But Details Don't Matter

I can imagine a nice Onion article that starts with this headline and then goes on, the way Onion articles do, repeating and amplifying this joke by quoting various mathematicians who say that that this discovery is incredibly important but the details aren't worth explaining.

The article below is a bit like that. It also seems to be confusing people. I've been running around trying to straighten things out. Let me try here.

It says mathematicians have just proved two infinities called p and t are equal. Then it goes into a long review of basics, like "how can one infinity be bigger than another?" and "is there an infinite set bigger than the set of integers and smaller than the set of real numbers?" This is great stuff, but it's old stuff. The first question was answered by Cantor. The second question was asked by Cantor. Gödel and Cohen answered it like this: you'll never know, you pathetic humans - and we can prove you'll never know! (Roughly speaking.)

Unfortunately, because most people have the attention span of a small bug when it comes to math, a lot of them quit around here and conclude that p must be the number of integers and t must be the number of real numbers... or something like that.

Or, they conclude that mathematicians have finally answered Cantor's question... showing up Gödel and Cohen for the arrogant bastards they were.

No, no, no. The infinities p and t are something else. Cantor's question is just as unanswerable as it always was.

So what are p and t?

If you read down far enough in the article, it says a few things about this:

Some problems remained, though, including a question from the 1940s about whether p is equal to t. Both p and t are orders of infinity that quantify the minimum size of collections of subsets of the natural numbers in precise (and seemingly unique) ways.

The details of the two sizes don’t much matter.

I really dislike this. Quanta is one of the very best magazines around when it comes to explaining math. They have very high standards. So I won't pull my punches here:

The details do matter! It's math, for god's sake! In math, the details matter!

What the author means is that:

The details matter, but if I explained them your eyeballs would fall out, so I'm not gonna.

If they just said that, preferably very early on, I'd like this article a lot more.

In fact the Quanta article does make a stab at explaining p and t. But it's in bunch of text to the right of the main article, so you know it's your own fault if you read it and your eyeballls fall out... you've voided the warrantee! It says this:

Briefly, p is the minimum size of a collection of infinite sets of the natural numbers that have a “strong finite intersection property” and no “pseudointersection,” which means the subsets overlap each other in a particular way; t is called the “tower number” and is the minimum size of a collection of subsets of the natural numbers that is ordered in a way called “reverse almost inclusion” and has no pseudointersection.

If you feel bad for not understanding this, don't. I'm a mathematician and I don't understand it either. The reason is that it's not an explanation.

Why not? Because "in a particular way" doesn't mean anything. Also, "in a way called reverse almost inclusion" doesn't mean anything unless you already know what "reverse almost inclusion" means. On top of that, they don't say what the "strong finite intersection property" is.

So this is a non-explanation. Perhaps that's why they say "Briefly" at the beginning. "It would take too long to explain this stuff, so we'll briefly not explain it."

It's like saying this:

Briefly, here is how you make shrimp jambalaya. You take shrimp and other ingredients and combine them in a particular way. Then you perform an activity called "jambalayafication".

Now, perhaps I'm being too grumpy about this half-hearted attempt at explanation. In fact I definitely am — I'm really getting into it, unleashing my inner curmudgeon, which I usually keep chained up in the cellar. I'm gonna regret saying all this, I can feel it already. It's like when you drink one beer too many, and you know, as you're taking the first sip of that one beer too many, that you'll regret it the next day. So I'll temper my remarks a bit now: this non-explanation does at least let the reader see that something is going on with infinite sets of natural numbers, and it's something fairly technical. With just a few changes this article could have been much better.

Scientific American is worse: they quote the article from Quanta magazine, and just leave out this non-explanation. So all the reader learns is that "the details don't much matter".

You can see the definition of p and t here:

• Maryanthe Malliaris and Saharon Shelah, General topology meets model theory, on p and t, Proceedings of the National Academy of Sciences, available for free at

Do you want me to explain them? Maybe it really doesn't matter. But I would be glad to give it a try.

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