# 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:

1. **Euclidean:** Given line l and point p ∉ l, exactly one line through p parallel to l
2. **Hyperbolic:** Given line l and point p ∉ l, infinitely many lines through p parallel to l
3. **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!)

## Related
- [[02-propositions|Propositions]]
- [[01-what-is-a-proof|What is a Proof?]]