Mathematicians Just Discovered Two New Types of Infinity And They Might Rewrite the Rules of Math

How two strange new infinities disrupt the fragile order of the mathematical universe

Introduction: A Puzzle That Never Stops Growing

Infinity has a strange way of slipping out of your hands just when you think you’ve got a grip on it. You might think of it as one big, boundless thing a sort of ultimate “largest number” floating out there in the abstract. But mathematicians learned long ago that infinity isn’t singular. It comes in layers, sizes, and sometimes downright bizarre shades.

If that already feels a bit mind bending, brace yourself: researchers from the Vienna University of Technology and the University of Barcelona have just uncovered two entirely new kinds of infinity. And these aren’t polite, well behaved infinities that fit neatly into the comfortable ladder mathematicians have been climbing for generations. These ones push back. They bend the rules. They suggest that the structure of mathematics may be far less orderly and far more wild than we hoped.

They’re called exacting cardinals and ultraexacting cardinals, and while their names may sound like something out of a sci fi glossary, their implications strike at the foundations of logic itself. If these results hold, our entire model of the mathematical universe may need to be redrawn.

Before we get to the chaos, though, it’s worth taking a short walk through the landscape of infinities we thought we understood.

1. How Many Infinities Do We Actually Have?

A ladder that keeps growing new rungs

Here’s a simple example that’s often used in classrooms:
You can count the natural numbers 1, 2, 3, 4, 5 forever. No surprise there. That’s one type of infinity.

But between 0 and 1, there are infinitely many decimals: 0.1, 0.01, 0.031, 0.00000049… and on and on. And it turns out that this second infinity is bigger. Not just metaphorically. Literally larger, in a rigorously defined way.

This shocked mathematicians in the 19th century. It meant infinity had layers, like geological strata each new one larger than the last. Eventually, these layers became a grand hierarchy of “large cardinals,” each described by special axioms that say, essentially, “Let’s assume this incredibly huge infinity exists, and see what happens next.”

Think of:

ℵ₀ (aleph null): the infinity of counting numbers.
Measurable cardinals: much bigger.
Supercompact cardinals: bigger still.
Huge cardinals: the name speaks for itself.

These aren’t casual numbers you can’t prove they exist using the standard mathematics taught in every university (the Zermelo–Fraenkel axioms plus the Axiom of Choice, collectively called ZFC). You need to extend the system with extra assumptions. To quote Joan Bagaria, one of the mathematicians involved, these cardinals are “numbers so large that one cannot prove they exist using the standard axioms of mathematics.”

In other words, mathematically speaking, we take a leap of faith and assume they exist. And shockingly, doing so opens doors to new regions of mathematics that are otherwise unreachable.

The hierarchy has always been tall, but relatively tidy a kind of cosmic library where each floor sits carefully above the last. At least… that was the hope.

2. Enter the Troublemakers: Exacting and Ultraexacting Cardinals

New infinities that don’t follow the script

The newly identified infinities exacting cardinals and ultraexacting cardinals don’t slot nicely into the existing structure. They’re powerful enough to alter the landscape in ways mathematicians didn’t anticipate.

Exacting cardinals

These are already stronger than many of the previously known giants. They seem to operate with rules that subtly shift how the mathematical universe behaves. Their strength isn't just size it's the way they influence the relationships between different infinite sets.

Ultraexacting cardinals

If exacting cardinals are strong, ultraexacting cardinals are… well, intense. They include all the strange features of exacting cardinals, but then layer on additional constraints and abilities. One mathematician described them as exacting cardinals “with superpowers,” which as informal as that sounds isn’t a bad way to picture them.

Imagine a chess piece that suddenly starts moving in patterns no one agreed on. Not just a queen that moves diagonally and straight, but one that occasionally teleports or nudges pieces on other boards. That’s the kind of disruption these cardinals introduce.

Juan Aguilera, another author of the research, put it more cautiously:
“Ultraexacting cardinals seem to be different. They interact very strangely with previous notions of infinity.”

And mathematicians don’t use the word “strangely” lightly.

3. The Hidden Battle for Order: The HOD Conjecture

A decades long hope starts to crack

One long standing vision in set theory is the HOD Conjecture, which, if true, would impose a kind of cosmic order on infinity.

HOD stands for Hereditary Ordinal Definability, a fancy name for a simple idea:
Maybe every set even gigantic, unimaginable ones can be defined in a systematic way using ordinal numbers (which are basically a way to count beyond the finite).

In plain terms, the HOD Conjecture imagines the universe of mathematics as a massive library. Some books may be hard to find, but they all exist somewhere on a shelf, labeled and catalogued. No chaos. No surprises that break the system.

But if exacting and ultraexacting cardinals exist, the library analogy collapses.

These new cardinals suggest that the mathematical universe (V) is “far from HOD” meaning the full universe is not a well organized library at all, but more like an infinite wilderness stretching beyond any map we can draw. The tiny piece we can catalog (the HOD part) may be just a sliver, like a well lit campsite surrounded by miles of uncharted forest.

When Aguilera says, “It could mean that the structure of infinity is more intricate than we thought,” what he’s really saying is:

We may have seriously underestimated how wild infinity actually is.

Mathematicians aren’t strangers to uncertainty, but this level of unpredictability hits at the foundation of everything they work with.

4. What Does Any of This Mean for the Real World?

Infinity never stays in its lane

You might wonder why someone outside a research department should care about cardinals with exotic names. After all, most people won’t be calculating with ultraexacting cardinals while doing their taxes or writing a phone app.

But the effects of this discovery echo far beyond pure mathematics.

Cryptography

Modern encryption relies heavily on assumptions about infinities, especially in number theory. When mathematicians tighten or loosen the structure of infinity, they alter the ground on which cryptographic systems stand.

Artificial intelligence

Machine learning theory, especially in the study of neural network capacity and limits, leans on set theoretic results involving infinite structures.

Cosmology and physics

Infinity appears at the edges of physics in black hole singularities, space time models, and even the mathematics used to describe quantum fields.

Even if these new cardinals never appear directly in a physics equation, they test whether the logic that supports all mathematical reasoning is stable.

It’s a bit like engineers discovering a new type of stress that can crack steel. You might never see that stress in your day to day life, but knowing it exists changes how you design skyscrapers.

Philosophy

There’s also a quieter, more introspective consequence. Every time mathematicians expose a new region of infinity, they’re asking us to reconsider what the universe and our reasoning about it can handle.

If infinity keeps revealing new, unruly structures, maybe we’ve been overly optimistic in thinking we can corral it into clean definitions.

5. A More Tangible Analogy (Because Infinity Is Hard to Picture)

Imagine sorting socks.

You start with a small drawer. Easy. One pair, two pairs, five pairs no big deal.

Then someone brings in a crate of socks. You sort them too. Takes longer, but fine.

Now imagine someone delivers a shipping container full of socks all colors, all sizes, massively jumbled. Then another container. Then a hundred containers. Each time, you invent new rules to keep the sorting manageable. Color first, then size. Or maybe you sort by material, then color.

Eventually, you develop a whole sorting theory.

Large cardinals are like those increasingly massive piles. They require new rules. New hierarchies. New assumptions.

But exacting and ultraexacting cardinals feel like discovering that some socks spontaneously appear or disappear, some duplicate themselves, and some refuse to be categorized by any rule you’ve come up with so far.

They force you to go back to your entire sorting system and say:
“Hold on. Maybe organizing all this was never possible in the neat way I imagined.”

6. Why This Discovery Matters for the Future of Mathematics

A structure we thought was solid turns out to have hidden instability

Mathematicians tend to treat large cardinal axioms like hypothetical but extremely useful tools. They’re not proven, but they unlock breakthroughs. They shape what can be proved and what remains forever undecidable.

The discovery of exacting and ultraexacting cardinals has two massive implications:

Implication 1: The hierarchy of infinities might be far more complicated than we thought.

The tidy ladder might be an illusion. Instead of straight rungs, we might be dealing with webs, branches, or structures we don’t have metaphors for yet.

Implication 2: The HOD Conjecture a major attempt to bring order to infinity may be false.

If that’s the case, then the mathematical universe is fundamentally disordered at its upper reaches.

This doesn’t mean mathematics breaks apart. But it does mean:

Our assumptions about “what can be known” might need revision.
Some questions previously thought solvable may never have definitive answers.
The boundary between provable and unprovable could shift in surprising ways.

That can be unsettling but also electrifying. Mathematicians thrive on uncertainty because that’s where the discoveries hide.

7. A Final Reflection: Infinity Is Bigger Than Our Imagination

Infinity has always been a kind of mirror. When we explore it, we’re really exploring the limits of human reasoning. Every time we stretch those limits, infinity responds by stretching too sometimes politely, sometimes with a rude shock.

The discovery of exacting and ultraexacting cardinals reminds us that the universe of mathematics is not a completed cathedral but a construction site, full of scaffolding, strange additions, and unexpected staircases.

And maybe that’s the best part.

Because if infinity were simple, we’d run out of surprises. Instead, it keeps opening new doors, challenging our confidence, and reminding us that the world even the abstract world of pure math is much stranger than we give it credit for.

And honestly? That’s a beautiful thing.

Open Your Mind !!!

Source: ZME Science

Open Your Mind