INFINITY is a concept that is easy to think about, but hard to understand. Who hasn’t looked up at the night sky and wondered whether space goes on forever? Is it an endless expanse, or does it eventually just stop? What does it mean if it doesn’t?
For trained mathematical brains, the infinite is if anything even more bamboozling. Mathematicians have known for well over a century now that infinity isn’t just one thing, it is infinitely many. There is an unending tower of ever greater infinities stretching up all the way to… well, whatever you’d like to call it.
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That isn’t even the worst of it. Although the existence of this tower of infinities is a logical consequence of mathematics as we know it, that same mathematics is powerless to describe it completely. Chip away at the plaster to reveal the structure underneath and you see that crucial load-bearing beams are missing in the lower levels, suggesting that the foundations of mathematics itself are unstable.
Mathematicians have long argued about how best to shore the infinite tower up. Some say we should simply leave well alone and hope for the best. Others have proposed fixes, variously deemed too costly, unlikely to work or not in keeping with the original style. No one has yet made anything like a breakthrough. Except, perhaps, until now. After decades of apparent stalemate, serious progress seems to have been made on the baffling question that lies at the heart of it all: a nearly 150-year-old unproven conjecture known as the continuum hypothesis.
Humans have probably been thinking about the unending for most of our existence. “Infinity is a very natural concept,” says . Jain mathematicians in India in the 4th and 3rd centuries BC believed that infinities come in more than one size, but it wasn’t until the 19th century that mathematician Georg Cantor really started to grasp infinity’s true, slippery nature.
To get a handle on his thought process, imagine drawing a number line. The first numbers you might add to it would be the natural numbers – the counting numbers that go 1, 2, 3 and so on. Although mathematically imprecise, the “and so on” means you could continue the counting process forever. You will never run out of natural numbers; their number is infinite.
That is where the weirdness starts, however. Now think of the even numbers: 2, 4, 6 and so on. Intuitively, you would say there are fewer even numbers than there are natural numbers – half as many, perhaps. But that “and so on” makes plain that there is no end point to them, either. In fact, you can pair up every natural number with an even number and vice versa – (1,2), (2, 4), (3, 6) and so on – so there must be the same “amount” of each. These two infinite sets of numbers have the same size. This size is written ℵ0 (pronounced aleph-null) and these sets are said to be “countably” infinite.
There are lots of countable sets of numbers. The integers, for example, which comprise the natural numbers, zero and all the negative natural numbers, are also countable. Perhaps more surprisingly, so are the fractions. There are clearly a lot of fractions, but Cantor used a cunning trick to prove that you can still perfectly pair them up with the natural numbers: they, too, are countably infinite.
Into the uncountable
That doesn’t exhaust all the numbers you might think to string along your number line, however. Numbers like π and √2 can’t be written as fractions, and when written in decimal continue forever after the decimal point without a repeating pattern. Add in these irrational numbers to the set of all numbers already discussed, and you have what is called the set of real numbers, or the continuum.
Cantor wondered whether, like the fractions, the real numbers were countably infinite too, but it turns out they aren’t. However you pair the real numbers with the countable numbers, there will always be real numbers that you have missed. “Cantor discovered that there are a lot more real numbers than natural numbers,” says . Cantor had discovered that infinity comes in different sizes. The countable infinity was the smallest one, but there was also a “continuum” infinity larger than it.
That wasn’t the half of it. If you take any set of numbers, there is another set that contains every possible combination of the elements in the original one. Cantor discovered that this “power set” of an infinite set was infinitely large, of a larger size. Because you can repeat this process, taking the power set of a power set ad infinitum, he had found a method to produce an infinite ladder of infinities.
This shocker just raised more questions. “Once you discover such a thing, you want to draw a map of the different kinds of infinity,” says Schindler. Cantor knew that the countable sets were the smallest infinity, but was the continuum infinity the next level up? This infinity is known as ℵ1, or aleph-one, and Cantor believed that it and the continuum infinity were one and the same. This assertion became known as the continuum hypothesis, but Cantor was never able to prove it.
Neither was anyone else. This all happened around 1878, and 22 years later, in 1900, mathematician David Hilbert put proving or disproving the continuum hypothesis top of his hit list of 23 problems for mathematicians to solve in the 20th century. Now, 22 years into the 21st century, it remains unsolved.
Granted, in the mid-20th century, an almighty spanner was thrown into the works. When Hilbert wrote his list, mathematicians believed that a logical conjecture, if built up rigorously on solid, agreed axioms of logic, could either be true or false. That changed in 1931 when Kurt Gödel produced his infamous incompleteness theorems. These showed that there was a third option: rather than being true or false, a conjecture could be “undecidable”. Even if you started with the right assumptions, in the form of logical axioms, and worked painstakingly through what they implied, there were some things you could never prove one way or the other.
In 1940, Gödel provided the first evidence that the continuum hypothesis might just be such a beast. He took the first step himself by proving that mathematics as it stood wasn’t strong enough to disprove the hypothesis: it couldn’t say definitively that the continuum wasn’t size ℵ1. In 1963, mathematician Paul Cohen landed what seemed like a final blow, showing that mathematics wasn’t strong enough to prove that they were the same size either. The continuum hypothesis was neither true nor false.
End of story? Not a bit of it. The inability of mathematics to say anything sensible about what, all things considered, seemed to be a relatively simple piece of mathematics was enough to convince many people that mathematics itself was the problem. After all, says Schindler, if infinity exists outside mathematics, then the continuum hypothesis must be decidable one way or another, even if we can’t figure it out yet. Some dispute that premise , but this doesn’t mean that strengthening the logical foundations of mathematics wouldn’t make an answer to the continuum hypothesis possible.
Most mathematicians work far enough away from the foundations that they don’t worry too much about what is going on underground. But dig a little and you find a collection of logical axioms underpinning set theory known as ZFC (for “Zermelo-Fraenkel, plus the axiom of choice”). These contain very basic assumptions about mathematical sets, such as the axiom that two sets are of equal size if they contain the same elements.
“You can essentially construct everything in mathematics using sets, and ZFC is powerful enough that you can do most things that mathematicians care about,” says David AsperÓ at the University of East Anglia, UK. It covers everything from building the real numbers to constructing a working theory of arithmetic. The hunch was that these foundations just weren’t strong enough to bear the full weight of the infinite tower.
One of the first proposals for a new supporting axiom emerged from a technique called forcing, introduced by Cohen in the 1960s. Loosely, Cohen started with a set of the real numbers of size ℵ1 – with no assumption that this equated to the continuum infinity or not – and then used this technique to cram more into it. Forcing was an extremely powerful tool for building interesting new mathematical sets. But it turned out that you could end up with the real numbers being any size of infinity – not just ℵ1 or the next one up, ℵ2, but ℵ42 or anything else. It was simply too powerful a technique to say anything useful about where the continuum infinity lay.
An answer seemed possible by restricting what forcing could do. In 1988, mathematicians Menachem Magidor, Matthew Foreman and Saharon Shelah proposed an axiom called Martin’s maximum, named after set theorist Donald Martin, that could do exactly that. And it gave, finally, an answer to the continuum hypothesis: that it was false. The continuum infinity isn’t ℵ1, but ℵ2; there is a level of infinity between the countable and the continuum. A decade later, in 1999, Hugh Woodin, then at the University of California, Berkeley, suggested another approach, called (*) and pronounced “star”. Like forcing, this also allowed mathematical objects to be built, but in a slightly different way. It, too, said that the continuum hypothesis is false, and that the continuum infinity lies at the third level up, ℵ2.
A new level of infinity?
There is no easy, intuitive example of what might sit between the countable and the continuum, but there are some complicated constructions that we know would do whether or not ℵ1 and the continuum are the same thing. The Hausdorff gap, for example, is a set involving sequences of numbers that has size ℵ1 regardless of the size of the continuum.
Still, with two competing ways to fortify the foundations of mathematics both giving the same answer to the continuum hypothesis, you might think it was game over. But then came another twist that could be loosely characterised as Woodin changing his mind. Both (*) and Martin’s maximum are axioms that help mathematicians to build infinite sets from the bottom up. Woodin instead started to advocate a top-down approach that he called ultimate L. This is a variant of an idea first suggested by Gödel, and is akin to building an observation deck high in the infinite tower to help you see what is going on down below. From above, ultimate L reaches the opposite conclusion to the others – it says that the continuum hypothesis is true.
Top down or bottom up, ultimate L, Martin’s maximum or (*)? Each of these approaches has its advantages and disadvantages, and it is up to their proponents to demonstrate why their idea is the right one. That is where one camp appears to have just edged in front. In 2021, AsperÓ and Schindler proved that if you are in the bottom-up camp, you don’t have to choose between Martin’s maximum and (*). With a few technical tweaks, if you choose Martin’s maximum as your additional axiom, that implies (*) is a legitimate approach too.
The proof took the best part of a decade to complete. AsperÓ and Schindler had originally hit on the idea in 2011, but when they first published their workings, Woodin spotted a mistake. Yet the has certainly rocked the world of mathematics. Not only does it convincingly fuse two approaches, but the very fact that they turn out to be so intertwined is seen by some as evidence that this is the right answer – and that Cantor was ultimately wrong in his conjecture. As Juliette Kennedy at the University of Helsinki, Finland, , absolutely dramatic things that has ever happened in the history of mathematics.”
“It’s remarkable,” says Woodin. “Typically, when two completely different lines of investigation converge on the same thing, that’s taken as evidence for truth.” Despite that, he still isn’t convinced, believing that ultimate L is a neater and better way forward. But his approach is still mired in the details. Since he first proposed the idea in 2010, he has yet to come up with the proof that ultimate L really works as he thinks it should. “I’m optimistic the proof is near,” he says. “The trouble is the community is getting very tired of me saying that. I’ve been optimistic for several years now.”
In some ways, an answer to the continuum hypothesis and the best way to firm up the foundations of mathematics seems closer than ever before. But it could just be an illusion. That is the thing with the infinite: you tend never to reach the end.
A RATIONAL PAIRING
The countable numbers (1, 2, 3 and so on) and the rational numbers (anything that can be written as a fraction) are clearly both infinite sets, but are they the same size? Because every countable number is also a rational number (1, 2, 3… is the same as 1/1, 2/1, 3/1…), the countables must be either smaller than the rationals or equal in size to them.
When he was exploring infinities in the 19th century, Georg Cantor worked out a clever method for proving that the opposite is also true: the rationals are either smaller than or equal in size to the countables. As both of these statements are true, this must mean the two sets are actually the same size.
To understand Cantor’s trick, first imagine putting together a grid consisting of fractions where the numerator (the number at the top of a fraction) is given by the number in the top row and the denominator (the bottom part) by the number in the left column (pictured, below).
This grid includes every possible fraction. Some will be there more than once – for example, 1/2, 2/4, 3/6 and so on are all really the same fraction – but the important thing is that we have caught them all.
Going on forever
This set is as big as the positive rational numbers, and it can be paired with the countables by moving through the grid in a diagonal pattern as shown below and assigning each rational number a countable number in order. This gives pairings (1/1, 1), (2/1, 2), (1/2, 3), (1/3, 4) and so on, and the process can continue forever.
A few technical conditions need to be checked, but essentially this shows that every rational can be nicely paired with a countable – and therefore that the two sets are the same size. This argument accounts only for the positive rationals, but with a few tweaks, it can easily also sweep up the rest.
IS INFINITY REAL?
Mathematics needs infinity. Two of the most practically useful branches of mathematics, trigonometry and calculus, use it, but in such a way that the infinities are conveniently hidden from the real world. You need the infinities involved in both to work out the optimum path of a rocket, but neither the trajectory nor the velocity, or any practically measurable quantity, will ever come close to the infinite. Mathematically, infinity is useful, but does it really exist in the physical world?
In many theories of physics, the appearance of actual infinities is a sign of things going awry. In Einstein’s equations of general relativity, for instance, the “singularity” at the centre of a black holes is an infinitely warped point in space-time; similarly, wind the clock of our expanding, cooling universe back some 13.8 billion years and you reach the point of the big bang where it is supposedly infinitely dense and hot.
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Actually, however, we know that general relativity breaks down when we reach these tiny, extreme scales where quantum theory also comes into play. The infinities seem to be the equivalent of a mathematical shrug: don’t know what’s going here, sorry.
The quantum field theories that underlie the standard model of particle physics provide another example. These were once filled with nonsensical predictions that, for example, the mass and charge of an electron were infinite. Over decades, physicists have steadily managed to remove many of these infinities, creating today’s highly successful model. If the standard model can one day be unified with general relativity to give a complete picture of the universe, perhaps the infinities will disappear completely – or of course, perhaps they won’t.
Accountably uncountable
The real numbers – every number that can be expressed as a decimal – aren’t a countable set. This means that however you pair them up with the countable numbers, you will always have some left over.
The proof goes something like this. First, imagine creating a table with all the countable numbers, 1, 2, 3, 4 and so on up to countable infinity, and pairing them in a second column with any randomly chosen real number with an infinite number of decimal places.
Now create a new number. Find your first digit by adding 1 to the first digit of the first real number in your list, and your second by adding 1 to the second digit of the second real number and so on. Here is an example:
1 3.153778425…
2 0.736785323…
3 7.270286930…
4 42.34603146…
… …
The new number is 4.885…
This process ensures our new real number with infinitely many decimal places has at least one digit different from every number in the (countably infinite) list, so can’t be in it. This reveals a contradiction: not every real number was on the original list paired off with all countable numbers. Therefore, these two sets don’t have the same size: there are more real numbers than countable numbers.
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