Godel's Incompleteness Theorems are among the most famous results in modern mathematics. They are often informally stated as follows:
1. Any formal system that can express elementary arithmetic contains statements that are "true but not provable".
2. Any consistent formal system that can express elementary arithmetic cannot prove its own consistency.
These informal statements, however, often raise more questions than they answer. What exactly is a "formal system"? What does it mean for a statement to be "true but not provable"? What does it mean to "express arithmetic", and why does arithmetic have anything to do with proof or consistency?
This talk will introduce the audience to the main ideas behind the incompleteness theorems. We'll discuss the historical context in which the incompleteness theorems were discovered, develop the concepts necessary to properly understand what the theorems actually say, and present a sketch of the theorems' proof.
Scott Sanderson is a Senior Engineer at Quantopian, where he is responsible for designing and implementing Python APIs for algorithmic trading. Prior to working at Quantopian, Scott studied Mathematics and Philosophy at Williams College, where he maintained an interest in the logical foundations of mathematics and computer science.