# Godel's Incompleteness Theorems

## Summary

Godel's Incompleteness Theorems are fundamental results showing the limitations of formal mathematics. The First Incompleteness Theorem shows that in any consistent logical system which includes arithmetic, some statements cannot be proved or disproved. Equivalently, the set of true statements about the natural numbers is undecidable. The Second Incompleteness Theorem states that no formal system which includes arithmetic can prove its own consistency, or that of a more powerful theory.

## Context

This concept has the prerequisites:

## Goals

• State Godel's First Incompleteness Theorem (in terms of some statements not being provable one way or the other)
• State the theorem in terms of the theory of natural numbers being undecidable
• State Godel's Second Incompleteness Theorem
• Be aware of Hilbert's program and why the second Incompleteness Theorem showed it to be impossible
• Prove the theorems

## -Free-

Notes on Logic (2013)
Lecture notes for a course on first order logic.
Author: Henry Cohn