Oops! Newly Discovered Infinities Might Have Broken The Mathematical Universe..."This changes completely the landscape of large cardinals."

IFL Science ^ | December 19, 2024 | Dr. Katie Spalding

[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 A new preprint (that has not yet been peer-reviewed) seems to have done just that, however – and it could have huge ramifications for how we ought to understand infinity.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.

[Red Badger]

Ours are gray...............

[lee martell]

Oh. Those Cardinals! Not the winter bird. Not the Baseball team. Not the opposite of Venial Sin.

[Chickensoup]

I think it is a car model..

[algore]

I can define infinity as “something the debt ceiling is quickly approaching”

[Repealthe17thAmendment]

I got through about 5 paragraphs before my head started to hurt.

[ImJustAnotherOkie]

Infinity cars are just Nissan’s after they go through a worm hole.

[philman_36]

Is one of the researchers named Jon Snow? He knows nothing.
/humor

[alancarp]

A mathematician walked into a bar and he told the bartender “I want a shot of whiskey for myself.”

Bartender says “Coming right up... anything else?”

The mathematician responded “Yes. I’ve got 50 friends with me. The first one wants half a shot. The second wants half of that - a quarter shot. The third wants half of _that_... and every succeeding friend wants half the amount of the previous one.”

The bartender immediately puts two shot glasses on the bar and fills both, saying “Here ya go — your last friend can finish off whatever’s left over.”

The mathematician paid for the drinks and thanked the bartender, who responded by saying “I can appreciate a man who knows his limits.”

[pfflier]

Boy did I get into trouble arguing that with my 10th grade geometry teacher.

If you add even infinity and odd infinity you are filling in the void spaces between numbers.

But with that stupid number line thingy If you count as far as you can in a positive direction you count to infinity. If you count as far as you can with negative numbers you can count to infinity.

If you add them both together you get either double infinity or zero which is the opposite of infinity.

That got me sent to the office.

[pfflier]

Boy did I get into trouble arguing that with my 10th grade geometry teacher.

If you add even infinity and odd infinity you are filling in the void spaces between numbers.

But with that stupid number line thingy If you count as far as you can in a positive direction you count to infinity. If you count as far as you can with negative numbers you can count to infinity.

If you add them both together you get either double infinity or zero which is the opposite of infinity.

That got me sent to the office.

[Bookshelf]

I say it’s spinach, and I say to Hell with it.

[Rockingham]

My head hurts from trying to make sense of the article.

[cgbg]

How many perverts are in that picture?

Infinity.

Lol.

[Ezekiel]

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.

This is so not complicated. I learned this particular secret of the universe from a space toy:

"To infinity... and beyond!" ~ Buzz Lightyear

"אל האינסוף... ומ*עבר* לו!"

As long as Hebrew has entered the room, "beyond" is based on עבר, the very root of "Hebrew".

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.

>>>

Hod (Hebrew הוֹד Hōḏ, lit. 'majesty, splendour, glory')[1][2] is the eighth sephira of the Kabbalistic Tree of Life. It is positioned on the left side of the tree beneath Gevurah (severity) and directly opposite Netzach (eternity).

Hod is associated with qualities such as submission, humility, and intellectual rigor. It represents the capacity to comprehend and articulate divine truths, balancing the emotive and instinctual energies of Netzach. This balance is crucial for maintaining the flow of divine energy through the sefirot and manifesting it in the material world. Hod is also linked to the planet Mercury and the archangel Michael in Western esoteric traditions.

Hod

<<<

Hod... is the eighth sephira:

∞

"Help, I've fallen and I can't get up!"

large cardinals

I can understand why this article is fixated. Tis the Season!

[BipolarBob]

https://www.youtube.com/watch?v=-PKpxebwbe0

[NorthMountain]

Your math teacher should have known that all infinities are not equal.

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. But the number of rational numbers is an integer ...

And I think that line of reasoning starts to get into the crazy territory of “large cardinals” that the OP is all about.

[17th Miss Regt]

A countable infinity. The same number as the number of all rational numbers!

[17th Miss Regt]

But there are more rational numbers than integers.

There are exactly as many rational numbers as there are integers. Look up Cantor's diagonal argument.

[suasponte137]

“ See? I’ve got it all figured out.”

You almost got it figured out, you forgot to carry the two.

[jonrick46]

These numerical “cardinals” are like trying to figure out the number of angels dancing on the head of a pin.