49 lines
1.7 KiB
Markdown
49 lines
1.7 KiB
Markdown
# Axioms and Gödel's Incompleteness Theorem
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**Source:** MIT 6.1200J Lecture 01
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## Axiom
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**Definition:** A proposition that is assumed to be True.
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### Key Insight
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- You MUST make assumptions in mathematics
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- The key is to **state them upfront**
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## Examples of Contradictory Axiom Systems
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All three are equally valid — they define different geometric worlds:
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1. **Euclidean:** Given line l and point p ∉ l, exactly one line through p parallel to l
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2. **Hyperbolic:** Given line l and point p ∉ l, infinitely many lines through p parallel to l
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3. **Spherical:** Given line l and point p ∉ l, no line through p parallel to l
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**Lesson:** Different axioms yield different proofs and theorems. Anyone who accepts your axioms must accept theorems derived from them.
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## Consistency and Completeness
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### Consistent
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Definition: A set of axioms is consistent if no proposition can be both proved and disproved.
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### Complete
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Definition: A set of axioms is complete if every proposition can be either proved or disproved.
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**Ideal scenario:** Both consistent AND complete.
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## Gödel's Incompleteness Theorem
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**Theorem (Kurt Gödel, 1930s):** No set of axioms is both complete and consistent.
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### Impact
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- Shocked the mathematical community
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- Logicians Russell and Whitehead spent careers trying to find complete + consistent axioms for arithmetic — impossible!
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### Corollary
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If axioms must be consistent (necessary for validity), then there exist **True statements that cannot be proved**.
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### Example
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- Goldbach's Conjecture might be true but unprovable
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- (If so, please don't assign it as homework!)
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## Related
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- [[02-propositions|Propositions]]
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- [[01-what-is-a-proof|What is a Proof?]]
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