What is Goethel's incomplete theorem?

The concept you're referring to is likely Gödel's Incompleteness Theorems, discussed in Dwarkesh Patel's conversation with Grant Sanderson on The Lunar Society podcast.

1. Understanding Gödel's Theorems: Gödel's Incompleteness Theorem posits that within any consistent axiomatic system, there are propositions that cannot be proven or disproven using the axioms within that system. This reflects the idea that no system can be both complete (able to prove every truth) and consistent (free of contradictions) simultaneously. The notion is akin to the paradoxical statement "this statement is a lie," which cannot be adjudicated as true or false without contradiction 1.

2. Practical Relevance: According to Sanderson, Gödel's Theorems don't frequently arise in practical, day-to-day mathematical work. Mathematicians aren't generally concerned about these limits when tackling natural questions, like the Riemann hypothesis or twin prime conjectures, which they expect to have definitive answers 1.

For further detailed exploration, you can check out the conversation between Dwarkesh Patel and Grant Sanderson in their episode of The Lunar Society Grant Sanderson (3Blue1Brown) - Past, Present, & Future of Mathematics.