Are two infinities equal? Two mathematicians claim to have proven it

The article is actually pretty good for the subject, though it gets itself contorted in a few weird ways. It ends strangely, saying:

While infinity is strange in many ways, it would be almost too strange if there weren’t many more sizes of it than the ones we’ve already found.

I think this misunderstands the whole thing. There are a number of sizes of infinity, and we've known that since Cantor. The question isn't the number of sizes of infinity (there are infinite sizes of infinity), but whether or not there exist intermediate sizes of infinity. I am not familiar with this work in particular, but Cantor thought that there was not an intermediate size (the author said he didn't know - he couldn't prove it, but he thought it was the case). I tend to agree with Cantor - the nature of infinity actually makes an intermediate size hard, because it can expand indefinitely through any intermediary you want to make. To put it simply, the set of integers is the same size as a number of other infinite sets, such as the number of rational numbers (numbers that can be expressed as a ratio), the number of positive integers, etc. However, the set of integers is smaller than the set of all real numbers. The question is whether there is an intermediate set size. Cantor answered "no" but couldn't prove it. These mathematicians proved that two sizes of infinities were the same that were previously unknown.johnnyb

September 17, 2017

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