Is there an infinity of infinities? The question sounds almost absurd, like a riddle designed to twist your brain into knots. But for mathematicians, it’s a serious — and endlessly fascinating — puzzle. What’s certain is that infinity doesn’t come in just one flavor.
For centuries, mathematicians have categorized infinities into a kind of ladder. The infinite set of natural numbers — 1, 2, 3, and so on — sits on one rung. On a higher rung, the infinite set of real numbers, which includes decimals and negatives, dwarfs it. And from there, infinities cascade upward, forming an endless hierarchy.
Recently, researchers from Vienna University of Technology and the University of Barcelona uncovered two new layers of this vastness, and they don’t quite play by the usual rules.
These new types of infinities are called exacting and ultra-exacting cardinals. Unlike their predecessors, these cardinals refuse to slot neatly into the established hierarchy of infinities. Their discovery forces mathematicians to reconsider what infinity really means — and whether chaos might lurk at its core.
How Many Infinities Are There?
Mathematicians have long categorized infinities into a hierarchy where some infinities are larger than others. For example, the infinity of counting numbers (1, 2, 3, …) is smaller than the infinity of real numbers, which includes an infinity of decimals between 0 and 1 (and beyond).
Mathematicians use “large cardinal axioms” to describe these layers, defining specific types of infinite numbers with unique and powerful properties. At the base of the ladder is the infinity of natural numbers, ℵ₀ (aleph-null). Climbing higher reveals infinities of increasing size and complexity: measurable cardinals, supercompact cardinals, and even so-called “huge” cardinals.
These axioms followed a predictable, linear progression. Each new “rung” of the ladder built on the one before it, creating a stable structure. But as these infinities grow, they stretch the foundational rules of mathematics to their limits. Large cardinals, for instance, exist outside ZFC — the Zermelo-Fraenkel set theory with the Axiom of Choice, the framework underpinning nearly all modern mathematics.
“Numbers ‘so large that one cannot prove they exist using the standard axioms of mathematics,’” is how Joan Bagaria, a mathematician at ICREA and the University of Barcelona, described these entities. Their existence must be assumed through new axioms. Yet their usefulness cannot be overstated — they allow mathematicians to explore regions of mathematics that would otherwise remain undecidable.
Exacting and ultraexacting cardinals are the newest additions to this pantheon. According to Bagaria, these cardinals “live in the uppermost region of the hierarchy of large cardinals” and appear compatible with the Axiom of Choice.
Exacting cardinals are stronger (or “bigger”) in their properties than many previously known large cardinals, meaning they can interact with the mathematical universe in new and unexpected ways. Ultraexacting cardinals are an even more powerful and restrictive version of exacting cardinals. Think of them as exacting cardinals with additional “superpowers” that make them interact with infinity in a way that amplifies their effect on the mathematical universe.
Order, Chaos, and the HOD Conjecture
For decades, mathematicians have debated whether infinity could ever be tamed. One guiding hope has been the HOD Conjecture, which suggests that even the most unruly infinities could fit within a broader order.
HOD, or Hereditary Ordinal Definability, proposes that infinitely large sets can be defined by “counting up to” them. If true, it would bring order to the mathematical universe, aligning the Axiom of Choice with the largest infinities.
But these new cardinals muddy the waters. Exacting and ultraexacting cardinals seem to break traditional patterns. “Typically, large notions of infinity ‘order themselves,’” explained Juan Aguilera, a co-author of the paper and mathematician at the Vienna University of Technology. “Ultraexacting cardinals seem to be different. They interact very strangely with previous notions of infinity.”
The implications are profound. If these new cardinals are accepted, they could provide strong evidence against the HOD Conjecture. “It could mean that the structure of infinity is more intricate than we thought,” Aguilera said.
Why Should You Care?
This isn’t just about adding a new number to a mathematical playbook. Discoveries like this ripple out into unexpected areas. Infinity lies at the heart of breakthroughs in cryptography, artificial intelligence, and cosmology. When mathematicians uncover new insights about the infinite, they pave the way for advances in fields as diverse as cybersecurity and the study of black holes.
Exacting cardinals also force us to confront deeper philosophical questions. Can we ever fully understand the universe if infinity keeps surprising us?
The findings appeared in the preprint server arXiv.
