## Friday, July 16, 2021

### Sizes of Infinity

I am floored. Here is a new mathematical result that sounds pretty important. I'm surprised I hadn't heard of it sooner. It was published online in April.

This Quanta article explains it pretty well. But if the article doesn't make sense to you, I can explain more. This is the field I had planned to go into when I was thinking I'd get a PhD. I loved my two logic courses at Eastern Michigan University.  But the one I took at UCSD was not fun. I think because it was too far above me, and I couldn't stay grounded.

The one problem with the article is that it made it sound like the big question was resolved. But it's not. I thought it was saying that the continuum hypothesis is false. The continuum hypothesis is about sizes of infinity. The smallest infinity is what you get when you count out all the infinite whole numbers (or all the fractions), and it is called the countable infinity. The continuum hypothesis says that the next size up is what you'd get "counting" the real numbers (like the number line). But there may be a size in between.

I hope there is a way to get a meaningful example of that in-between size of infinity. (The are bigger and bigger infinities, but the two things grounded in numbers we know well, integers and real numbers, are the most interesting to me.)

A fun way to start thinking about infinity is a book that's accessible even to young kids. It's a five chapter picture book titled The Cat in Numberland. Sadly, it doesn't seem to be available (unless you want to pay ridiculous prices). My publisher, Natural Math, tried to help the author get it reprinted, but Cricket books (Carus publishing) wouldn't give up their rights, and won't republish. (Maybe we should look into that again...)

[The Quanta article links to the proof that was published online in April. I don't expect to understand that, but I'll try reading it. I might quit very quickly.]