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Is mathematics mostly chaos or mostly order?

(www.quantamagazine.org)
116 points baruchel | 2 comments | 20 Jun 25 15:21 UTC | HN request time: 0.396s | source
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Sniffnoy ◴[24 Jun 25 04:10 UTC] No.44362793[source]▶
> 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.)

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bubblyworld ◴[24 Jun 25 06:42 UTC] No.44363447[source]▶
I suspect they mean "add" in the sense of "add in an axiom asserting the existence of another cardinal". Things like consistency strength of the resulting theory seem to vary wildly depending on what other cardinals you throw into the mix (if I understood the article correctly, haven't read the paper, mea culpa).
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1. gowld ◴[24 Jun 25 13:47 UTC] No.44366228[source]▶
I think you are right. Plain English is a terrible language for higher math. It's extremely misleading (especially when talking about infinite objects and probability, where most of the mystery and confusion comes from terms that aren't precisely defined in the reader's mind).
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2. jerf ◴[24 Jun 25 14:13 UTC] No.44366479[source]▶
I enjoyed quanta magazine for a while. But I think they ended up pretty much exhausting the set of things that can be explained to an interested audience with a bit more English, but doesn't take actual mathematics. I know enough about this topic to know that reading the article didn't teach me anything useful because I don't know enough to understand what they are saying. And if I did know enough, this article probably still wouldn't effectively teach me because it's too simplified.

I haven't seen a quanta article in a while that I found useful. I appreciate the attempt but I don't think it works anymore. And I don't think it's their "fault"... I just think the slice of things this works for was smaller than we might have liked.