Is Mathematics Mostly Chaos or Mostly Order?

If I had a semester or two of free time I'd love to hit this subject again. I once told my math prof (logician) who made a comment about super cardinals: careful it's powerful but it's power from the devil. I half regret that comment in retrospect.

I've never made peace with Cantor's diagonaliztion argument because listing real numbers on the right side (natural number lhs for the mapping) is giving a real number including transedentals that pre-bakes in a kind of undefined infinite.

Maybe it's the idea of a completed infinity that's my problem; maybe it's the fact I don't understand how to define (or forgot cauchy sequences in detail) an arbitrary real.

In short, if reals are a confusing inginite you can only tie yourself up in knots using confusing.

Sigh - wish I could do better!

> But add a smaller cardinal to one of the new infinities, and “they kind of blow up,” Bagaria said. “This is a phenomenon that had never appeared before.”

I have to wonder just what is meant by this, because in ZFC, a sum of just two (or any finite number) of cardinals can't "blow up" like this; you need an infinite sum. I mean, presumably they're referring to such an infinite sum, but they don't really explain, and they make it sound like it's just adding two even though that can't be what is meant.

(In ZFC, if you add two cardinals, of which at least one is infinite, the sum will always be equal to the maximum of the two. Indeed, the same is true for multiplication, as long as neither of the cardinals is zero. And of course both of these extend to any finite sum. To get interesting sums or products that involve infinite cardinals, you need infinitely many summands or factors.)

I’ve always considered math is something that is discovered, neither chaotic or orderly, it just… is. Really brilliant people make new discoveries, but they were there the whole time waiting to be found.

This article seems to kind of dance around yet agree with the discovery thing, but in an indirect way.

Math is just math. Music is just music. Even seemingly-random musical notes played in a “song” has a rational explanation relative to the instrument. It isn’t the fault of music that a song might sound chaotic, it’s just music. Bad music maybe. This analogy can break down quickly, but in my head it makes sense.

Disclaimer - the most advanced math classes I’ve taken: calc3/linear/diffeq.

how much of modern set theory is reverse engineered from axioms rather than discovered. we're always building highways through a forest we haven't mapped, assuming every tree will fall in line. and suddenly these new large cardinals show up that don't even sit neatly in the ladder. it's maynot be failure of math,but failure of narrative. we thought the infinite was climbable, now it's folding sideways. maybe the math we're building is just a subset of what's possible, shaped by what's provable under our current tools. lot of deep shit probably hiding in the unprovable.