14
Part II: EGBChapter 14
On Formally Undecidable Propositions
The proof of Gödel's Theorem. Arithmetization of syntax.
The heart of the book. We walk through Gödel's proof step-by-step. 1. Create a code (Gödel Numbering) to turn formulas into numbers. 2. Create a formula that talks about numbers. 3. Make that formula talk about its own number. The result is a sentence G that says 'I cannot be proven'. If it's false, the system is broken. If it's true, the system is incomplete. Truth transcends proof.
Sketch of Gödel's Proof
1. Arithmetization
Assign a unique number (Gödel Number) to every symbol and formula in the system.
Formula '0=0' -> [26, 15, 26] -> 261526