Posted on 12/20/2024 1:14:13 PM PST by Red Badger
It’s an idea straight out of the schoolyard: that you might one day accidentally count so high that you break the laws of math. A new preprint (that has not yet been peer-reviewed) seems to have done just that, however – and it could have huge ramifications for how we ought to understand infinity.
It’s fitting that such a baffling result would have come from set theory: it’s an area with a reputation for being abstract and often counter-intuitive; it has its own esoteric alphabet and language; and it’s famous for results that seem either too basic to have even bothered proving (see: 1 + 1 = 2) or so patently absurd that you figure they must have made a mistake somewhere along the way (see: 1 + 1 = 1).
The trouble is, we really can’t do without it. At the heart of set theory is the hunt for a way to tame math once and for all – to figure out what we can prove, and what we can only assume. To do that, mathematicians sometimes need to look for the edge cases: the bits of math where things are so huge, weird, or fundamental, that all the rules we take for granted start breaking down.
Unfortunately, sometimes they succeed.
The infinity ladder
“Infinity” is an unintuitive and at times baffling concept. It’s not enough to say, for example, that “infinity is the number of natural numbers there are” – because if that’s the case, how many even numbers are there? How many fractions? How many if you include irrational numbers as well?
The answer to all of the above is, unsurprisingly, also “infinity” – but there are at least two different sizes of it on show there. Mathematicians can prove, it turns out, that the sets of even numbers, whole numbers, and fractions are all the same size – an infinite number known as ℵ0 (pronounced “aleph-null”). The set of reals, on the other hand – that is, all rational and irrational numbers – is much bigger.
Exactly how much bigger, though, is a question that is already pushing at the limits of what we know and can prove. We’re into the world of “large cardinals” now: numbers “so large that one cannot prove they exist using the standard axioms of mathematics,” explained Joan Bagaria, one of the three coauthors of the new paper and a mathematician, logician, and set theorist at ICREA and the University of Barcelona in Spain.
It’s a fact that’s both a limitation and a strength. Existing outside of ZFC – the initialism stands for “Zermelo-Fraenkel plus Axiom of Choice”, two minimal sets of rules that form the foundation of just about all math in the world – means the very existence of large cardinals “has to be postulated as new axioms,” Bagaria told IFLScience. In other words, it cannot be proved – only supposed true the same way we take it for granted that x = x.
But this position outside of normal rules also makes large cardinals a valuable tool for dealing with the more hinky areas of math. They “give us a deeper understanding of the structure and the nature of […] the mathematical universe,” Bagaria said. “They allow us to prove many new theorems, and therefore to decide many mathematical questions that are undecidable using only the ZFC axioms.”
For example: even in this intangible world of unprovable infinities, some kind of order can be felt out – at least, to an extent. There are the inaccessible cardinals, Bagaria explains – the smallest of the large cardinals (the word “small” is somewhat load-bearing here, as you can imagine). Above those, there are the measurable cardinals; eventually, we reach compact, supercompact, and perhaps modestly named “huge” cardinals.
But go much further, and even these esoteric classifications start to break down. “Eventually, the large cardinals become so strong that they become in contradiction with the Axiom of Choice,” Bagaria says. “This is the world of Large Cardinals Beyond Choice, which can hardly be accepted as true since the Axiom of Choice is needed in most areas of mathematics.”
Welcome to the jungle
It’s into this ever-weirder hierarchy that the new numbers have been thrown. Labeled by their discoverers as “exacting” and “ultraexacting” cardinals, they “live in the uppermost region of the hierarchy of large cardinals,” Bagaria explains; “they are compatible with the Axiom of Choice, and they have very natural formulations, so they can be readily accepted.”
So far, so reasonable – but the new cardinals nevertheless spell trouble for some mathematicians’ pictures of infinity. The problem lies in a property called Hereditary Ordinal Definability, or “HOD” – the idea that a set, even an infinitely large one, can be understood by sort of “counting up to” it.
It’s a handy tool for infinity-wrangling – and some mathematicians had hoped that it was more generally applicable. If all, or at least basically all, sets – including these infinitely large ones – could be defined in this way, it would mean that the chaos of the large cardinals was a blip rather than an unraveling; that the Axiom of Choice would become justified again even at the top of the hierarchy.
That’s why, for the last decade or so, set theorists have been debating the so-called “HOD conjecture”. It’s essentially a formalization of that wish: “The HOD conjecture tells us that the mathematical universe is orderly and ‘close’ to the universe of definable mathematical objects,” coauthor of the new paper Juan Aguilera, a mathematical logician at the Vienna University of Technology in Austria, explained to IFLScience.
Solving the conjecture one way or the other would be tricky, to say the least. Thanks to the weirdness of large cardinals, it would theoretically require less effort to prove true than false – but definitive answers in either direction were elusive. The evidence, however, was less so: “Many people thought, until now, that the HOD Conjecture was probably true,” Bagaria said, “with evidence coming from the work on canonical inner models for large cardinals carried out over the last decades.”
In “all those models,” Bagaria explains, the HOD Conjecture seemed to hold. So what’s changed?
An exacting question
In an area already defined by counter-intuitiveness and intangibility, the exacting and ultraexacting cardinals introduced in the new preprint still manage to be notably weird.
“Typically, large notions of infinity ‘order themselves’ in the sense that even if they are discovered in different contexts, one is always clearly bigger or smaller than the others,” Aguilera told us. “Ultraexacting cardinals seem to be different.”
It’s not just that they don’t quite fit themselves – they make otherwise well-behaved cardinals act out as well, he explains. “They interact very strangely with previous notions of infinity,” explained Aguilera. “They amplify other infinities: cardinals that are considered ‘mildly large’ behave as much larger infinities in the presence of ultraexacting cardinals.”
It’s an unexpected tangle in what we thought was a fairly well-laid-out hierarchy – and it has profound implications for how we might envision infinity going forward. “In my opinion it shows that there is some revision to be made,” Aguilera said. “Maybe the structure of infinity is more intricate than we thought, and this warrants deeper and more careful exploration.”
Still, it’s bad news for the HOD conjecture. If exacting and ultraexacting cardinals are accepted, it’s just a short jump to then show that the HOD conjecture is false – that ultimately, chaos, not order, wins out.
It’s not a killing blow – remember, the existence of these large cardinals has to be introduced via axiom rather than proved rigorously, so the results “do not directly disprove the HOD Conjecture,” Bagaria cautioned. “But [they] provide very strong evidence against it, contrary to the prevailing intuitions.”
But here’s the question: after so many years of hope that the HOD conjecture would eventually prevail, is it really such a bad thing that it may not? What Bagaria and colleagues have found may temporarily disorient, but it also opens up a rich new world of large cardinals, with behaviors and implications that are ripe for new research.
“The three of us and other colleagues will continue studying exacting and ultraexacting cardinals,” Aguilera told IFLScience. “It could be that these are the first instances of a new kind of infinity.”
“This is something to be clarified,” he said. “Maybe this is just the beginning.”
The preprint is available on arXiv.
You say “infinities” and I say “vanities.”
My cardinals are smaller and like black sunflower seeds.
You can’t get closer than the Planck distance, right? Convince me I’m wrong. Don’t actually attempt a closer approach or you might inadvertantly create a black hole.
LOL! You are absolutely right!
I was a math major at a university that prided itself on its Mathematics discipline.
Mathematics is a tool. Not always perfect, there is always error, but it will get the job done if you use it within it’s limits just as any tool.
Cardinal Gate at The Ohio State Fair.
Go Vols!
Are any of the cardinals named Biggles or Fang?
Nobody expects the large cardinals.
Memorize, learn to keep a straight face, tell it when you want to tease out who in a room has the math mind.
Those people (if any) will say “duh.” No telling what the others will do; but any who laugh (nervously) are probably afraid of you.
Words, words and nothing but words…
Diet of Worms?
Those who wrote this have too much time on their hands.
👍
It reminds me of the attempting to reach the speed of light. Theoretically it’s impossible, isn’t it? If we could generate enough energy, we could go to 99.9% of the speed of, but never quite reach it. Or so they say.
“There are infinite integers. There are also infinite rational numbers. But there are more rational numbers than integers. So the number of rational numbers is a greater infinity than the number of integers.”
It’s all semantics. Infinity is greater than infinity?
Ri....ght!
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