How Can Infinity Come in Many Sizes? -- Quanta Magazine

[erronis]

https://www.quantamagazine.org/how-can-infinity-come-in-many-sizes-20260223/
Mark Belan, Jordana Cepelewicz

Intuition breaks down once we're dealing with the endless. To begin with: Some infinities are bigger than others.

An interactive article - well constructed but hard to excerpt.

nfinity invites resistance. Aristotle rejected the existence of the infinite entirely; to him, infinity was simply a limit that could never be reached, not a true mathematical entity. In the early 17th century, Galileo wrote that typical ways of thinking about sets and numbers held no meaning in the realm of the infinite, and that mathematicians would only find paradoxes if they tried to apply their usual tool kit to it. And when, 200 years later, Georg Cantor formalized the idea that infinity comes in many sizes, he was met with anger and fear. His colleagues dismissed his work as that of a madman.

But in time, Cantor's work on sets and infinity would form the bedrock of modern mathematics. As David Hilbert, another mathematical great, later wrote: "No one shall expel us from the paradise that Cantor has created for us."

So how can infinity have different sizes?

Welcome to Cantor's paradise.

. . .

[Doodley]

1. Fascinating!

[hvn_nbr_2]

2. I studied this stuff when I was a math major 50+ years ago

Fascinating stuff.

BTW, the referenced article is well-written and accessible, I think. It only explains the first two levels of infinity, but that's enough to show that infinity is not one single entity. I'm not generally very impressed by scientific writing on the net--it's often written by journalists/writers who don't really understand what they're talking about. But these two writers both know their topic and can explain it clearly.

[erronis]

3. I've been enjoying Quanta Magazine's articles for several years

They do a good job of bridging the deep theoretical and exploratory nature of mathematics along with a writing style that makes it approachable to even someone like me.

And then there are the references/cites for those that can dig deeper!

'hvn_nbr_2'? Seems there must be a story behind your handle.