Lex Fridman

All mathematics could not be built from sets, giving math its first truly rigorous foundation.

the axiomatization of mathematics.

The paradox has forced mathematicians to develop ZFC and other axiomatic systems, and mathematical logic emerged.

Gerrit O. Turing and others created entire new fields.

So can you explain what set theory is and how does it serve as a foundation of modern mathematics and maybe even the foundation of truth?

And axioms are, I guess, facts that we assume are true based on which we then build the ideas of mathematics.

So there's a bunch of facts, axioms about sets that we can put together, and if they're sufficiently

powerful.

We can then build on top of that a lot of really interesting mathematics.

And going to perplexity, the axiom of choice is a fundamental principle in set theory, which states that for any collection of non-empty sets, it is possible to select exactly one element from each set, even if no explicit rule to make the choice is given.

This axiom allows the construction of a new set containing one element from each original set, even in cases where the collection is infinite or where there is no natural way to specify a selection rule.

So this was controversial, and this was described before there was even a language for axiomatic systems.

So we're going to say the three letters of ZFC may be a lot in this conversation.

You already mentioned Zermelo-Fraenkel set theory.

That's the Z and the F, and the C in that comes from this axiom of choice.

So ZFC sounds like a super technical thing, but it is the set of axioms that's the foundation of modern mathematics.

Can we maybe speak to the axioms that underlie ZSC?

So going to perplexity, ZSC, or as Amela Frankel said, theory with the axiom of choice, as we mentioned, is the standard foundation for most modern mathematics.

It consists of the following main axioms.

axiom of extensionality, axiom of empty set, axiom of pairing, axiom of union, axiom of power set, axiom of infinity, axiom of separation, axiom of replacement, axiom of regularity, and axiom of choice.