The hot inner core of the mathematical universe

Set theory starts out as a very simple way of organizing our thoughts — something every student should learn. But it gets more tricky when we start pondering infinite sets. And when we start pondering the universe — the collection of all sets — it gets a lot harder. Mathematicians have learned that there are obstacles to fully understanding the universe.

The collection of all sets can't be a set — Bertrand Russell and other logicians discovered this over a century ago. But more importantly, Gödel's theorem puts limits on how well any axioms can pin down the properties of the universe. Most mathematicians like to use the Zermelo-Fraenkel axioms together with the axiom of choice. But there are many questions left unsettled by these axioms.

Knowing this, you might give up on trying to fully understand the universe. That's actually what most mathematicians do. Frankly, the questions left unsettled by the ZFC axioms don't seem very urgent to most of us!

But set theorists don't give up. They've developed a lot of fascinating ways to make progress despite the obstacles.

In the 1960s, Paul Cohen introduced forcing. This is a way to make the universe larger, by making up a bunch of new sets, without violating the axioms you're using.

If I think the universe is U, you can use forcing to say "fine, but it's equally consistent to assume the universe is some larger collection V". Cohen used this to show the axiom of choice couldn't be proved from the other axioms in ZFC. Given a universe U where the Zermelo-Fraenkel axioms hold, he used forcing to build a bigger universe V where those axioms still hold, but the axiom of choice does not!

As an undergrad, I gave up my studies of set theory before I learned forcing. It was too hard to understand, and probably too badly explained: I don't think anyone even said what I just told you! I moved on to other things - there's a lot of fun stuff to learn. But for modern set theorists, forcing is utterly basic.

So what's new?

One new thing is set-theoretic geology. In this approach to set theory, instead of making the universe larger, you make it smaller. You try to 'dig down' and find the smallest possible universe!

So, starting with some universe V, we look for a smaller universe U that can give rise to V by forcing. If this is true, we call U a ground for V.

There can be lots of grounds for a universe V. This raises a big question: if we have two grounds for V, is there a ground that's contained in both?

In 2015, Toshimichi Usuba showed this is true! In fact he showed that for any set of grounds of V, there's a ground contained in all of these.

This raises another big question: is there a smallest ground, a ground contained in all other grounds? If so, this is called the bedrock of our universe V.

Usuba showed that the bedrock exists if a certain kind of infinite number exists! There are different sizes of infinity, and this particular kind is called 'hyper-huge'. It's so huge that it's not even explained in the Wikipedia article on huge cardinals. So, I can't explain it to you, or even to myself.

But still, I think I get the basic idea: if we have a large enough infinity, digging down infinitely far that much will get us down to the bedrock of the universe.

Naively, I tend to favor small universes. So, the bedrock appeals to me. However, you need a big universe to have large infinities like 'hyper-huge cardinals'. So, my minimalist philosophy runs into a problem, because your universe needs to contain big infinities for you to 'have time' to dig deep enough to hit bedrock!

Is this a paradox? Certainly not in the literal sense of a logical contradiction. But how about in the sense of something bizarre that makes no sense?

Probably not. There's a way to take the universe and divide into 'levels', called the von Neumann hierarchy. If you assert the existence of large cardinals, you're making the universe 'taller' — you're adding extra levels. But if you stick in extra sets by forcing, you might be making the universe 'wider' — that is, adding more sets at existing levels. So, you may need a super-huge cardinal to have enough time to chip away at the stuff in all these levels until you hit bedrock.

This is just my guess; I'm no expert. For more information from an actual expert try the blog article I'm linking to, by Joel David Hamkins.

He talks about a concept called the 'mantle', without explaining it. But he explains it in a comment to his post: the mantle of the universe is the intersection of all grounds. If there's a hyper-huge cardinal, this must be the bedrock. If not, other things can happen.

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