! Disclaimer: Ironically these notes are incomplete.

Overview

Kurt Godel proved that any formulation of logic that can represent arithmetic can encode statements and proofs as numbers, through Godel Numbering. With this Godel encoded a derivation of the Liars Paradox - ‘This problem is not provable in T’ - proving that not all mathematical statements are Decidable.

This proved that there is no solution to the Decision Problem (Entscheidungsproblem)

Takeways

There is a gap between truth and proof. Truth and provability are not the same thing.
If we want mathematics to be consistent, we must accept that it is incomplete and undecidable.
Any Formal System that is powerful enough to describe basic arithmetic is either incomplete or inconsistent.
Gödel’s Second Incompleteness Theorem demonstrates that a consistent formal system cannot even prove its own consistency, let alone its completeness (? | salt).

Remember

Consistency ensures no contradiction.
Decidability ensures that there is a finite, mechanical way to determine the truth or falsehood of any statement within the system.
Completeness guarantees that every true statement has a proof or counterexample within the system.
- Completeness guarantees consistency, but consistency does not guarantee completeness. (?)