My Little 2021 Mathematics A-to-Z: Zorn’s Lemma

The joke to which I alluded last week was a quick pun. The setup is, “What is yellow and equivalent to the Axiom of Choice?” It’s the topic for this week, and the conclusion of the Little 2021 Mathematics A-to-Z. I again thank Mr Wu, of Singapore Maths Tuition, for a delightful topic.

Zorn’s Lemma

Max Zorn did not name it Zorn’s Lemma. You expected that. He thought of it just as a Maximal Principle when introducing it in a 1934 presentation and 1935 paper. The word “lemma” connotes that some theorem is a small thing. It usually means it’s used to prove some larger and more interesting theorem. Zorn’s Lemma is one of those small things. With the right background, a rigorous proof is a couple not-too-dense paragraphs. Without the right background? It’s one of those proofs you read the statement of and nod, agreeing, that sounds reasonable.

The lemma is about partially ordered sets. A set’s partially ordered if it has a relationship between pairs of items in it. You will sometimes see a partially ordered set called a “poset”, a term of mathematical art which make me smile too. If we don’t know anything about the ordering relationship we’ll use the ≤ symbol, just like this was ordinary numbers. To be partially ordered, whenever x ≤ y and y ≤ x, we know that x and y must be equal. And the converse: if x = y then x ≤ y and y ≤ x. What makes this partial is that we’re not guaranteed that every x and y relate in some way. It’s a totally ordered set if we’re guaranteed that at least one of x ≤ y and y ≤ x is always true. And then there is such a thing as a well-ordered set. This is a totally ordered set for which every subset (unless it’s empty) has a minimal element.

If we have a couple elements, each of which we can put in some order, then we can create a chain. If x ≤ y and y ≤ z, then we can write x ≤ y ≤ z and we have at least three things all relating to one another. This seems like stuff too basic to notice, if we think too literally about the relationship being “is less than or equal to”. If the relationship is, say, “divides wholly into”, then we get some interesting different chains. Like, 2 divides into 4, which divides into 8, which divides into 24. And 3 divides into 6 which divides into 24. But 2 doesn’t divide into 3, nor 3 into 2. 4 doesn’t divide into 6, nor 6 into either 8 or 4.

So what Zorn’s Lemma says is, if all the chains in a partially ordered set each have an upper bound, then, the partially ordered set has a maximal element. “Maximal element” here means an element that doesn’t have a bigger comparable element. (That is, m is maximal if there’s no other element b for which m ≤ b. It’s possible that m and b can’t be compared, though, the way 6 doesn’t divide 8 and 8 doesn’t divide 6.) This is a little different from a “maximum” . It’s possible for there to be several maximal elements. But if you parse this as “if you can always find a maximum in a string of elements, there’s some maximum element”? And remember there could be many maximums? Then you’re getting the point.

You may also ask how this could be interesting. Zorn’s Lemma is an existence proof. Most existence proofs assure us a thing we thought existed does, but don’t tell us how to find it. This is all right. We tend to rely on an existence proof when we want to talk about some mathematical item but don’t care about fussy things like what it is. It is much the way we might talk about “an odd perfect number N”. We can describe interesting things that follow from having such a number even before we know what value N has.

A classic example, the one you find in any discussion of using Zorn’s Lemma, is about the basis for a vector space. This is like deciding how to give directions to a point in space. But vector spaces include some quite abstract things. One vector space is “the set of all functions you can integrate”. Another is “matrices whose elements are all four-dimensional rotations”. There might be literally infinitely many “directions” to go. How do we know we can find a set of directions that work as well as, for guiding us around a city, the north-south-east-west compass rose does? So there’s the answer. There are other things done all the time, too. A nontrivial ring-with-identity, for example, has to have a maximal ideal. (An ideal is a subset of the ring that’s still a ring.) This is handy to know if you’re working with rings a lot.

The joke in my prologue was built on the claim Zorn’s Lemma is equivalent to the Axiom of Choice. The Axiom of Choice is a piece of set theory that surprised everyone by being independent of the Zermelo-Fraenkel axioms. The Axiom says that, if you have a collection of disjoint nonempty sets, then there must exist at least one set with exactly one element from each of those sets. That is, you can pick one thing out of each of a set of bins. It’s easy to see how this has in common with Zorn’s Lemma being too obvious to imagine proving. That’s the sort of thing that makes a good axiom. Thing about a lemma, though, is we do prove it. That’s how we know it’s a lemma. How can a lemma be equivalent to an axiom?

I’l argue by analogy. In Euclidean geometry one of the axioms is this annoying statement about on which side of a line two other lines that intersect it will meet. If you have this axiom, you can prove some nice results, like, the interior angles of a triangle add up to two right angles. If you decide you’d rather make your axiom that bit about the interior angles adding up? You can go from that to prove the thing about two lines crossing a third line.

So it is here. If you suppose the Axiom of Choice is true, you can get Zorn’s Lemma: you can pick an element in your set, find a chain for which that’s the minimum, and find your maximal element from that. If you make Zorn’s Lemma your axiom? You can use x ≤ y to mean “x is a less desirable element to pick out of this set than is y”. And then you can choose a maximal element out of your set. (It’s a bit more work than that, but it’s that kind of work.)

There’s another theorem, or principle, that’s (with reservations) equivalent to both Zorn’s Lemma and the Axiom of Choice. It’s another piece that seems so obvious it should defy proof. This is the well-ordering theorem, which says that every set can be well-ordered. That is, so that every non-empty subset has some minimum element. Finally, a mathematical excuse for why we have alphabetical order, even if there’s no clear reason that “j” should come after “i”.

(I said “with reservations” above. This is because whether these are equivalent depends on what, precisely, kind of deductive logic you’re using. If you are not using ordinary propositional logic, and are using a “second-order logic” instead, they differ.)

Ermst Zermelo introduced the Axiom of Choice to set theory so that he could prove this in a way that felt reasonable. I bet you can imagine how you’d go from “every non-empty set has a minimum element” right back to “you can always pick one element of every set”, though. And, maybe if you squint, can see how to get from “there’s always a minimum” to “there has to be a maximum”. I’m speaking casually here because proving it precisely is more work than we need to do.

I mentioned how Zorn did not name his lemma after himself. Mathematicians typically don’t name things for themselves. Nor did he even think of it as a lemma. His name seems to have adhered to the principle in the late 30s. Credit the nonexistent mathematician Bourbaki writing about “le théorème de Zorn”. By 1940 John Tukey, celebrated for the Fast Fourier Transform, wrote of “Zorn’s Lemma”. Tukey’s impression was that this is how people in Princeton spoke of it at the time. He seems to have been the first to put the words “Zorn’s Lemma” in print, though. Zorn isn’t the first to have stated this. Kazimierez Kuratowski, in 1922, described what is clearly Zorn’s Lemma in a different form. Zorn remembered being aware of Kuratowski’s publication but did not remember noticing the property. The Hausdorff Maximal Principle, of Felix Hausdorff, has much the same content. Zorn said he did not know about Hausdorff’s 1927 paper until decades later.

Zorn’s lemma, the Axiom of Choice, the well-ordering theorem, and Hausdorff’s Maximal Principle all date to the early 20th century. So do a handful of other ideas that turn out to be equivalent. This was an era when set theory saw an explosive development of new and powerful ideas. The point of describing this chain is to emphasize that great concepts often don’t have a unique presentation. Part of the development of mathematics is picking through several quite similar expressions of a concept. Which one do we enshrine as an axiom, or at least the canonical presentation of the idea?

We have to choose.

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And with this I at last declare the hard work Little 2021 Mathematics A-to-Z at an end. I plan to follow up, as traditional, with a little essay about what I learned while doing this project. All of the Little 2021 Mathematics A-to-Z essays should be at this link. And then all of the A-to-Z essays from all eight projects should be at this link. Thank you so for your support in these difficult times.

From my Third A-to-Z: Zermelo-Fraenkel Axioms

The close of my End 2016 A-to-Z let me show off one of my favorite modes, that of amateur historian of mathematics who doesn’t check his primary references enough. So far as I know I don’t have any serious errors here, but then, how would I know? … But keep in mind that the full story is more complicated and more ambiguous than presented. (This is true of all histories.) That I could fit some personal history in was also a delight.

I don’t know why Thoralf Skolem’s name does not attach to the Zermelo-Fraenkel Axioms. Mathematical things are named with a shocking degree of arbitrariness. Skolem did well enough for himself.

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gaurish gave me a choice for the Z-term to finish off the End 2016 A To Z. I appreciate it. I’m picking the more abstract thing because I’m not sure that I can explain zero briefly. The foundations of mathematics are a lot easier.

Zermelo-Fraenkel Axioms

I remember the look on my father’s face when I asked if he’d tell me what he knew about sets. He misheard what I was asking about. When we had that straightened out my father admitted that he didn’t know anything particular. I thanked him and went off disappointed. In hindsight, I kind of understand why everyone treated me like that in middle school.

My father’s always quick to dismiss how much mathematics he knows, or could understand. It’s a common habit. But in this case he was probably right. I knew a bit about set theory as a kid because I came to mathematics late in the “New Math” wave. Sets were seen as fundamental to why mathematics worked without being so exotic that kids couldn’t understand them. Perhaps so; both my love and I delighted in what we got of set theory as kids. But if you grew up before that stuff was popular you probably had a vague, intuitive, and imprecise idea of what sets were. Mathematicians had only a vague, intuitive, and imprecise idea of what sets were through to the late 19th century.

And then came what mathematics majors hear of as the Crisis of Foundations. (Or a similar name, like Foundational Crisis. I suspect there are dialect differences here.) It reflected mathematics taking seriously one of its ideals: that everything in it could be deduced from clearly stated axioms and definitions using logically rigorous arguments. As often happens, taking one’s ideals seriously produces great turmoil and strife.

Before about 1900 we could get away with saying that a set was a bunch of things which all satisfied some description. That’s how I would describe it to a new acquaintance if I didn’t want to be treated like I was in middle school. The definition is fine if we don’t look at it too hard. “The set of all roots of this polynomial”. “The set of all rectangles with area 2”. “The set of all animals with four-fingered front paws”. “The set of all houses in Central New Jersey that are yellow”. That’s all fine.

And then if we try to be logically rigorous we get problems. We always did, though. They’re embodied by ancient jokes like the person from Crete who declared that all Cretans always lie; is the statement true? Or the slightly less ancient joke about the barber who shaves only the men who do not shave themselves; does he shave himself? If not jokes these should at least be puzzles faced in fairy-tale quests. Logicians dressed this up some. Bertrand Russell gave us the quite respectable “The set consisting of all sets which are not members of themselves”, and asked us to stare hard into that set. To this we have only one logical response, which is to shout, “Look at that big, distracting thing!” and run away. This satisfies the problem only for a while.

The while ended in — well, that took a while too. But between 1908 and the early 1920s Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem paused from arguing whose name would also be the best indie rock band name long enough to put set theory right. Their structure is known as Zermelo-Fraenkel Set Theory, or ZF. It gives us a reliable base for set theory that avoids any contradictions or catastrophic pitfalls. Or does so far as we have found in a century of work.

It’s built on a set of axioms, of course. Most of them are uncontroversial, things like declaring two sets are equivalent if they have the same elements. Declaring that the union of sets is itself a set. Obvious, sure, but it’s the obvious things that we have to make axioms. Maybe you could start an argument about whether we should just assume there exists some infinitely large set. But if we’re aware sets probably have something to teach us about numbers, and that numbers can get infinitely large, then it seems fair to suppose that there must be some infinitely large set. The axioms that aren’t simple obvious things like that are too useful to do without. They assume stuff like that no set is an element of itself. Or that every set has a “power set”, a new set comprising all the subsets of the original set. Good stuff to know.

There is one axiom that’s controversial. Not controversial the way Euclid’s Parallel Postulate was. That’s the ugly one about lines crossing another line meeting on the same side they make angles smaller than something something or other. That axiom was controversial because it read so weird, so needlessly complicated. (It isn’t; it’s exactly as complicated as it must be. Or for a more instructive view, it’s as simple as it could be and still be useful.) The controversial axiom of Zermelo-Fraenkel Set Theory is known as the Axiom of Choice. It says if we have a collection of mutually disjoint sets, each with at least one thing in them, then it’s possible to pick exactly one item from each of the sets.

It’s impossible to dispute this is what we have axioms for. It’s about something that feels like it should be obvious: we can always pick something from a set. How could this not be true?

If it is true, though, we get some unsavory conclusions. For example, it becomes possible to take a ball the size of an orange and slice it up. We slice using mathematical blades. They’re not halted by something as petty as the desire not to slice atoms down the middle. We can reassemble the pieces. Into two balls. And worse, it doesn’t require we do something like cut the orange into infinitely many pieces. We expect crazy things to happen when we let infinities get involved. No, though, we can do this cut-and-duplicate thing by cutting the orange into five pieces. When you hear that it’s hard to know whether to point to the big, distracting thing and run away. If we dump the Axiom of Choice we don’t have that problem. But can we do anything useful without the ability to make a choice like that?

And we’ve learned that we can. If we want to use the Zermelo-Fraenkel Set Theory with the Axiom of Choice we say we were working in “ZFC”, Zermelo-Fraenkel-with-Choice. We don’t have to. If we don’t want to make any assumption about choices we say we’re working in “ZF”. Which to use depends on what one wants to use.

Either way Zermelo and Fraenkel and Skolem established set theory on the foundation we use to this day. We’re not required to use them, no; there’s a construction called von Neumann-Bernays-Gödel Set Theory that’s supposed to be more elegant. They didn’t mention it in my logic classes that I remember, though.

And still there’s important stuff we would like to know which even ZFC can’t answer. The most famous of these is the continuum hypothesis. Everyone knows — excuse me. That’s wrong. Everyone who would be reading a pop mathematics blog knows there are different-sized infinitely-large sets. And knows that the set of integers is smaller than the set of real numbers. The question is: is there a set bigger than the integers yet smaller than the real numbers? The Continuum Hypothesis says there is not.

Zermelo-Fraenkel Set Theory, even though it’s all about the properties of sets, can’t tell us if the Continuum Hypothesis is true. But that’s all right; it can’t tell us if it’s false, either. Whether the Continuum Hypothesis is true or false stands independent of the rest of the theory. We can assume whichever state is more useful for our work.

Back to the ideals of mathematics. One question that produced the Crisis of Foundations was consistency. How do we know our axioms don’t contain a contradiction? It’s hard to say. Typically a set of axioms we can prove consistent are also a set too boring to do anything useful in. Zermelo-Fraenkel Set Theory, with or without the Axiom of Choice, has a lot of interesting results. Do we know the axioms are consistent?

No, not yet. We know some of the axioms are mutually consistent, at least. And we have some results which, if true, would prove the axioms to be consistent. We don’t know if they’re true. Mathematicians are generally confident that these axioms are consistent. Mostly on the grounds that if there were a problem something would have turned up by now. It’s withstood all the obvious faults. But the universe is vaster than we imagine. We could be wrong.

It’s hard to live up to our ideals. After a generation of valiant struggling we settle into hoping we’re doing good enough. And waiting for some brilliant mind that can get us a bit closer to what we ought to be.

The End 2016 Mathematics A To Z: Zermelo-Fraenkel Axioms

gaurish gave me a choice for the Z-term to finish off the End 2016 A To Z. I appreciate it. I’m picking the more abstract thing because I’m not sure that I can explain zero briefly. The foundations of mathematics are a lot easier.

Zermelo-Fraenkel Axioms

I remember the look on my father’s face when I asked if he’d tell me what he knew about sets. He misheard what I was asking about. When we had that straightened out my father admitted that he didn’t know anything particular. I thanked him and went off disappointed. In hindsight, I kind of understand why everyone treated me like that in middle school.

My father’s always quick to dismiss how much mathematics he knows, or could understand. It’s a common habit. But in this case he was probably right. I knew a bit about set theory as a kid because I came to mathematics late in the “New Math” wave. Sets were seen as fundamental to why mathematics worked without being so exotic that kids couldn’t understand them. Perhaps so; both my love and I delighted in what we got of set theory as kids. But if you grew up before that stuff was popular you probably had a vague, intuitive, and imprecise idea of what sets were. Mathematicians had only a vague, intuitive, and imprecise idea of what sets were through to the late 19th century.

And then came what mathematics majors hear of as the Crisis of Foundations. (Or a similar name, like Foundational Crisis. I suspect there are dialect differences here.) It reflected mathematics taking seriously one of its ideals: that everything in it could be deduced from clearly stated axioms and definitions using logically rigorous arguments. As often happens, taking one’s ideals seriously produces great turmoil and strife.

Before about 1900 we could get away with saying that a set was a bunch of things which all satisfied some description. That’s how I would describe it to a new acquaintance if I didn’t want to be treated like I was in middle school. The definition is fine if we don’t look at it too hard. “The set of all roots of this polynomial”. “The set of all rectangles with area 2”. “The set of all animals with four-fingered front paws”. “The set of all houses in Central New Jersey that are yellow”. That’s all fine.

And then if we try to be logically rigorous we get problems. We always did, though. They’re embodied by ancient jokes like the person from Crete who declared that all Cretans always lie; is the statement true? Or the slightly less ancient joke about the barber who shaves only the men who do not shave themselves; does he shave himself? If not jokes these should at least be puzzles faced in fairy-tale quests. Logicians dressed this up some. Bertrand Russell gave us the quite respectable “The set consisting of all sets which are not members of themselves”, and asked us to stare hard into that set. To this we have only one logical response, which is to shout, “Look at that big, distracting thing!” and run away. This satisfies the problem only for a while.

The while ended in — well, that took a while too. But between 1908 and the early 1920s Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem paused from arguing whose name would also be the best indie rock band name long enough to put set theory right. Their structure is known as Zermelo-Fraenkel Set Theory, or ZF. It gives us a reliable base for set theory that avoids any contradictions or catastrophic pitfalls. Or does so far as we have found in a century of work.

It’s built on a set of axioms, of course. Most of them are uncontroversial, things like declaring two sets are equivalent if they have the same elements. Declaring that the union of sets is itself a set. Obvious, sure, but it’s the obvious things that we have to make axioms. Maybe you could start an argument about whether we should just assume there exists some infinitely large set. But if we’re aware sets probably have something to teach us about numbers, and that numbers can get infinitely large, then it seems fair to suppose that there must be some infinitely large set. The axioms that aren’t simple obvious things like that are too useful to do without. They assume stuff like that no set is an element of itself. Or that every set has a “power set”, a new set comprising all the subsets of the original set. Good stuff to know.

There is one axiom that’s controversial. Not controversial the way Euclid’s Parallel Postulate was. That’s the ugly one about lines crossing another line meeting on the same side they make angles smaller than something something or other. That axiom was controversial because it read so weird, so needlessly complicated. (It isn’t; it’s exactly as complicated as it must be. Or for a more instructive view, it’s as simple as it could be and still be useful.) The controversial axiom of Zermelo-Fraenkel Set Theory is known as the Axiom of Choice. It says if we have a collection of mutually disjoint sets, each with at least one thing in them, then it’s possible to pick exactly one item from each of the sets.

It’s impossible to dispute this is what we have axioms for. It’s about something that feels like it should be obvious: we can always pick something from a set. How could this not be true?

If it is true, though, we get some unsavory conclusions. For example, it becomes possible to take a ball the size of an orange and slice it up. We slice using mathematical blades. They’re not halted by something as petty as the desire not to slice atoms down the middle. We can reassemble the pieces. Into two balls. And worse, it doesn’t require we do something like cut the orange into infinitely many pieces. We expect crazy things to happen when we let infinities get involved. No, though, we can do this cut-and-duplicate thing by cutting the orange into five pieces. When you hear that it’s hard to know whether to point to the big, distracting thing and run away. If we dump the Axiom of Choice we don’t have that problem. But can we do anything useful without the ability to make a choice like that?

And we’ve learned that we can. If we want to use the Zermelo-Fraenkel Set Theory with the Axiom of Choice we say we were working in “ZFC”, Zermelo-Fraenkel-with-Choice. We don’t have to. If we don’t want to make any assumption about choices we say we’re working in “ZF”. Which to use depends on what one wants to use.

Either way Zermelo and Fraenkel and Skolem established set theory on the foundation we use to this day. We’re not required to use them, no; there’s a construction called von Neumann-Bernays-Gödel Set Theory that’s supposed to be more elegant. They didn’t mention it in my logic classes that I remember, though.

And still there’s important stuff we would like to know which even ZFC can’t answer. The most famous of these is the continuum hypothesis. Everyone knows — excuse me. That’s wrong. Everyone who would be reading a pop mathematics blog knows there are different-sized infinitely-large sets. And knows that the set of integers is smaller than the set of real numbers. The question is: is there a set bigger than the integers yet smaller than the real numbers? The Continuum Hypothesis says there is not.

Zermelo-Fraenkel Set Theory, even though it’s all about the properties of sets, can’t tell us if the Continuum Hypothesis is true. But that’s all right; it can’t tell us if it’s false, either. Whether the Continuum Hypothesis is true or false stands independent of the rest of the theory. We can assume whichever state is more useful for our work.

Back to the ideals of mathematics. One question that produced the Crisis of Foundations was consistency. How do we know our axioms don’t contain a contradiction? It’s hard to say. Typically a set of axioms we can prove consistent are also a set too boring to do anything useful in. Zermelo-Fraenkel Set Theory, with or without the Axiom of Choice, has a lot of interesting results. Do we know the axioms are consistent?

No, not yet. We know some of the axioms are mutually consistent, at least. And we have some results which, if true, would prove the axioms to be consistent. We don’t know if they’re true. Mathematicians are generally confident that these axioms are consistent. Mostly on the grounds that if there were a problem something would have turned up by now. It’s withstood all the obvious faults. But the universe is vaster than we imagine. We could be wrong.

It’s hard to live up to our ideals. After a generation of valiant struggling we settle into hoping we’re doing good enough. And waiting for some brilliant mind that can get us a bit closer to what we ought to be.