Lex Fridman

In essence...

The goal was to formalize all of mathematics in precise axiomatic systems and then prove, using only very elementary finitary reasoning about symbols, that these systems are free of contradiction.

Right, exactly right.

Just for clarification, you use the word theory.

Is it in this context synonymous with axiomatic system?

So theory includes both the axioms and the consequences of the axioms, and you use it interchangeably, and the context is supposed to help you figure out which of the two you're talking about, the axioms or the consequences.

Or maybe to you, they're basically the same.

And that's, of course, connected to the halting problem.

All of these contradictions and paradoxes are all nicely, beautifully interconnected.

So can we just linger on Gerdes' Completeness Theorem?

You mentioned the two components there.

There's so many questions to ask.

What is the difference between provability and truth?

What is true and what is provable?

Maybe that's a good line to draw.

Yeah, it has the disquotation.

Yeah, a mathematician comes to mind that said he has a proof, but the margins are too small to continue.

So that doesn't count as a proof.

So what is, again, the tension between truth and proof, which is more powerful, and how do the two interplay with the contradictions that we've been discussing?

How traumatic is that, that there are statements that are independent from the theory?