How to pronounce “set theory” in English

initial main results in set theory, and it's profound and amazing and insightful and the beginning point of so many later arguments.

of two ways that set theory emerges. On the one hand, set theory is its own subject of mathematics, with its

- Right - ...Zermelo-Fraenkel set theory, that's the Z and the F and the C in that is this...

the Zermelo set theory and gave the proof that in that theory, you can prove that every set admits a well ordering.

nature of set theory into our fundamental understanding of how sets are, and it's not confusing anymore.

You know set theory.

That's the central idea of set theory.

forms of- of Zermelo-Fraenkel set theory and so on.

so captured by the idea of using set theory as a foundation of mathematics and it was so powerful and convenient and unifying in a

and one has the sense that he had in mind set theory in which all the questions are going to be answered.

disturbing paradox okay that showed that uh all of set theory and certainly Fger's

- And we should say, in set theory, consistency means that it is impossible to derive a contradiction from the axioms of the theory.

So the fact that set theory is able to serve as a foundation means that mathematics can be founded on logic.

Then in 1963, Paul Cohen proved there's also a world where all the axioms of set theory hold true except for the axiom of choice.

- The following is a conversation with Joel David Hamkins, a mathematician and philosopher specializing in set theory, the foundation of mathematics and the nature of

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

Set theory really has two roles that it's serving.

- 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

We wanna use the set theory foundations, but we want to do it in a way that is trustworthy and reliable.

The Austrian mathematician, Kurt Godel, proved there is a world all the other already accepted axioms of set theory hold true, and so does the axiom of choice.

Paul Cohen was awarded the Fields Medal three years later for his groundbreaking result, as well as his other work in set theory, and after Godel

And so really, from this point of view, set theory is about the transfinite recursive constructions or well-founded definitions and

It's very common to hear things said about set theory that really aren't taking account of this distinction between

So, I mean, the history of the current set theory axioms, known as the Zermelo-Fraenkel axioms, came out in the

He didn't have a list of axioms in the way that we have for set theory

—that he was pressed to produce the theory in which his argument could be formalized, and that was the origin of what's known as Zermelo set theory.

So cone of perplexity, ZFC, or Zermelo-Fraenkel set theory with the Axiom of Choice, as we mentioned, is the standard foundation for most

Just put yourself in the mindset of people at the beginning of this, of trying to formalize set theory.

because at that time, you know, with the Russell Paradox and so on, there were these various contradictions popping up in various parts of set theory and the Burali-Forti

And, and Hilbert was famously supportive of set theory.

And so he proposed, "Look, we're going to have this strong theory, this set theory that we want to be proving