C10-1 元数学结构形式化规范

模块依赖

from typing import List, Set, Dict, Tuple, Optional, Union, Callable
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
from enum import Enum
import hashlib
from no11_number_system import No11Number
from test_C9_1 import SelfReferentialArithmetic
from test_C9_2 import RecursiveNumberTheory
from test_C9_3 import SelfReferentialGroup, SelfReferentialRing, SelfReferentialField

核心数据结构

形式语言定义

class SymbolType(Enum):
    """符号类型枚举"""
    VARIABLE = "variable"
    CONSTANT = "constant"
    FUNCTION = "function"
    RELATION = "relation"
    LOGICAL = "logical"
    AUXILIARY = "auxiliary"

@dataclass(frozen=True)
class Symbol:
    """形式符号"""
    name: str
    type: SymbolType
    arity: int = 0  # 函数或关系的元数
    
    def encode(self) -> No11Number:
        """符号的No-11编码"""
        # 使用符号的哈希值生成唯一编码
        hash_val = hash((self.name, self.type, self.arity))
        return No11Number(abs(hash_val) % 1000)  # 限制大小

class Term(ABC):
    """项的抽象基类"""
    @abstractmethod
    def free_variables(self) -> Set[Symbol]:
        """返回自由变量集合"""
        pass
    
    @abstractmethod
    def substitute(self, var: Symbol, replacement: 'Term') -> 'Term':
        """变量替换"""
        pass
    
    @abstractmethod
    def encode(self) -> No11Number:
        """项的No-11编码"""
        pass

@dataclass(frozen=True)
class VariableTerm(Term):
    """变量项"""
    symbol: Symbol
    
    def free_variables(self) -> Set[Symbol]:
        return {self.symbol}
    
    def substitute(self, var: Symbol, replacement: Term) -> Term:
        if self.symbol == var:
            return replacement
        return self
    
    def encode(self) -> No11Number:
        return self.symbol.encode()

@dataclass(frozen=True)
class ConstantTerm(Term):
    """常量项"""
    symbol: Symbol
    
    def free_variables(self) -> Set[Symbol]:
        return set()
    
    def substitute(self, var: Symbol, replacement: Term) -> Term:
        return self
    
    def encode(self) -> No11Number:
        return self.symbol.encode()

@dataclass(frozen=True)
class FunctionTerm(Term):
    """函数项"""
    function: Symbol
    arguments: Tuple[Term, ...]
    
    def __post_init__(self):
        if len(self.arguments) != self.function.arity:
            raise ValueError(f"Function {self.function.name} expects {self.function.arity} arguments")
    
    def free_variables(self) -> Set[Symbol]:
        return set().union(*(arg.free_variables() for arg in self.arguments))
    
    def substitute(self, var: Symbol, replacement: Term) -> Term:
        new_args = tuple(arg.substitute(var, replacement) for arg in self.arguments)
        return FunctionTerm(self.function, new_args)
    
    def encode(self) -> No11Number:
        # 编码为: [函数符号编码, 参数1编码, 参数2编码, ...]
        func_code = self.function.encode()
        arg_codes = [arg.encode() for arg in self.arguments]
        # 组合编码（简化实现）
        combined = func_code.value
        for arg_code in arg_codes:
            combined = combined * 1000 + arg_code.value
        return No11Number(combined)

class Formula(ABC):
    """公式的抽象基类"""
    @abstractmethod
    def free_variables(self) -> Set[Symbol]:
        """返回自由变量集合"""
        pass
    
    @abstractmethod
    def substitute(self, var: Symbol, replacement: Term) -> 'Formula':
        """变量替换"""
        pass
    
    @abstractmethod
    def encode(self) -> No11Number:
        """公式的No-11编码"""
        pass
    
    @abstractmethod
    def is_well_formed(self) -> bool:
        """检查是否是良构公式"""
        pass

@dataclass(frozen=True)
class AtomicFormula(Formula):
    """原子公式"""
    relation: Symbol
    arguments: Tuple[Term, ...]
    
    def free_variables(self) -> Set[Symbol]:
        return set().union(*(arg.free_variables() for arg in self.arguments))
    
    def substitute(self, var: Symbol, replacement: Term) -> Formula:
        new_args = tuple(arg.substitute(var, replacement) for arg in self.arguments)
        return AtomicFormula(self.relation, new_args)
    
    def encode(self) -> No11Number:
        rel_code = self.relation.encode()
        arg_codes = [arg.encode() for arg in self.arguments]
        combined = rel_code.value
        for arg_code in arg_codes:
            combined = combined * 1000 + arg_code.value
        return No11Number(combined)
    
    def is_well_formed(self) -> bool:
        return len(self.arguments) == self.relation.arity

@dataclass(frozen=True)
class NegationFormula(Formula):
    """否定公式"""
    formula: Formula
    
    def free_variables(self) -> Set[Symbol]:
        return self.formula.free_variables()
    
    def substitute(self, var: Symbol, replacement: Term) -> Formula:
        return NegationFormula(self.formula.substitute(var, replacement))
    
    def encode(self) -> No11Number:
        # 否定编码为: 1 + 子公式编码
        return No11Number(1 + self.formula.encode().value)
    
    def is_well_formed(self) -> bool:
        return self.formula.is_well_formed()

@dataclass(frozen=True)
class BinaryFormula(Formula):
    """二元逻辑公式"""
    left: Formula
    operator: str  # "∧", "∨", "→", "↔"
    right: Formula
    
    def free_variables(self) -> Set[Symbol]:
        return self.left.free_variables() | self.right.free_variables()
    
    def substitute(self, var: Symbol, replacement: Term) -> Formula:
        return BinaryFormula(
            self.left.substitute(var, replacement),
            self.operator,
            self.right.substitute(var, replacement)
        )
    
    def encode(self) -> No11Number:
        op_code = {"∧": 2, "∨": 3, "→": 4, "↔": 5}[self.operator]
        left_code = self.left.encode().value
        right_code = self.right.encode().value
        return No11Number(op_code * 1000000 + left_code * 1000 + right_code)
    
    def is_well_formed(self) -> bool:
        return self.left.is_well_formed() and self.right.is_well_formed()

@dataclass(frozen=True)
class QuantifiedFormula(Formula):
    """量化公式"""
    quantifier: str  # "∀" or "∃"
    variable: Symbol
    formula: Formula
    
    def free_variables(self) -> Set[Symbol]:
        return self.formula.free_variables() - {self.variable}
    
    def substitute(self, var: Symbol, replacement: Term) -> Formula:
        if var == self.variable:
            return self  # 约束变量不能替换
        if self.variable in replacement.free_variables():
            # 需要重命名以避免变量捕获
            new_var = self._fresh_variable()
            renamed_formula = self.formula.substitute(self.variable, VariableTerm(new_var))
            return QuantifiedFormula(self.quantifier, new_var, 
                                   renamed_formula.substitute(var, replacement))
        return QuantifiedFormula(self.quantifier, self.variable,
                               self.formula.substitute(var, replacement))
    
    def encode(self) -> No11Number:
        q_code = 6 if self.quantifier == "∀" else 7
        var_code = self.variable.encode().value
        formula_code = self.formula.encode().value
        return No11Number(q_code * 1000000 + var_code * 1000 + formula_code)
    
    def is_well_formed(self) -> bool:
        return (self.variable.type == SymbolType.VARIABLE and 
                self.formula.is_well_formed())
    
    def _fresh_variable(self) -> Symbol:
        """生成新的变量符号"""
        i = 0
        while True:
            new_var = Symbol(f"x_{i}", SymbolType.VARIABLE)
            if new_var not in self.formula.free_variables():
                return new_var
            i += 1

形式系统定义

@dataclass
class Axiom:
    """公理"""
    name: str
    formula: Formula
    
    def verify_well_formed(self) -> bool:
        """验证公理是良构公式"""
        return self.formula.is_well_formed()

@dataclass
class InferenceRule:
    """推理规则"""
    name: str
    premises_pattern: List[Formula]  # 前提模式
    conclusion_pattern: Formula      # 结论模式
    validity_check: Callable[[List[Formula], Formula], bool]
    
    def apply(self, premises: List[Formula]) -> Optional[Formula]:
        """应用推理规则"""
        if len(premises) != len(self.premises_pattern):
            return None
        
        # 尝试模式匹配和实例化
        substitution = self._match_premises(premises)
        if substitution is None:
            return None
        
        # 生成结论
        conclusion = self._instantiate_conclusion(substitution)
        
        # 验证有效性
        if self.validity_check(premises, conclusion):
            return conclusion
        return None
    
    def _match_premises(self, premises: List[Formula]) -> Optional[Dict[Symbol, Term]]:
        """模式匹配"""
        # 简化实现：这里需要完整的模式匹配算法
        return {}
    
    def _instantiate_conclusion(self, substitution: Dict[Symbol, Term]) -> Formula:
        """实例化结论"""
        result = self.conclusion_pattern
        for var, term in substitution.items():
            result = result.substitute(var, term)
        return result

class FormalSystem:
    """形式系统"""
    def __init__(self, name: str = "MetaMathSystem"):
        self.name = name
        self.symbols: Dict[str, Symbol] = {}
        self.axioms: Dict[str, Axiom] = {}
        self.rules: Dict[str, InferenceRule] = {}
        self.theorems: Set[Formula] = set()
        self._initialize_basic_elements()
    
    def _initialize_basic_elements(self):
        """初始化基本元素"""
        # 基本符号
        self.add_symbol(Symbol("0", SymbolType.CONSTANT))
        self.add_symbol(Symbol("1", SymbolType.CONSTANT))
        self.add_symbol(Symbol("=", SymbolType.RELATION, 2))
        self.add_symbol(Symbol("<", SymbolType.RELATION, 2))
        self.add_symbol(Symbol("+", SymbolType.FUNCTION, 2))
        self.add_symbol(Symbol("*", SymbolType.FUNCTION, 2))
        self.add_symbol(Symbol("collapse", SymbolType.FUNCTION, 1))
        
        # 基本推理规则
        self._add_modus_ponens()
        self._add_generalization()
    
    def add_symbol(self, symbol: Symbol):
        """添加符号"""
        self.symbols[symbol.name] = symbol
    
    def add_axiom(self, axiom: Axiom):
        """添加公理"""
        if axiom.verify_well_formed():
            self.axioms[axiom.name] = axiom
            self.theorems.add(axiom.formula)
        else:
            raise ValueError(f"Axiom {axiom.name} is not well-formed")
    
    def add_rule(self, rule: InferenceRule):
        """添加推理规则"""
        self.rules[rule.name] = rule
    
    def _add_modus_ponens(self):
        """添加分离规则"""
        def mp_check(premises: List[Formula], conclusion: Formula) -> bool:
            if len(premises) != 2:
                return False
            # 检查是否形如 [A, A→B] ⊢ B
            if isinstance(premises[1], BinaryFormula) and premises[1].operator == "→":
                return premises[1].left == premises[0] and premises[1].right == conclusion
            return False
        
        self.add_rule(InferenceRule(
            "modus_ponens",
            [Formula(), BinaryFormula(Formula(), "→", Formula())],
            Formula(),
            mp_check
        ))
    
    def _add_generalization(self):
        """添加全称概括规则"""
        def gen_check(premises: List[Formula], conclusion: Formula) -> bool:
            if len(premises) != 1:
                return False
            # 检查是否形如 A(x) ⊢ ∀x.A(x)
            if isinstance(conclusion, QuantifiedFormula) and conclusion.quantifier == "∀":
                return conclusion.formula == premises[0]
            return False
        
        self.add_rule(InferenceRule(
            "generalization",
            [Formula()],
            QuantifiedFormula("∀", Symbol("x", SymbolType.VARIABLE), Formula()),
            gen_check
        ))
    
    def is_theorem(self, formula: Formula) -> bool:
        """检查是否是定理"""
        return formula in self.theorems
    
    def encode_self(self) -> No11Number:
        """Gödel编码整个形式系统"""
        # 编码所有组成部分
        symbol_codes = [s.encode().value for s in self.symbols.values()]
        axiom_codes = [a.formula.encode().value for a in self.axioms.values()]
        
        # 组合编码（简化版）
        combined = 0
        for code in symbol_codes + axiom_codes:
            combined = combined * 10000 + code
        
        return No11Number(combined % 1000000)  # 限制大小

证明系统定义

@dataclass
class ProofStep:
    """证明步骤"""
    formula: Formula
    justification: Union[str, Tuple[str, List[int]]]  # "axiom:name" 或 ("rule:name", [前提索引])
    
    def is_valid(self, system: FormalSystem, previous_steps: List['ProofStep']) -> bool:
        """验证步骤的有效性"""
        if isinstance(self.justification, str) and self.justification.startswith("axiom:"):
            axiom_name = self.justification[6:]
            return (axiom_name in system.axioms and 
                   system.axioms[axiom_name].formula == self.formula)
        
        elif isinstance(self.justification, tuple):
            rule_name = self.justification[0][5:]  # 去掉"rule:"前缀
            premise_indices = self.justification[1]
            
            if rule_name not in system.rules:
                return False
            
            rule = system.rules[rule_name]
            premises = [previous_steps[i].formula for i in premise_indices]
            
            result = rule.apply(premises)
            return result is not None and result == self.formula
        
        return False

class Proof:
    """形式证明"""
    def __init__(self, goal: Formula):
        self.goal = goal
        self.steps: List[ProofStep] = []
    
    def add_step(self, step: ProofStep):
        """添加证明步骤"""
        self.steps.append(step)
    
    def verify(self, system: FormalSystem) -> bool:
        """验证证明的有效性"""
        for i, step in enumerate(self.steps):
            if not step.is_valid(system, self.steps[:i]):
                return False
        
        # 检查最后一步是否是目标
        return len(self.steps) > 0 and self.steps[-1].formula == self.goal
    
    def encode(self) -> No11Number:
        """证明的Gödel编码"""
        step_codes = [step.formula.encode().value for step in self.steps]
        combined = 0
        for code in step_codes:
            combined = combined * 10000 + code
        return No11Number(combined % 1000000)
    
    def collapse(self) -> 'Proof':
        """证明的collapse操作：移除冗余步骤"""
        # 找出被使用的步骤
        used_indices = {len(self.steps) - 1}  # 最后一步总是需要的
        
        changed = True
        while changed:
            changed = False
            for i, step in enumerate(self.steps):
                if i in used_indices and isinstance(step.justification, tuple):
                    for premise_idx in step.justification[1]:
                        if premise_idx not in used_indices:
                            used_indices.add(premise_idx)
                            changed = True
        
        # 构造新证明
        collapsed = Proof(self.goal)
        index_map = {}
        
        for i, step in enumerate(self.steps):
            if i in used_indices:
                new_index = len(collapsed.steps)
                index_map[i] = new_index
                
                # 调整引用
                if isinstance(step.justification, tuple):
                    rule_name, premise_indices = step.justification
                    new_indices = [index_map[idx] for idx in premise_indices]
                    new_step = ProofStep(step.formula, (rule_name, new_indices))
                else:
                    new_step = step
                
                collapsed.add_step(new_step)
        
        return collapsed
    
    def is_self_verifying(self) -> bool:
        """检查证明是否自验证"""
        # 证明包含对自身有效性的证明
        verification_formula = self._construct_verification_formula()
        for step in self.steps:
            if step.formula == verification_formula:
                return True
        return False
    
    def _construct_verification_formula(self) -> Formula:
        """构造验证公式"""
        # 简化实现：返回一个表示"此证明有效"的公式
        return AtomicFormula(
            Symbol("Valid", SymbolType.RELATION, 1),
            (ConstantTerm(Symbol(f"proof_{id(self)}", SymbolType.CONSTANT)),)
        )

class ProofSearcher:
    """证明搜索器"""
    def __init__(self, system: FormalSystem, max_depth: int = 10):
        self.system = system
        self.max_depth = max_depth
    
    def search(self, goal: Formula) -> Optional[Proof]:
        """搜索证明"""
        # 如果目标已经是定理，直接返回
        if self.system.is_theorem(goal):
            proof = Proof(goal)
            for axiom_name, axiom in self.system.axioms.items():
                if axiom.formula == goal:
                    proof.add_step(ProofStep(goal, f"axiom:{axiom_name}"))
                    return proof
        
        # 深度优先搜索
        return self._dfs_search(goal, 0, set())
    
    def _dfs_search(self, goal: Formula, depth: int, 
                    visited: Set[Formula]) -> Optional[Proof]:
        """深度优先搜索证明"""
        if depth > self.max_depth or goal in visited:
            return None
        
        visited.add(goal)
        
        # 尝试每个推理规则
        for rule_name, rule in self.system.rules.items():
            # 尝试反向应用规则
            premises = self._find_premises_for_conclusion(rule, goal)
            if premises is not None:
                # 递归证明每个前提
                premise_proofs = []
                for premise in premises:
                    sub_proof = self._dfs_search(premise, depth + 1, visited.copy())
                    if sub_proof is None:
                        break
                    premise_proofs.append(sub_proof)
                
                if len(premise_proofs) == len(premises):
                    # 构造完整证明
                    proof = Proof(goal)
                    
                    # 添加所有前提的证明步骤
                    premise_indices = []
                    for sub_proof in premise_proofs:
                        for step in sub_proof.steps:
                            proof.add_step(step)
                        premise_indices.append(len(proof.steps) - 1)
                    
                    # 添加最终步骤
                    proof.add_step(ProofStep(goal, (f"rule:{rule_name}", premise_indices)))
                    return proof
        
        return None
    
    def _find_premises_for_conclusion(self, rule: InferenceRule, 
                                    conclusion: Formula) -> Optional[List[Formula]]:
        """寻找能推出结论的前提"""
        # 简化实现：需要完整的反向推理算法
        return None

Gödel编码系统

class GödelEncoder:
    """Gödel编码器"""
    def __init__(self):
        self.symbol_map: Dict[str, int] = {}
        self.reverse_map: Dict[int, str] = {}
        self._initialize_basic_encoding()
    
    def _initialize_basic_encoding(self):
        """初始化基本符号编码"""
        basic_symbols = [
            "0", "1", "=", "<", "+", "*", "collapse",
            "¬", "∧", "∨", "→", "↔", "∀", "∃",
            "(", ")", ",", "x", "y", "z"
        ]
        
        for i, symbol in enumerate(basic_symbols):
            self.symbol_map[symbol] = i + 1
            self.reverse_map[i + 1] = symbol
    
    def encode_symbol(self, symbol: Symbol) -> No11Number:
        """编码单个符号"""
        if symbol.name in self.symbol_map:
            return No11Number(self.symbol_map[symbol.name])
        else:
            # 动态分配新编码
            new_code = len(self.symbol_map) + 1
            self.symbol_map[symbol.name] = new_code
            self.reverse_map[new_code] = symbol.name
            return No11Number(new_code)
    
    def encode_formula(self, formula: Formula) -> No11Number:
        """编码公式"""
        return formula.encode()
    
    def encode_proof(self, proof: Proof) -> No11Number:
        """编码证明"""
        return proof.encode()
    
    def encode_system(self, system: FormalSystem) -> No11Number:
        """编码整个形式系统"""
        return system.encode_self()
    
    def diagonal_lemma(self, system: FormalSystem, 
                      property_formula: Formula) -> Formula:
        """对角化引理：构造自引用公式"""
        # 构造表示"公式x满足性质P"的公式
        x = Symbol("x", SymbolType.VARIABLE)
        
        # 构造"∃y(y是x的编码 ∧ P(y))"
        y = Symbol("y", SymbolType.VARIABLE)
        encoding_relation = AtomicFormula(
            Symbol("encodes", SymbolType.RELATION, 2),
            (VariableTerm(y), VariableTerm(x))
        )
        
        # 应用性质到编码
        property_applied = property_formula.substitute(x, VariableTerm(y))
        
        # 构造存在量化
        exists_formula = QuantifiedFormula(
            "∃", y,
            BinaryFormula(encoding_relation, "∧", property_applied)
        )
        
        # 使用不动点组合子构造自引用
        # G ≡ ∃y(y encodes G ∧ P(y))
        return self._fixed_point_combinator(exists_formula)
    
    def _fixed_point_combinator(self, formula: Formula) -> Formula:
        """不动点组合子"""
        # 简化实现：实际需要更复杂的构造
        return formula
    
    def construct_gödel_sentence(self, system: FormalSystem) -> Formula:
        """构造Gödel句子：我不可证明"""
        # 构造"x是可证明的"性质
        x = Symbol("x", SymbolType.VARIABLE)
        provable = AtomicFormula(
            Symbol("Provable", SymbolType.RELATION, 1),
            (VariableTerm(x),)
        )
        
        # 否定
        not_provable = NegationFormula(provable)
        
        # 应用对角化引理
        return self.diagonal_lemma(system, not_provable)

模型论实现

@dataclass
class Interpretation:
    """解释函数"""
    domain: Set[No11Number]
    constant_interp: Dict[Symbol, No11Number]
    function_interp: Dict[Symbol, Callable]
    relation_interp: Dict[Symbol, Set[Tuple[No11Number, ...]]]

class Model:
    """形式系统的模型"""
    def __init__(self, domain: Set[No11Number], interpretation: Interpretation):
        self.domain = domain
        self.interpretation = interpretation
    
    def evaluate_term(self, term: Term, assignment: Dict[Symbol, No11Number]) -> No11Number:
        """在赋值下计算项的值"""
        if isinstance(term, VariableTerm):
            return assignment.get(term.symbol, No11Number(0))
        elif isinstance(term, ConstantTerm):
            return self.interpretation.constant_interp.get(term.symbol, No11Number(0))
        elif isinstance(term, FunctionTerm):
            func = self.interpretation.function_interp.get(term.function)
            if func is None:
                raise ValueError(f"Function {term.function.name} not interpreted")
            args = [self.evaluate_term(arg, assignment) for arg in term.arguments]
            return func(*args)
        else:
            raise ValueError(f"Unknown term type: {type(term)}")
    
    def satisfies(self, formula: Formula, assignment: Dict[Symbol, No11Number]) -> bool:
        """检查公式在赋值下是否满足"""
        if isinstance(formula, AtomicFormula):
            relation = self.interpretation.relation_interp.get(formula.relation, set())
            args = tuple(self.evaluate_term(arg, assignment) for arg in formula.arguments)
            return args in relation
        
        elif isinstance(formula, NegationFormula):
            return not self.satisfies(formula.formula, assignment)
        
        elif isinstance(formula, BinaryFormula):
            left = self.satisfies(formula.left, assignment)
            right = self.satisfies(formula.right, assignment)
            
            if formula.operator == "∧":
                return left and right
            elif formula.operator == "∨":
                return left or right
            elif formula.operator == "→":
                return not left or right
            elif formula.operator == "↔":
                return left == right
        
        elif isinstance(formula, QuantifiedFormula):
            if formula.quantifier == "∀":
                return all(self.satisfies(formula.formula, 
                          {**assignment, formula.variable: d})
                          for d in self.domain)
            else:  # ∃
                return any(self.satisfies(formula.formula,
                          {**assignment, formula.variable: d})
                          for d in self.domain)
        
        return False
    
    def is_model_of(self, system: FormalSystem) -> bool:
        """检查是否是系统的模型"""
        for axiom in system.axioms.values():
            if not self.satisfies(axiom.formula, {}):
                return False
        return True
    
    def is_self_referential(self) -> bool:
        """检查模型是否自引用"""
        # 检查模型的编码是否在论域中
        model_encoding = self._encode_self()
        return model_encoding in self.domain
    
    def _encode_self(self) -> No11Number:
        """模型的自编码"""
        # 简化实现
        return No11Number(hash(tuple(self.domain)) % 1000000)

class ModelConstructor:
    """模型构造器"""
    def __init__(self, system: FormalSystem):
        self.system = system
    
    def construct_standard_model(self) -> Model:
        """构造标准模型"""
        # 论域：所有有效的No-11数
        domain = {No11Number(i) for i in range(100)}
        
        # 解释函数
        constant_interp = {
            Symbol("0", SymbolType.CONSTANT): No11Number(0),
            Symbol("1", SymbolType.CONSTANT): No11Number(1)
        }
        
        function_interp = {
            Symbol("+", SymbolType.FUNCTION, 2): lambda x, y: x + y,
            Symbol("*", SymbolType.FUNCTION, 2): lambda x, y: x * y,
            Symbol("collapse", SymbolType.FUNCTION, 1): lambda x: x  # 简化
        }
        
        relation_interp = {
            Symbol("=", SymbolType.RELATION, 2): {(x, x) for x in domain},
            Symbol("<", SymbolType.RELATION, 2): {(x, y) for x in domain for y in domain if x < y}
        }
        
        interpretation = Interpretation(domain, constant_interp, 
                                      function_interp, relation_interp)
        
        return Model(domain, interpretation)
    
    def construct_self_model(self) -> Model:
        """构造包含系统自身的模型"""
        standard = self.construct_standard_model()
        
        # 添加系统的编码到论域
        system_encoding = self.system.encode_self()
        extended_domain = standard.domain | {system_encoding}
        
        # 扩展解释以包含元语言谓词
        extended_interp = Interpretation(
            extended_domain,
            standard.interpretation.constant_interp,
            standard.interpretation.function_interp,
            standard.interpretation.relation_interp
        )
        
        # 添加"是定理"关系
        theorem_relation = {
            (formula.encode(),) 
            for formula in self.system.theorems
        }
        extended_interp.relation_interp[Symbol("IsTheorem", SymbolType.RELATION, 1)] = theorem_relation
        
        return Model(extended_domain, extended_interp)

元定理证明器

class MetaTheoremProver:
    """元定理证明器"""
    def __init__(self, system: FormalSystem):
        self.system = system
        self.encoder = GödelEncoder()
    
    def internalize_meta_statement(self, meta_statement: str) -> Formula:
        """将元语言陈述内部化为对象语言公式"""
        # 简化实现：实际需要完整的解析器
        if meta_statement == "system is consistent":
            # "系统一致"内部化为"¬∃x(x是矛盾的证明)"
            x = Symbol("x", SymbolType.VARIABLE)
            is_proof_of_contradiction = AtomicFormula(
                Symbol("ProvesContradiction", SymbolType.RELATION, 1),
                (VariableTerm(x),)
            )
            return NegationFormula(
                QuantifiedFormula("∃", x, is_proof_of_contradiction)
            )
        
        # 其他元陈述的内部化...
        return AtomicFormula(Symbol("True", SymbolType.RELATION, 0), ())
    
    def prove_reflection_principle(self) -> Proof:
        """证明反射原理"""
        # 如果系统证明"若系统一致则φ"，那么系统证明φ
        # 这需要构造一个复杂的证明
        pass
    
    def prove_fixed_point_theorem(self) -> Proof:
        """证明不动点定理"""
        # 对每个性质P，存在句子G使得G↔P(⌜G⌝)
        pass
    
    def prove_incompleteness(self) -> Tuple[Formula, Proof]:
        """证明不完备性"""
        # 构造Gödel句子
        gödel_sentence = self.encoder.construct_gödel_sentence(self.system)
        
        # 证明：如果系统一致，则Gödel句子不可证明
        # 这是一个元定理，需要在元层次证明
        
        return gödel_sentence, None  # 简化实现

接口规范

形式系统接口

class MetaMathInterface:
    """元数学系统的标准接口"""
    def create_formal_system(self, name: str) -> FormalSystem:
        """创建形式系统"""
        pass
    
    def add_axioms(self, system: FormalSystem, axioms: List[Axiom]):
        """批量添加公理"""
        pass
    
    def prove_theorem(self, system: FormalSystem, theorem: Formula) -> Optional[Proof]:
        """尝试证明定理"""
        pass
    
    def verify_proof(self, system: FormalSystem, proof: Proof) -> bool:
        """验证证明"""
        pass
    
    def encode_object(self, obj: Union[Formula, Proof, FormalSystem]) -> No11Number:
        """Gödel编码任意对象"""
        pass
    
    def construct_model(self, system: FormalSystem) -> Model:
        """构造系统的模型"""
        pass
    
    def check_independence(self, system: FormalSystem, formula: Formula) -> bool:
        """检查公式是否独立于系统"""
        pass

验证规范

自指性验证

def verify_self_reference(system: FormalSystem) -> bool:
    """验证系统的自指性质"""
    # 1. 系统能编码自身
    # 2. 系统能表达关于自身的陈述
    # 3. 存在自引用的定理
    pass

完备性检查

def check_completeness(system: FormalSystem) -> Tuple[bool, Optional[Formula]]:
    """检查系统的完备性"""
    # 寻找既不可证明也不可反驳的公式
    pass

一致性验证

def verify_consistency(system: FormalSystem) -> bool:
    """验证系统的一致性"""
    # 检查是否能推出矛盾
    pass

错误处理规范

所有元数学操作必须进行严格的错误检查：

1. 语法错误: 公式必须是良构的
2. 类型错误: 符号使用必须符合其类型
3. 证明错误: 每个证明步骤必须有效
4. 编码错误: Gödel编码必须可逆
5. 模型错误: 解释必须满足所有公理

性能要求

1. 公式解析: O(n) 其中n是公式长度
2. 证明验证: O(m*n) 其中m是步骤数，n是平均公式长度
3. 定理搜索: 指数级，但有深度限制
4. 模型检查: O(|D|^k) 其中|D|是论域大小，k是量词嵌套深度

测试规范

每个元数学组件必须通过以下测试：

1. 语法测试: 公式构造和解析
2. 证明测试: 证明的构造和验证
3. 编码测试: Gödel编码的双向性
4. 模型测试: 模型的构造和验证
5. 元定理测试: 关键元定理的证明