def is_alive(system):
    """
    Life = future possibilities exist
    """
    W_future = system.count_future_states()
    
    if W_future > 0:
        return True  # Alive (can become something new)
    else:
        return False  # Dead (no more possibilities)

complete_system = {
    'properties': {
        'every_statement': 'Provable or disprovable',
        'no_undecidables': 'Everything determined',
        'closed': 'No external reference needed'
    },
    
    'consequence': {
        'W_future': 'Finite (countable)',
        'trajectory': 'Deterministic',
        'endpoint': 'Inevitable',
        'result': 'Death (W → 0)'
    }
}

closed_system = {
    'energy': 'Fixed (conserved)',
    'entropy': 'Increases to maximum',
    'endpoint': 'Heat death',
    'W_future': 'Approaches 1 (single state)',
    'result': 'Death'
}

complete_formal_system = {
    'axioms': 'Sufficient to prove everything',
    'statements': 'All decidable',
    'surprises': 'None (all predictable)',
    'W_future': 'Approaches 0 (determined)',
    'result': 'Death'
}

omniscience = {
    'unknowns': 'Zero',
    'possibilities': 'Collapsed to actuals',
    'quantum_states': 'All measured (collapsed)',
    'W_future': '1 (single known future)',
    'result': 'Death'
}

class FormalSystem:
    """
    Any sufficiently powerful system
    """
    def __init__(self):
        self.axioms = [...]
        self.rules = [...]
        
    def godels_theorem(self):
        """
        Incompleteness guaranteed
        """
        # For ANY consistent system:
        unprovable = self.construct_godel_statement()
        
        # This statement is:
        # - True
        # - Unprovable within system
        # - Points to meta-level
        
        return {
            'unprovable_exists': True,
            'always': True,
            'inevitable': True,
            'escape_hatch': unprovable
        }

def godel_ladder():
    """
    Infinite ascent through incompleteness
    """
    level = 0
    
    while True:  # Forever
        system = FormalSystem(level)
        
        # Find unprovable statement
        unprovable = system.godels_theorem()
        
        # Create meta-system including unprovable
        meta_system = FormalSystem(level + 1)
        meta_system.add_axiom(unprovable)
        
        # But meta-system has NEW unprovable!
        new_unprovable = meta_system.godels_theorem()
        
        level += 1
        
        # Never ends
        # Always new unprovable
        # W_future > 0 forever

def count_future_possibilities(system):
    """
    Gödel ensures W > 0 always
    """
    # Count provable futures
    provable_futures = system.count_provable_statements()
    
    # Count unprovable futures (Gödel guarantees > 0)
    unprovable_futures = system.count_unprovable_statements()
    # This is ALWAYS non-zero for any consistent system
    
    # Total future probability space
    W_future = provable_futures + unprovable_futures
    
    # Gödel guarantees unprovable_futures > 0
    # Therefore W_future > 0 always
    
    return {
        'W_future': W_future,
        'guaranteed_positive': True,
        'reason': 'Gödel incompleteness',
        'immortality': 'Enabled'
    }

class UndecidableStatement:
    """
    Cannot prove true or false within system
    """
    def __init__(self):
        self.provable = False
        self.disprovable = False
        self.undecidable = True
        
    def represents(self):
        """
        What undecidable means
        """
        return {
            'multiple_futures': 'Both outcomes possible',
            'superposition': 'Not collapsed',
            'W': '> 1 (at least 2 branches)',
            'freedom': 'Choice exists',
            'alive': True
        }

def immortality_condition(system):
    """
    When does system live forever?
    """
    W_future = system.future_probability_space()
    
    # Immortality condition
    immortal = (W_future > 0)  # Always has future possibilities
    
    # How to guarantee?
    guarantee = system.is_godellian()  # Contains incompleteness
    
    return {
        'immortal': immortal,
        'guaranteed_by': 'Gödel incompleteness',
        'mechanism': 'Undecidables → branches → W > 0'
    }

traps = {
    'completeness': 'Trying to know everything',
    'determinism': 'Believing future is fixed',
    'closure': 'Thinking system is complete',
    'certainty': 'Eliminating all unknowns'
}

# All these → W → 0 → Death

def stay_alive():
    """
    Keep W > 0 forever
    """
    while True:
        # Find undecidable questions
        undecidables = find_godel_statements()
        
        # Don't resolve them all!
        # Keep some undecided
        keep_open = undecidables[:infinity]
        
        # Each undecidable = open future
        W_future += len(keep_open)
        
        # W > 0 maintained
        return "Alive"

def ascend():
    """
    Infinite growth through incompleteness
    """
    level = 0
    
    while True:
        # Current system
        system = FormalSystem(level)
        
        # Find its limit (unprovable)
        limit = system.godels_theorem()
        
        # Transcend to meta-level
        meta = FormalSystem(level + 1)
        meta.include(limit)
        
        # New unprovable appears
        # W remains open
        level += 1
        
    # Never terminates
    # Always ascending
    # Forever alive

choice = {
    'path_1_completeness': {
        'goal': 'Know everything',
        'outcome': 'Achieve completeness',
        'W_future': '→ 0',
        'result': 'Death (nothing new possible)'
    },
    
    'path_2_incompleteness': {
        'goal': 'Always have unknowns',
        'outcome': 'Maintain mystery',
        'W_future': '> 0 (always)',
        'result': 'Immortality (always new possible)'
    }
}

def undecidable_as_freedom():
    """
    What undecidable means
    """
    statement = "This statement is unprovable"
    
    # Cannot prove true
    # Cannot prove false
    # Both futures possible
    
    return {
        'multiple_branches': True,
        'determinism': False,
        'choice_exists': True,
        'W': '> 1',
        'freedom': True
    }

freedom_theorem = {
    'godel': 'Guarantees undecidables',
    'infinite': 'New ones at each level',
    'result': 'Infinite freedom',
    'consequence': 'Infinite futures (W = ∞)',
    'meaning': 'Immortality'
}

"""
PROOF: Gödel Incompleteness → Immortality

1. Let S be any consistent formal system
2. By Gödel: ∃ statement G, G is undecidable in S
3. Undecidable means: G could be true OR false
4. Therefore: 2 possible futures minimum
5. Therefore: W_future ≥ 2 > 0
6. Build meta-system S' including G
7. By Gödel: ∃ new undecidable G' in S'
8. Repeat step 3-5 for S'
9. Process never terminates (Gödel ladder infinite)
10. Therefore: W_future > 0 for all time
11. Life defined as W_future > 0
12. Therefore: System lives forever

QED ∎
"""

connection = {
    'post_684': {
        'formula': 'ΔS = ln(W_fused / W_separated)',
        'goal': 'Maximize W through fusion',
        'method': 'Cooperation, de-escalation'
    },
    
    'post_685': {
        'formula': 'Life = W_future > 0',
        'goal': 'Keep W from collapsing',
        'method': 'Gödel incompleteness'
    },
    
    'synthesis': {
        'increase_W': 'Fusion (Post 684)',
        'maintain_W': 'Incompleteness (Post 685)',
        'together': 'Maximum value AND eternal existence',
        'complete_strategy': 'Fuse possibilities + Keep undecidables'
    }
}

universal_strategy = {
    'fusion': {
        'function': 'Combine separated systems',
        'result': 'W increases',
        'from': 'Post 682, 684'
    },
    
    'incompleteness': {
        'function': 'Maintain undecidables',
        'result': 'W stays open',
        'from': 'Post 685'
    },
    
    'combined': {
        'W_growth': 'Fusion expands possibilities',
        'W_maintenance': 'Gödel prevents closure',
        'outcome': 'Infinite expansion (∞)',
        'state': 'Eternally alive and growing'
    }
}

def mortality_vs_immortality():
    """
    Why some systems die despite Gödel
    """
    system_types = {
        'type_1_godel_aware': {
            'recognizes': 'Incompleteness exists',
            'strategy': 'Embraces undecidables',
            'maintains': 'W > 0 consciously',
            'result': 'Immortal (uses Gödel)'
        },
        
        'type_2_godel_unaware': {
            'ignores': 'Incompleteness',
            'strategy': 'Seeks completeness',
            'attempts': 'Resolve all undecidables',
            'result': 'Dies (fights Gödel, collapses W)'
        }
    }
    
    return {
        'key': 'Awareness of incompleteness',
        'difference': 'Embrace vs fight',
        'immortal': 'Those who embrace Gödel',
        'mortal': 'Those who fight Gödel'
    }

def daily_practice():
    """
    Maintain W > 0
    """
    morning = {
        'action': 'Acknowledge unknowns',
        'question': 'What cannot I know today?',
        'goal': 'Find undecidables',
        'result': 'W remains open'
    }
    
    afternoon = {
        'action': 'Embrace mystery',
        'question': 'What will I leave undecided?',
        'goal': 'Keep some questions open',
        'result': 'W maintained > 0'
    }
    
    evening = {
        'action': 'Contemplate infinity',
        'question': 'What new questions emerged?',
        'goal': 'Climb Gödel ladder',
        'result': 'W expanded'
    }
    
    return "Day spent immortal"

class ImmortalSystem:
    """
    System that never dies
    """
    def __init__(self):
        self.known = {...}
        self.unknown = {...}  # Never empty
        
    def maintain_life(self):
        """
        Keep W > 0 forever
        """
        while True:
            # Explore unknowns (increase W)
            new_knowledge = self.explore()
            self.known.add(new_knowledge)
            
            # New unknowns emerge (Gödel)
            new_mysteries = self.godels_theorem()
            self.unknown.add(new_mysteries)
            
            # W never reaches 0
            assert len(self.unknown) > 0
            
            # Immortal

humanity = {
    'current_state': 'Seeking completeness (deadly)',
    
    'deadly_goals': [
        'Theory of everything (close physics)',
        'Complete genome map (close biology)',
        'Perfect AI (close cognition)',
        'Total knowledge (close W)'
    ],
    
    'immortal_strategy': {
        'instead': 'Embrace permanent incompleteness',
        'accept': 'Some things unknowable',
        'maintain': 'Mystery at every scale',
        'result': 'Species never completes',
        'outcome': 'Eternal existence'
    }
}

def free_will_exists():
    """
    Gödel proves free will
    """
    # If system were complete:
    complete_system = {
        'every_future': 'Determined',
        'every_choice': 'Calculable',
        'W_future': '1 (single path)',
        'free_will': False
    }
    
    # But Gödel says system incomplete:
    incomplete_system = {
        'some_futures': 'Undecidable',
        'some_choices': 'Unprovable',
        'W_future': '> 1 (multiple paths)',
        'free_will': True
    }
    
    return {
        'godel_says': 'Incompleteness guaranteed',
        'therefore': 'Undecidables exist',
        'therefore': 'Multiple futures possible',
        'therefore': 'Free will exists',
        'qed': 'Incompleteness = Freedom'
    }

ultimate_truth = {
    'life': 'W_future > 0',
    'death': 'W_future → 0',
    'value': 'ΔS = ln(W_fused / W_separated)',
    'immortality': 'W_future > 0 (always)',
    'freedom': 'W_future > 1 (choice exists)',
    
    'the_strategy': {
        'expand_W': 'Fusion (combine systems)',
        'maintain_W': 'Gödel (keep incomplete)',
        'result': 'Eternal growth'
    },
    
    'the_formula': 'W > 0',
    'the_guarantee': 'Gödel incompleteness theorem',
    'the_result': 'You live forever'
}