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== Gödel’s incompleteness theorems ==
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__NOTOC__
  
 
'''Gödel’s incompleteness theorems''' are two famous results in mathematical logic (proved by Kurt Gödel in 1931). In plain language, they show that any reasonable “rulebook” for doing arithmetic has built-in limits: it can’t prove everything that’s true, and it can’t fully prove its own reliability from within itself.
 
'''Gödel’s incompleteness theorems''' are two famous results in mathematical logic (proved by Kurt Gödel in 1931). In plain language, they show that any reasonable “rulebook” for doing arithmetic has built-in limits: it can’t prove everything that’s true, and it can’t fully prove its own reliability from within itself.
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Math proofs inside a formal system are like playing by the rules—no intuition, no hand-waving, just legal moves.
 
Math proofs inside a formal system are like playing by the rules—no intuition, no hand-waving, just legal moves.
  
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=== The First Incompleteness Theorem (informal) ===
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<youtube>u3GYrEOoKGk</youtube>
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= The First Incompleteness Theorem =
 
If a formal system is:
 
If a formal system is:
  
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* free of contradictions ('''consistent'''),
 
* free of contradictions ('''consistent'''),
  
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then:
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then there will be some statements that are '''true''' but '''not provable''' using that system’s rules.
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* there will be some statements that are '''true''' but '''not provable''' using that system’s rules.
 
  
 
'''High-school analogy:'''   
 
'''High-school analogy:'''   
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*not everything true can be proven inside it.*
 
*not everything true can be proven inside it.*
  
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=== The Second Incompleteness Theorem (informal) ===
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= The Second Incompleteness Theorem =
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For the same kind of system (arithmetic + consistency), Gödel also showed:
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For the same kind of system (arithmetic + consistency), Gödel also showed the system cannot prove its own consistency using only its own rules. It’s like a rulebook that can’t provide a totally trustworthy “certificate” that the rulebook will never lead to contradictions, without relying on something outside itself.
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* the system cannot prove its own consistency using only its own rules.
 
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'''High-school analogy:''' 
 
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It’s like a rulebook that can’t provide a totally trustworthy “certificate” that the rulebook will never lead to contradictions, without relying on something outside itself.
 
  
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=== Why people talk about this in consciousness debates ===
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[[Creatives#Roger Penrose |Sir Roger Penrose]]’s rough line of thought (simplified) is:
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Penrose’s rough line of thought (simplified) is:
 
  
 
* Computers follow formal rules (algorithms).
 
* Computers follow formal rules (algorithms).
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* Therefore, human understanding may not be purely algorithmic.
 
* Therefore, human understanding may not be purely algorithmic.
  
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=== Important caution (what Gödel does NOT automatically prove) ===
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<youtube>gd6-12ClN14</youtube>
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= Important caution (what Gödel does NOT automatically prove) =
 
Gödel’s theorems do '''not''' automatically prove that:
 
Gödel’s theorems do '''not''' automatically prove that:
  
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Those are additional philosophical steps, and many mathematics
 
Those are additional philosophical steps, and many mathematics
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= Liar's Paradox =
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A classic logic puzzle that creates a contradiction using self-reference.
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The most famous version is a sentence like:
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: '''“This statement is false.”'''
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=== Why it’s a paradox ===
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Try to label the sentence as either '''true''' or '''false''':
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* If the statement is '''true''', then what it says must be correct — so it is '''false'''.
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* If the statement is '''false''', then what it says is incorrect — which would make it '''true'''.
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Either choice flips into the other. That’s the paradox.
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=== A simpler everyday version ===
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* “I am lying.”
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If the speaker is lying, then they’re telling the truth. 
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If they’re telling the truth, then they’re lying.
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Same trap.
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=== What the paradox teaches ===
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The Liar’s Paradox shows that:
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* '''self-reference''' can break ordinary true/false logic
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* some statements can’t be consistently assigned a truth value within a simple system
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It’s one of the reasons logicians became careful about:
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 +
* how languages refer to themselves
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* what counts as a valid “statement” in a formal system
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=== Connection (loosely) to Gödel ===
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Gödel’s work is not the same as the Liar’s Paradox, but it uses a related move. Gödel constructed mathematical statements that “talk about” provability inside a formal system.

Latest revision as of 15:20, 5 January 2026

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Gödel’s incompleteness theorems are two famous results in mathematical logic (proved by Kurt Gödel in 1931). In plain language, they show that any reasonable “rulebook” for doing arithmetic has built-in limits: it can’t prove everything that’s true, and it can’t fully prove its own reliability from within itself.

These theorems come up in Orch-OR discussions because Roger Penrose argues that the limits of formal rule systems hint that human understanding may involve something beyond standard computation. (That claim is debated.)

Key idea: a “formal system”

A formal system is like a super-strict game with:

  • Axioms: starting rules you accept without proof (your “rulebook”)
  • Rules of inference: allowed step-by-step moves for proving new statements

Math proofs inside a formal system are like playing by the rules—no intuition, no hand-waving, just legal moves.

The First Incompleteness Theorem

If a formal system is:

  • strong enough to do basic arithmetic, and
  • free of contradictions (consistent),

then there will be some statements that are true but not provable using that system’s rules.

High-school analogy: Imagine a language that can describe lots of facts about numbers. Gödel showed that you can always craft a “tricky sentence” that basically says:

“This statement cannot be proven in this rule system.”

If the system could prove it, it would create a contradiction. So a consistent system can’t prove it—even though (from the outside) we can see what’s going on.

So the system is inevitably incomplete:

  • not everything true can be proven inside it.*

The Second Incompleteness Theorem

For the same kind of system (arithmetic + consistency), Gödel also showed the system cannot prove its own consistency using only its own rules. It’s like a rulebook that can’t provide a totally trustworthy “certificate” that the rulebook will never lead to contradictions, without relying on something outside itself.

Sir Roger Penrose’s rough line of thought (simplified) is:

  • Computers follow formal rules (algorithms).
  • Gödel shows formal rule systems have limits.
  • Humans seem able to “see” truths that no fixed formal system can prove.
  • Therefore, human understanding may not be purely algorithmic.


Important caution (what Gödel does NOT automatically prove)

Gödel’s theorems do not automatically prove that:

  • humans are “beyond computation,” or
  • minds must use quantum physics, or
  • brains access a special non-computational feature of reality.

Those are additional philosophical steps, and many mathematics

Liar's Paradox

A classic logic puzzle that creates a contradiction using self-reference.

The most famous version is a sentence like:

“This statement is false.”

Why it’s a paradox

Try to label the sentence as either true or false:

  • If the statement is true, then what it says must be correct — so it is false.
  • If the statement is false, then what it says is incorrect — which would make it true.

Either choice flips into the other. That’s the paradox.

A simpler everyday version

  • “I am lying.”

If the speaker is lying, then they’re telling the truth. If they’re telling the truth, then they’re lying.

Same trap.

What the paradox teaches

The Liar’s Paradox shows that:

  • self-reference can break ordinary true/false logic
  • some statements can’t be consistently assigned a truth value within a simple system

It’s one of the reasons logicians became careful about:

  • how languages refer to themselves
  • what counts as a valid “statement” in a formal system

Connection (loosely) to Gödel

Gödel’s work is not the same as the Liar’s Paradox, but it uses a related move. Gödel constructed mathematical statements that “talk about” provability inside a formal system.

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