The Man Who Almost Broke Math (And Himself...) - Axiom of Choice

- There is a rule in mathematics that is so simple, you would think it obviously must be true, but if you accept it, you find there are now some line segments that have no length. A sphere without adding anything to it can be turned into two identical spheres. A hundred plus years of mathematics has been built on this axiom.

It seems intuitive and it works, but it also creates ridiculous paradoxes. So, is it right? Well, it all starts with the issue of choice.

Try this, choose a number. I can just pluck a random number from my head, like 37 or 42, but that is the human brain at work, not a mathematical process. In math, you can't truly pick things at random because formulas always give the same result, which is why computers don't have true random number generators.

Instead, they usually run an algorithm on your current local time to generate numbers that appear random. So if we can't pick randomly, how do we select anything in math? Well, the only way is to follow a rule of some sort.

So a rule could be always choose the smallest thing. For example, if we're looking at whole positive integers, the smallest is one. For prime numbers it would be two, easy, but what about the real numbers?

That's any number, positive, negative, whole, fraction, even irrational like pi or the square root of two. Now try to choose the smallest one. It's impossible.

The real numbers stretch off to negative infinity. Even if we try to fix our rule by making it super specific, like choose the smallest number after one, we still get stuck. There's 1.

01 and then 1. 0001, then 1. 00000000001 and so on.

So really what number comes after one? If we can't begin to specify the order of the real numbers, next and previous, first and last, we're stuck. The ridiculous part is we know we have infinite options, but despite that, we can't figure out how to just pick one.

The mission to resolve this began with one man in 1870. He took on the task of putting the real numbers in a definitive order, even if it killed him and it nearly did. Georg Cantor was a talented German mathematician who found himself at the center of a firestorm after publishing one of his very first papers at the age of 29.

For centuries, our understanding of infinity was heavily influenced by Galileo's 1638 book. It raised a key question, are there more natural numbers or are there more square numbers? Just looking at them, the square numbers are more spaced out and they only become more sparse the higher you go.

So it would appear there are fewer squares than natural numbers, but Galileo realized he could draw a line matching every natural number with its own square. And since he could make this one-to-one mapping, that he knew that the two sets must be exactly the same size. So there are actually just as many square numbers as there are natural numbers.

From this counterintuitive result, Galileo concluded that terms like more than or less than don't apply to infinity, how we normally use them. It's all just one big concept of foreverness and this view prevailed for centuries. In fact, it's how many people still understand infinity today, but 200 years on, Cantor wasn't satisfied.

In 1874, he wondered what if there were two infinite sets out there that didn't map perfectly to each other? Would they be different infinities? So he set out to compare the natural numbers and the real numbers between zero and one.

Cantor started by assuming he could perfectly map these sets to each other, one-to-one. So he imagined writing down an infinite list with a natural number on one side and a real number between zero and one on the other. Since there is no smallest real number, he would just write them down in any order.

Assuming he now has a complete infinite list, Cantor writes down another real number and to do it, he takes the first digit of the first number and adds one, then the second digit of the second number, and again, he adds one. He keeps doing this all the way down the list. If the digit is an eight or a nine, he subtracts one instead of adding to avoid duplicates, and by the end of this process, he has written down a real number between zero and one, but that number doesn't appear anywhere in his list.

It's different from the first number in the first decimal place, different from the second number in the second decimal place and so on down the line. It has to be different from every number on the list by at least one digit, the digit on the diagonal. That's why this is called Cantor's Diagonalization Proof and it shows there must be more real numbers between zero and one than there are natural numbers extending out to infinity.

Cantor had revealed something remarkable. Infinity doesn't come in just one size. Some infinities like the set of square numbers, integers or rational numbers can be paired perfectly with the natural numbers.

You can literally count them, one, two, three and so on. So Cantor called these countable infinities, but then there are bigger infinities, Cantor called them uncountable. These infinities like the set of all real numbers, the complex numbers, they can't be matched one-to-one with the natural numbers.

Cantor's results rocked the mathematical community. After all, how can something that continues forever be bigger than something else that continues forever? His work was labeled a horror and a grave disease, but Cantor wasn't discouraged.

His success only spurred him to pursue his even grander goal to show that even uncountably infinite sets could be placed in a definitive order. What Cantor called a well-order. For a set to be well ordered, he required two conditions.

First, the set must have a clear starting point. And second, every subset, a collection of items from that set, must also have a clear starting point. So for example, the natural numbers are well ordered, there's a starting point, one, and any subset, say six, seven, eight, also has a clear starting point.

In this case, six, you always know which number comes before and which comes next. But what about the integers? Integers stretch off to infinity in both the positive and negative directions.

Well Cantor realized he could just pick zero as the starting point and from there his ordering went one, negative one, two, negative two, ranking the integers by their absolute value, their distance from zero. It doesn't matter if you put the positives first or the negatives first, as long as you are consistent. Ordering them this way is actually what allows us to map the integers to the natural numbers and see that both sets are the same size, but there are other ways we could well order the integers.

We could start with zero and then have one, two, three, all the way to positive infinity and then negative one, negative two, negative three, all the way to negative infinity. This is not how we're used to counting, but both of these options fit the definition of a well ordering. There's a clear starting point, zero, and all their subsets also have a definitive starting point.

Cantor had successfully well ordered a set that was infinite in both directions, but it was only countably infinite. In his next book, he published his well ordering theorem. It claimed that every set, even the uncountably infinite ones like the real numbers could be well ordered.

The problem was he hadn't actually proven this because he couldn't, every method he tried had failed, but there was one big reason that Cantor was so confident in his theorem. Cantor was a devout Lutheran and he believed God was speaking through him. He said, my theory stands as firm as a rock.

Every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years and above all because I have followed its roots so to speak, to the first infallible cause of all created things.

Belief notwithstanding, the well ordering theorem was a lofty claim to make without any mathematical proof. And so for the second time, the mathematical community attacked and ostracized Cantor. Leading the charge was Leopold Kronecker, the head of mathematics at the University of Berlin.

Kronecker completely dismissed Cantor's work, labeling him a scientific charlatan and a corrupter of the youth. And Kronecker used to be Cantor's teacher. Cantor dreamed of joining him at the University of Berlin, but all his applications were mysteriously denied.

So Cantor took the rejection personally. In 1884, he wrote 52 letters to a friend and every one of them bemoaned Kronecker. Soon Cantor suffered what would be the first of many nervous breakdowns.

He was confined to a sanitarium for recovery. The only way he could prove everyone wrong was by well ordering the real numbers, but he couldn't find a starting point, literally. Once Cantor was released from the sanatorium, he stepped away from math, a broken man.

And over the next 15 years he taught philosophy and rarely dabbled in his old pursuits. Perhaps his greatest challenge came at the 1904 International Congress of mathematicians. There, Julius König, a respected professor from Budapest, announced he had proof that Cantor's well ordering theorem was wrong.

In the audience was not only Cantor but also his wife, two of his daughters and his colleagues. He felt utterly humiliated, but there was also another in attendance. Ernst Zermelo.

Zermelo was a German mathematician who had recently developed a keen interest in Cantor's work and as he listened to König's presentation, something felt off. Within 24 hours, Zermelo had pinpointed the problem. König's proof contained a damning contradiction, and within a month, Zermello published a three-page article titled "Proof That Every Set Can Be Well-Ordered" and it was flawless.

Zermelo's breakthrough came when he discovered something profound in Cantor's work, a mechanism which Cantor uses unconsciously and instinctively everywhere, but formulates explicitly nowhere. See, all along, Cantor had been assuming that he could make an infinite number of choices at once from any set, including uncountable infinite sets like the real numbers, but this was just an assumption. Nowhere in the mathematical rule book was this explicitly permitted and math is built on rules, specifically axioms.

Axioms are simple statements we accept as true without proof. Zermelo realized Cantor's assumption needed to be formalized into something that holds up in a system of proof. A new axiom that said, making all of those choices was possible.

He needed the axiom of choice. - The axiom of choice can be said in the sense that if you have infinitely many sets and each set is not empty, then there is a way to choose one element from each of the sets. - For finite sets, this seems obvious, just go set by set and pick something.

Even for infinite sets, it's easy if there's a clear rule, like always choose the smallest thing, but sometimes there is no natural rule. In those cases when you're choosing from infinitely many sets, including the uncountable ones, you need the axiom of choice. We can't say how we're choosing, but the axiom makes all of these choices all at once.

The axiom doesn't allow you to say which element you've chosen, only that infinitely many choices are possible. So how does this new axiom enable us to well order the real numbers. Zermelo uses the axiom of choice to choose a number from the set of all real numbers.

He places this number, let's call it X1 into a new set, R. The axiom then allows him to choose another number from the subset of all reals minus the one taken out. He calls this number X2 and places it as the next number in his set and he keeps doing this, taking the chosen number and placing it next, X3, X4, X5.

Now it feels like he's choosing these numbers one at a time, but in reality the choices are made from all possible subsets at the same time. As Zermelo indexes each number with the natural numbers, at first it might seem like he'd run into a problem because the natural numbers are only accountably infinite, whereas there are way more reals. So he should eventually run out of labels, but we can count beyond infinity.

We did it earlier when we counted past positive infinity to get to negative one, negative two and so on. So we just need a new set of numbers that extends past the naturals, call the next number omega, then omega plus one, omega plus two, and so on. These omega numbers are not bigger than infinity.

They just come after infinity. They don't tell us how many things are there, but they do tell us their order. So the next number we pull out, we'll label it X omega, then X omega plus one, X omega plus two, and so on.

This will continue until we match the size of the real numbers and our original set is empty. Now every real number is in our new set. There is a first number, X1, and every subset also has a first number.

And just like that, we have successfully well ordered the real numbers. This order looks nothing like our familiar ordering. A billion could come before 0.

2, but with this process, we can prove that a well ordering exists. And more than that, we now have a way to resolve our issue of how to choose mathematically. We can't pick a smallest real number, but now we can pick a first real number, our starting point, and we can do this for any set, meaning all sets can be well ordered no matter the infinity.

So Cantor's well-ordering theorem and Zermelo's axiom of choice are equivalent. Cantor was so relieved. Zermelo had proved the well-ordering theorem and well ordered the real numbers all in under a month.

Zermelo took something mathematicians had unknowingly relied on for decades and turned it into a formal axiom. He showed that understanding math isn't just about numbers, it's about the logic behind them. And lately I've been trying to do a similar thing, but with AI where I'm trying to understand the logic behind how models like ChatGPT work, and if you've also wanted to learn how generative AI actually works, well then you can do that with today's sponsor, Brilliant.

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The axiom of choice may have been a new idea, but its use was anything but. Zermelo scanned dozens of papers from other mathematicians and realized they had also been using the axiom all along, even those who had criticized Cantor's work. - It just goes to show how unintuitive it is that it's even an axiom.

People had been using it for like a decade, unknowingly. - But this almost seems too obvious. Zermelo's proof didn't actually construct a well order.

It just said one must exist, but can something exist if we can't actually build it? His proof also used an uncountable number of steps, was that even allowed? Some mathematicians argued proofs should be finite, others accepted infinity, but only the countable kind and then things got worse.

When mathematicians played around with the axiom of choice, it created disturbing results. One of the first came from Giuseppe Vitali in 1905. Vitali used the axiom of choice to build a set of numbers that shattered our idea of what it means for something to have length.

So what Vitali does is he takes every real number between zero and one and assigns it to one of an infinite number of bins. Let's call these bins groups. So we want each real number to end up in exactly one of our infinite bins.

So how does he do it? Well, let's say we have two numbers, X and Y. If their difference, X minus Y is equal to a rational, that is one integer divided by another integer, well then both X and Y will go into the same bin.

But if we have two other numbers, let's say P and Q and their difference is not a rational, so it's an irrational difference, well then those two numbers will go into separate bins. So let's do some examples. If this is 3/4 minus a half, then we get a quarter and so both 3/4 and a half will go into the same bin.

In fact, you can see that all rational numbers from this span, zero to one, they'll all end up in the same group. Now if you have irrational numbers, well it's not clear whether they will go into the same bin or not because for example, if we have the number root two over two minus say root two over two minus a quarter, well then that does have a rational difference even though each of these numbers is irrational. So these two numbers will go into the same group, but if we have irrational numbers root two over two minus root two over three, well that gives an irrational difference.

So root two over three will have to go into a different bin and it will be joined by all of the numbers it has a rational difference from. And in this way, you can assign each real number to exactly one of these bins. Next, Vitali used the axiom of choice to reach into each group and select exactly one number, which would be a representative of the group.

So we could pull out 3/4 from the rational group, root two over two from this group, root two over three from that group and so on. Though of course because we're using the axiom of choice, you don't actually know what that representative number is, just that you have one. So we could write it down like this.

We have these representatives from each group and together they form the Vitali set. You can visualize this set as a collection of points between zero and one. Next, Vitali makes infinite copies of his set and each one he shifts by a different rational number between negative one and positive one.

So if you think about what that does, it's gonna move each representative number to be at the position of every other number in its group. If we just had the one rational number that we plucked out as a representative from the rational group, now we're gonna shift it by every possible rational number between negative one and positive one. So it's going to end up at every other position occupied by the other members of its group, at least on the span between zero and one.

So if you imagine now merging all of these infinite sets together, there's gonna be no overlap between the points. And second, we are going to have every real number between zero and one because on that span we have every member of every group. So now the question is what is the size of the Vitali set?

Now we know that the union of those sets must be greater than or equal to one because we have every real number between zero and one, but also these points only extend out as far as negative one or positive two. So it must be less than or equal to three, but this is where the problem arises because what number for the size of the Vitali set could you add to itself infinitely many times and end up with a value between one and three? There is no number like that.

I mean if the size of the Vitali set was zero, you add it up infinitely, many times you still get zero. If the size of the Vitali set is a small positive value, then you add it up infinitely many times, you're gonna get infinity, not three. So we have a contradiction and the only way out is if the Vitali set itself is unmeasurable, which seems crazy.

Non-measurable sets like the Vitali set have no consistent definition of size or length or area or even probability. But math is built on the idea that everything can be quantified, whether it's distance, time, or weight, except now there are non-measurable sets and it seems like the axiom of choice is to blame. This was just the start of the uproar caused by the axiom.

In 1924, two mathematicians, Stefan Banach and Alfred Tarski used it to show something that looks like a magic trick. They proved you could take a single solid ball and split it into just five pieces, and then by carefully rotating and moving those pieces, you could reassemble them into two balls each identical to the one we started with, and you could keep going until eventually you have an infinite number of balls, infinity all from one. This sounds absurd, but we can actually see how it works by building a graph.

Imagine you can move in four directions, up, down, left and right. After taking a step, say to the left, you get the same four choices, up, down, left and right, but if you go to the right, you'll end up back where you started. So the only rule we're gonna have is that you can't immediately reverse a move, and we'll keep repeating this at every step, drawing each new line, half the size of the previous one so it all fits on the screen.

If we keep going, we'll end up with this infinitely branching graph. Looking at our graph, we can break it into five sections. There's the middle section where we started, and then there are four other sections that are all identical, just rotated.

So if we take this section to the left and we move everything one step to the right, the top part ends up here, the bottom part here, and the leftmost part here, then we've almost recreated the entire graph. The only thing we're missing is this section, so let's add it back in, but we could have done the same thing in a completely different way by taking the bottom section and moving it one step up. Now the leftmost part ends up here, the rightmost part here and the bottom here.

Again, we're just missing one section, so let's add it back in. But this means I can recreate the entire original graph in two completely different ways. We took one graph, split it into sections, shifted the sections, so the left section went to the right and the down section up and somehow ended up with two identical copies.

This is exactly what Banach and Tarski did, but with a ball, like our graph, we again have four moves. We can rotate the ball up, down, left or right, and again, our only rule is that we can't immediately reverse a move. And to make sure we never come back to the same point, every rotation will be by the same irrational portion of a circle.

We can pick a random starting point, mark it and then start rotating the ball. Each point is colored based on the direction of rotation used to get there. If we do this an infinite number of times, we end up with this collection of points.

This is a countably infinite collection because we could list each rotation and assign it a natural number, but the surface of a ball has uncountably infinite points just like the real number line. So if we want to cover the entire surface, we would need to repeat this process, but where do we start next? Since there are uncountably infinite possible starting points, we can't list them all and we wanna be sure to avoid any points we've already colored.

So the solution is to use the axiom of choice. With it, we can choosing unique starting points, even though we can't say exactly how we are choosing them. Once we've colored every point on the ball, we can split the points into five groups, one for the starting points and four others based on the final rotation used to arrive at those points.

These groups can now be treated just like the sections of our graph. We can take the group of points that end with a left rotation and rotate it to the right. Then we add in the group that ends with a right rotation, and just like that, we've recreated our original ball and we can do it again making an extra move to account for the starting points.

We can equally take the group that ends with a down rotation and rotate it upwards. Then we add in the group that ends with an up rotation and our starting points, and now we've recreated our original ball a second time. Now, this is a bit of an oversimplification, but it gives you the essence of how this is done.

From one ball, we have created two identical balls of the same volume, and nothing stops us from doing this again. Two balls can become four, four become eight, and before you know it, you've got infinite balls. - The axiom of choice is something that's so obviously true and it's consequences are so obviously false that you're like, what the hell is going on?

- This infinite duplication is theoretically possible, but the catch is the groups we split the ball into aren't simple shapes. They're actually non-measurable, just like the Vitali set, although the original ball has a volume and the duplicated balls have a volume, the step in between violates our understanding of size. This is what allows the paradox to happen.

- Of course, those are not physically plausible cuts, but like there's a more meta physical question like, should this even remotely be possible if we could make such cuts? And the answer to almost every human I know is absolutely not. - The truth is no one knew what was going on.

That same year, Tarski tried to push the axiom of choice further proving it is equivalent to the statement that squaring any infinite set would not increase its size. When Tarski first submitted this work to a journal in Paris, the editor Lebesgue responded dismissively, nobody's interested in the equivalence between two false statements. Not to be deterred, Tarski sent it to a different editor at the same journal, Freche, his response, nobody's interested in the equivalence of two obviously true statements.

Tarski never submitted a paper there again. So math was in crisis for over 30 years with people not knowing what to believe. - The question is, wait a second, is this really an axiom or is this something that you can prove?

- In 1938, we finally started getting some answers. The Austrian mathematician, Kurt Godel, proved there is a world all the other already accepted axioms of set theory hold true, and so does the axiom of choice. Then in 1963, Paul Cohen proved there's also a world where all the axioms of set theory hold true except for the axiom of choice.

This is kind of like the parallel postulate in geometry. You can think of geometry as a game. The first four postulates or axioms are like the minimum rules required to play that game, and then the fifth axiom selects the universe that you wanna play in.

If you choose that the fifth axiom doesn't hold, so there are no parallel lines, then you're playing in spherical geometry. If you choose one parallel line, you're playing in flat geometry, and if you choose more than one parallel line, then you're playing in hyperbolic geometry. All of these geometries are valid.

It just depends on the math you want to do, and it's the same for the axiom of choice. The axiom of choice can neither be proven nor disproven from the other axioms. So as long as the other axioms are consistent, adding choice won't lead to any contradictions.

Paul Cohen was awarded the Fields Medal three years later for his groundbreaking result, as well as his other work in set theory, and after Godel and Cohen's work, most of the debates about the axiom of choice died down. - In the end, what the hell is going on is that it's up to you whether you want to choose for the axiom of choice to be a part of your system or not, and face the consequences of either having it or not having it. - Despite the counterintuitive results created by the axiom of choice, like non-measurable sets and infinite duplication, it is incredibly useful, choice allows mathematicians to replace lengthy explicit proofs with more concise arguments.

By proving statements in the finite case, many proofs can be extended to any infinite case in just one line. This reduces proofs that could have been 20 pages to just half a page. And the axiom of choice doesn't just make math easier.

It is essential to some proofs. There are many theorems where the general case can't be proven without using choice somewhere. Now, some mathematicians still prefer proofs without choice, even if it's harder, the proof has to be spelled out step by step to generalize to infinite cases, and this provides additional information.

Some mathematicians spend their time studying universes without the axiom of choice to understand what happens when we remove it. But today, the axiom of choice is almost universally accepted. For the past 80 plus years, generations of mathematicians have been taught with choice as a given to the point where many who use the axiom of choice might not even realize when they're doing it.

- If you don't include the axiom of choice, then you're kind of working with both hands tied behind your back. It's very hard to make any progress on modern math. - So the question was never really is the axiom of choice right?

But rather is the axiom of choice right for what you want to do?