Lex Fridman

Both Russell's Paradox and the proof.

Yeah, that's a profound question.

And does a committee with just one member meet also?

It's heartbreaking.

I mean, there's nothing more traumatic to a person who dreams of constructing mathematics all from logic to get a very clean, simple contradiction.

What do you think about the Frege project, the philosophy of logic, the dream of the power of logic to construct the mathematical universe?

I think this is a good moment to talk about Gödel's incompleteness theorems.

So can you explain them and what they teach us about the nature of mathematical truth?

This minefield of paradoxes.

So we kind of think of proof as, maybe it's fair to say, almost like outside of math.

It's like tools operating on math.

And then for Hilbert, he thought proof is inside the axiomatic system.

Something like this.

What does the word finitary and finitary theory mean?

So going to perplexity, piano arithmetic is a foundational system for formalizing the properties and operations of natural numbers using a set of axioms called the piano axioms.

Piano arithmetic provides a formal language and axioms for arithmetic operations such as addition and multiplication over the natural numbers.

The axioms define the existence

of a first natural number, usually zero or one, the concept of successor function, which generates the next natural number, rules for addition and multiplication built from these concepts, the principle of induction along proofs around all natural numbers, and it goes on.

So it's a very particular kind of arithmetic that is affinitary.

And if I may go into a perplexity definition of Hilbert's program, it was David Hilbert's early 20th century project to give all of classical mathematics a completely secure, finitary foundation.