Lex Fridman

This is the Lex Friedman Podcast.

To support it, please check out our sponsors in the description, where you can also find ways to contact me, ask questions, give feedback, and so on.

And now, dear friends, here's Joe David Hempkins.

Some infinities are bigger than others.

This idea from Cantor at the end of the 19th century, I think it's fair to say broke mathematics before rebuilding it.

And I also read that this was a devastating and transformative discovery for several reasons.

So one, it created a theological crisis because infinity is associated with God.

How could there be multiple infinities?

and also Cantor was deeply religious himself.

Second, there's a mathematical civil war.

The leading German mathematician, Kronecker, called Cantor a corrupter of youth and tried to block his career.

Third, many fascinating paradoxes emerged from this.

like Russell's paradox about the set of all sets that don't contain themselves, and those threatened to make all of mathematics inconsistent.

And finally, on the psychological side, on the personal side, Cantor's own breakdown.

He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis.

So laying that all out on the table,

Can you explain the idea of infinity, that some infinities are larger than others, and why was this so transformative to mathematics?

Yeah, let's talk about the Hilbert's Hotel.

Like I mentioned, I'm a programmer, so starting at zero is a wonderful place to start.

Exactly.