From New Scientist.
What the significance of this is, or if it is significant at all, I have no idea.
It may sound strange, but mathematicians have created an entire ladder of infinities, each larger than the next. Now a new kind of infinity threatens to upset that order, and perhaps redefine the structure of the mathematical universe
By Alex Wilkins
6 December 2024
Infinity is more complex than you might think
A new kind of infinity appears to break the rules of how extremely large numbers behave, and could redraw the way the mathematical universe is ordered.
It may come as a surprise, but mathematicians have long known that there is more than one kind of infinity. In 1878, Georg Cantor first showed that the infinite set of real numbers, which includes negatives and decimals, is actually larger than the infinite set of natural, or whole, numbers. Proving this involves a careful comparison between the two sets, rather than attempting the impossible task of counting them all.
Incredible maths proof is so complex that almost no one can explain it
Following Cantor’s ideas, mathematicians soon realised that they could construct ever-larger infinite sets, creating a hierarchical ladder of sets that is itself infinite. “People have been coming up with larger and larger notions of infinity,” says Juan Aguilera at the Vienna University of Technology in Austria. “You can look at all the previous ones that people have come up with and you can fit them in a hierarchy.”
But now, Aguilera and his colleagues have proposed two new sizes of infinity, called exacting and ultra-exacting cardinals, that don’t obey the rules. “They don’t quite fit in this linear hierarchy,” says Aguilera. “They interact very, very strangely with other notions of infinity.”
Aguilera and his team defined these sets by making them so large that they must both contain mathematically exact copies of their entire structure – a bit like a house that contains multiple, full-scale models of itself – and also contain small versions of larger sets, like adding models of the surrounding neighbourhood or city to the house. Ultra-exacting cardinals have one further rule, which says that these sets must also contain the mathematical rules of how to make them – as if the nested house was also wallpapered with blueprints of itself.
These unusual properties are what cause these sets to fall off the infinity ladder, as they play havoc with some of the deepest rules of mathematics. Around the beginning of the 20th century, mathematicians began trying to find a rigorous foundation for their entire field, defining a basic set of rules, or axioms, that could be used to build and prove any other theory. The most widely accepted form of this foundation today, known as Zermelo-Fraenkel set theory, included a controversial rule called the axiom of choice that is at the root of the problem.
This axiom says that you can always build a new set of numbers by picking out numbers from other sets, but doesn’t tell you explicitly how to do it. Some mathematicians felt that this didn’t work when considering infinite sets, because it would require asserting the existence of mathematical objects without offering a way to prove them. However, over time, they came to accept the rule, and it is now used as a key measuring stick in organising the infinite ladder, carving it into three broad regions.
At the bottom of the ladder, the first and smallest region contains infinities that adhere to the set theory axioms – these are the real and natural number infinities studied by Cantor. At the top, the third and largest region has numbers so huge that all set theory axioms break down, including the axiom of choice. It is a region of “chaos”, says Aguilera.
Many infinities fit somewhere between these, in a second region. Exacting and ultra-exacting cardinals initially appeared to as well – but when the team actually tried to pin them down, they found that it wasn’t possible. “It’s not quite clear if they are at the top of this middle region, where the axioms are still compatible with all the other axioms of set theory, or whether they are forming a fourth region that is kind of to the side of the chaotic region, but on top of the previous ones,” says Aguilera.
Resolving the question of where these new sets fit is more than just academic tidiness, however – the entire mathematical universe could be at stake. This is because of a key unsolved problem called the Hereditarily Ordinal Definable (HOD) conjecture, which proposes that as you get to the very largest infinities, the axiom of choice starts to make sense instead of leading to contradictions. This would suggest that mathematics becomes more ordered at the largest scales, says Gabriel Goldberg at the University of California, Berkeley, who wasn’t involved in the work. “The large cardinals are sort of justifying the axiom of choice for you.”
If these exacting cardinals are accepted by the wider mathematical community – which isn’t necessarily certain, given their existence straddles the borders of maths and philosophy – then “it strongly suggests that maybe the HOD conjecture is false, so maybe chaos rules”, says team member Philipp Lücke at the University of Hamburg, Germany.
Or perhaps not. “It’s a little too early to say that that’s the picture,” says Goldberg, who thinks order may yet win out. “It seems like actually there’s a lot of structure that emerges.”