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