**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.

#bigness

View 45 previous comments

- +John Baez I believe that set theory may be more often used in other parts of mathematics than your comment about "working mathematician" suggests.

One thinks, for example, of set-theoretic topology, where the researchers routinely use sophisticated set theoretic ideas such as Martin's axiom, PFA and large cardinals, and essentially all the fundamental questions are independent of ZFC. Those researchers, in my experience, generally consider themselves topologists rather than set theorists.

Another case would be the work on infinite combinatorics, such as infinitary Ramsey theory, which uses many set-theoretic ideas.

The subject of cardinal characteristics of the continuum in set theory touches many other areas of mathematics, as those characteristics generally arise naturally in those other areas.

The research area known as Borel equivalence relation theory is deeply connected with huge parts of mathematics, and practioners are generally expert not only the descriptive set theoretic tools, where the subject was born, but also in the areas where their target classifications problems arise, as well as methods from ergodic theory and others. The goal of this area is to provide an analysis of the relative difficulty of the naturally arising classification problems in mathematics, and a rich hierarchy has emerged. Those researchers include some very impressive generalists, and it would be hard to describe them as isolated.

Lastly, of course set theory is deeply connected with essentially all parts of mathematical logic, including especially model theory, where many of the fundamental questions and answers have a set-theoretic nature, but also computability theory, both in the analysis of higher computation and connections with descriptive set theory, but also in the new focus on randomness notions in computability theory, many of which are inspired by ideas in forcing.51w - Just to highlight two points that +David Roberts already made:

1) Invoking topos theory here is not about an alternative to set theory or opposing set theory, but about broadening the perspective. Sometimes broadening the perspective helps clarify aspects of a special case. The claim is that from the broader perspective of topos theory it is easier to get a conceptual grasp on forcing in ZFC. Because in topos theory we recognize forcing as a special case of one the most fundamental operations that governs topos theory in the first place: passage to sheaf toposes.

2) Those "well-rooted trees" that appear in models of ZFC inside topos theory are not an artifact arising in the translation. Instead, they are what ZF set theory is all about in the first place. The global membership relation which is the hallmark of ZF-type set theory over "structural" set theory, that's the encoding of tree-like structures. (A set whose elements are sets whose elements are sets... is a trunk whose branches are trunks whose branches..., hence is a tree)

That the central role of such trees in the foundations may feel irksome, as John says above, that's why some people feel they'd rather have the option to discuss foundations not necessarily involving such trees. Because it's not necessary.51w - +Joel David Hamkins - your examples are good. There are indeed people who apply sophisticated set theory in other branches of mathematics. However, your viewpoint makes you see more of them than most of us see, just as I know lots of mathematical physicists and easily slip into feeling that everyone knows some quantum field theory, when in reality few mathematicians do.

It's not surprising that quantum field theory seems obscure and esoteric to most mathematicians: most students don't see any in their math courses. What's surprising, given the logically central nature of the concept of 'set' in mathematics, is how little most mathematicians know about set theory. In my department full of algebraists, geometers, topologists and analysts, I doubt any has ever used any sophisticated set theory in their work. I suppose I should check: I used some effective descriptive set theory in my first paper, but I don't go around talking about this, so nobody would know; there could be other people like that. But I see the phrase "projective resolution" on the blackboard a whole lot, and never any buzzwords from set theory.

Yesterday I suddenly decided the reason for this must be sitting in plain sight: the axioms of set theory, unlike the axioms of other subjects, were originally designed to be*categorical*. [1] Of course they're not, but deviations from this are still perceived by most mathematicians as 'pathological': Goedel's theorem is still considered a kind of tragic shock - or, more likely, a scary rumble at the horizon. I think we need to get past that if we ever want to make set theory part of 'ordinary mathematics', instead of the 'queen and handmaiden' it is now: seemingly respected, but usually a hard-working and ignored humble servant.~~----------------------~~

[1]*Categorical*. I feel I have to explain this peculiar term to bystanders, though certainly not the people in this conversation. It has nothing to do with category theory, it's more closely related to phrases like "you are categorically forbidden to drink before 4 pm". An axiom system is**categorical**if it has a unique model (up to isomorphism).

A good example is the usual second-order axiomatization of the real number system as a complete ordered field. Most mathematicians want to speak of 'the' real line, most of us don't want to study the subject of 'real lines', with a long list of more or less exotic examples. So, we are happy to learn that there's a categorical axiom system for the real line, and initially disturbed to learn that (because it's second-order) this fact doesn't carry as much weight as you might hope: there are in fact lots of undecidable questions about the real line, and thus different 'versions' of the real line living in different universes, where these answers have different answers.

Similarly, most mathematicians would like to believe there is 'the' universe of sets.51w - +John Baez Yes, of course you are right; mathematicians who don't engage in set theory much tend to know little about it.

As a possible counterpoint, however, let me say that when I first joined MathOverflow, I had expected that there would be essentially no way for a logician or set theorist to rise very high, because I expect that mathematicians generally wouldn't be so interested in set-theoretic questions. But I found just the opposite! I was very pleased to discover on MathOverflow that mathematicians from all parts of the subject have questions about and are interested in issues connected with logic and set theory. People have questions about the axiom of choice, the continuum hypothesis, about infinite combinatorics, about the nature of logical independence and so on. Their questions also helped broaden me as a mathematician, since I had to learn a little about their area in order to answer.

I find your remarks on categoricity to be spot-on, and I believe that this difference in viewpoint is the main point of contention in the current philosophical debate on set-theoretic pluralism, on whether set theory is about a single universe of sets or about multiple possible set-theoretic universes. In my own writing, I have pointed to the work on set-theoretic geology, the topic of your post, as an example illustrating how the pluralist perspective leads to interesting or even profound mathematical questions. Another case would be the modal logic of forcing, as the pluralist perspective leads one naturally to a modal perspective on mathematical truth.51w - I know this is and old post, but I'd like to add something that helped me understand forcing and the axiom of choice. Cohen used forcing to construct a model that had an inner model in which the axiom of choice didn't hold. In fact, all the examples such as Solovay's model with all sets Lebesgue measurable, Blass's model with no nonprincipal ultrafilters, and various models of the axiom of determinacy follow the general scheme:

1. Use forcing to make certain sets, implied to exist by AC, hard to define.

2. Restrict to an inner model of "easily defined" sets, in which the axiom of choice therefore fails.

This happens for two reasons:

1. Forcing preserves AC, so some other step is needed to get a model in which it fails.

2. In L, every set has a definable well ordering. So these "choicy" objects (unmeasurable sets, nonprincipal ultrafilters, undetermined games) are actually definable.

So neither forcing nor inner models suffices on its own.

This somewhat goes against a common way of thinking, as expounded by Schechter in his handbook on analysis, that these choicy objects are intangible. It seems to me that it is only the case that they can consistently be supposed to be intangible.49w - +Robert Furber - thanks for those insights, which are very helpful to me. I still haven't gotten around to thinking about how people use forcing to break AC.

By the way, this is not an old post by my standards, it's about the fifth-to-last of several hundred.49w

Add a comment...