熵
        return entropy2 > entropy1

6. 完备性检测器

@dataclass
class CompletenessChecker:
    """完备性检测器"""
    theory: Theory
    
    def is_complete(self) -> bool:
        """检测理论是否完备"""
        # 采样一些公式
        analyzer = IncompletenessAnalyzer(self.theory)
        formulas = analyzer._generate_formulas(100)
        
        for formula in formulas:
            # 检查是否 T ⊢ φ 或 T ⊢ ¬φ
            proof_pos = self.theory.prove(formula)
            proof_neg = self.theory.prove(NotFormula(formula))
            
            if proof_pos is None and proof_neg is None:
                # 找到不可判定的公式，理论不完备
                return False
        
        # 注意：这只是近似检测
        return True
    
    def is_consistent(self) -> bool:
        """检测理论是否一致"""
        # 检查是否能证明⊥
        bottom = AtomicFormula(
            Symbol("⊥", SymbolType.RELATION, 0),
            ()
        )
        
        proof = self.theory.prove(bottom)
        return proof is None
    
    def verify_incompleteness_dilemma(self) -> bool:
        """验证完备性与一致性不可兼得"""
        is_complete = self.is_complete()
        is_consistent = self.is_consistent()
        
        # 不应该既完备又一致
        return not (is_complete and is_consistent)

7. 不完备性定理验证器

class IncompletenessTheoremVerifier:
    """不完备性定理的完整验证"""
    
    def __init__(self, theory: Theory):
        self.theory = theory
        self.analyzer = IncompletenessAnalyzer(theory)
        self.entropy_calc = EntropyCalculator()
        self.completeness_checker = CompletenessChecker(theory)
    
    def verify_all_theorems(self) -> Dict[str, bool]:
        """验证所有不完备性定理"""
        results = {}
        
        # 第一不完备性
        results["first_incompleteness"] = self.analyzer.verify_first_incompleteness()
        
        # 第二不完备性
        results["second_incompleteness"] = self.analyzer.verify_second_incompleteness()
        
        # 熵增原理
        reflected = ReflectionOperator().reflect(self.theory)
        results["entropy_increase"] = self.entropy_calc.verify_entropy_increase(
            self.theory, reflected
        )
        
        # 完备性困境
        results["completeness_dilemma"] = self.completeness_checker.verify_incompleteness_dilemma()
        
        return results

算法规范

1. Gödel句构造算法

def construct_godel_sentence(theory: Theory) -> GodelSentence:
    """
    构造理论的Gödel句
    
    步骤：
    1. 定义可证性谓词
    2. 构造否定可证性的公式模式
    3. 应用对角化
    4. 验证自引用性质
    """
    return GodelSentence.construct(theory)

2. 不可判定性检测算法

def detect_undecidability(theory: Theory, formula: Formula) -> bool:
    """
    检测公式的不可判定性
    
    算法：
    1. 尝试证明公式
    2. 尝试证明公式的否定
    3. 如果都失败，则不可判定
    """
    analyzer = IncompletenessAnalyzer(theory)
    return analyzer.is_undecidable(formula)

3. 熵计算算法

def calculate_theory_entropy(theory: Theory, sample_size: int = 1000) -> float:
    """
    计算理论的熵
    
    算法：
    1. 生成公式样本
    2. 统计不可判定公式
    3. 计算比例作为熵
    """
    calculator = EntropyCalculator()
    return calculator.compute_entropy(theory, sample_size)

正确性要求

1. Gödel句必须真正自引用
2. 不可判定性检测必须完整
3. 熵计算必须反映真实复杂度
4. 所有编码保持No-11约束
5. 对角化过程必须正确

测试规范

每个组件都需要完整的单元测试，验证：

• Gödel句的构造和性质
• 不完备性定理的成立
• 熵的严格递增
• 完备性与一致性的不可兼得
• No-11约束的保持